Categorical absorption of a non-isolated singularity

Aporva Varshney

doi:10.4171/jncg/684

JournalsjncgVol. 20, No. 4pp. 1419–1464

Categorical absorption of a non-isolated singularity

Aporva Varshney
University College London, UK
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Abstract

We study an example of a projective threefold with a non-isolated singularity and its derived category. The singular locus can be locally described as a line of surface nodes compounded with a threefold node at the origin. We construct a semiorthogonal decomposition where one component absorbs the singularity in the sense of Kuznetsov–Shinder, and the other components are equivalent to the derived categories of smooth projective varieties. The absorbing category is seen to be closely related to the absorbing category constructed for nodal varieties by Kuznetsov–Shinder, reflecting the geometry of the singularity. We further show that the semiorthogonal decomposition is induced by one on a geometric resolution and briefly consider the properties of the absorbing category under smoothing.

1. Introduction

Studying the bounded derived category of coherent sheaves

𝐃^{b} (X)

of a variety

X

over a field

k

can often reveal information about the geometry of

X .

One way in which we may study the structure of

𝐃^{b} (X)

is by considering how it can be broken down into smaller pieces: a semiorthogonal decomposition

𝐃^{b} (X) = ⟨ 𝒜_{1}, \dots, 𝒜_{n} ⟩

is an ordered collection of full triangulated subcategories

𝒜_{1}, \dots, 𝒜_{n}

generating

𝐃^{b} (X)

such that ‘there are no morphisms from right to left’. Precisely, for

B \in 𝒜_{j}

and

A \in 𝒜_{i}

with

j > i,

Hom (B, A) = 0 .

Recently, there has been interest in studying

𝐃^{b} (X)

when

X

is a singular variety. One approach we may take is to consider the full triangulated subcategory of perfect complexes,

𝐃^{perf} (X),

that is, the complexes quasi-isomorphic to a bounded complex of locally free sheaves. When

X

is smooth,

𝐃^{perf} (X)

coincides with

𝐃^{b} (X),

since in this setting all coherent sheaves have a finite resolution by locally free sheaves, and so we can think of

𝐃^{perf} (X)

as capturing smoothness. It then makes sense to define the singularity category [19

D. O. Orlov, Triangulated categories of singularities and D-branes in Landau–Ginzburg models (in Russian). Tr. Mat. Inst. Steklova 246 (2004), 240–262. English translation. Proc. Steklov Inst. Math. 246 (2004), 227–248 Zbl 1101.81093 MR 2101296

]

𝐃_{sg} (X) := 𝐃^{b} (X) / 𝐃^{perf} (X) .

This category exhibits locality in the sense that it depends only on an open neighbourhood of the singular locus.

On the other hand, one can take a more global approach to studying the singularity. Suppose

X

is a projective variety with a single isolated singularity. Then, one can consider whether the singularity is captured at the level of the derived category by asking if there exists a semiorthogonal decomposition

𝐃^{b} (X) = ⟨ 𝒜, ℬ ⟩,

where

ℬ \subset 𝐃^{perf} (X),

so that

𝒜

is, in some sense, responsible for the singularity. Following [15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

], we say that

𝒜

provides a categorical absorption of singularities. We can motivate this further as follows: there does not exist a semiorthogonal decomposition of the form

𝐃^{b} (X) = ⟨ 𝐃_{sg} (X), 𝐃^{perf} (X) ⟩,

but we can try to approximate it by asking that

ℬ \subset 𝐃^{perf} (X)

is as large as possible. Examples studied in the literature also have the property that

𝒜

is equivalent to

𝐃^{b} (A)

for some dg-algebra

A

with finite-dimensional cohomology.

The existence of a categorical absorption depends not only on the singularity but also on the global structure of the variety; indeed, there are several obstructions as a result of this. Nevertheless, we briefly review several instances where such a semiorthogonal decomposition has been found. For the rest of the section, we work over

k = ℂ .

In [8

J. Karmazyn, A. Kuznetsov, and E. Shinder, Derived categories of singular surfaces. J. Eur. Math. Soc. (JEMS) 24 (2022), no. 2, 461–526 Zbl 1493.14023 MR 4382477

], the authors study surfaces with cyclic quotient singularities. These include the singularities of the weighted projective space

ℙ (1, 1, n),

which are shown to have a semiorthogonal decomposition

𝐃^{b} (ℙ (1, 1, n)) = ⟨ 𝐃^{b} (ℂ [z_{1}, \dots, z_{n - 1}] / {(z_{1}, \dots, z_{n - 1})}^{2}), 𝐃^{b} (ℂ), 𝐃^{b} (ℂ) ⟩,

where

\deg z_{i} = 0 .

For surfaces with type

A_{n}

singularities, it is shown that the singularity is often absorbed by the algebra

ℂ [w] / w^{n + 1}

with

\deg w = 0,

with a simple example being the blow-up of a smooth surface

S

in a non-reduced point

Spec ℂ [w] / w^{n + 1} \subset S .

A semiorthogonal decomposition is produced for the threefold

ℙ (1, 1, 1, 3)

in [9

Y. Kawamata, On the derived category of a weighted projective threefold. Boll. Unione Mat. Ital. 15 (2022), no. 1–2, 245–252 Zbl 1492.14028 MR 4390551

]:

𝐃^{b} (ℙ (1, 1, 1, 3)) = ⟨ 𝐃^{b} (R), 𝐃^{b} (ℂ), 𝐃^{b} (ℂ) ⟩,

where this time

R

is an algebra with

dim R = 45 .

The most relevant case for us will be categorical absorption of nodal varieties. Nodal threefolds are considered in various incarnations in [10

Y. Kawamata, Semi-orthogonal decomposition of a derived category of a 3-fold with an ordinary double point. In H. Abban, G. Brown, A. Kasprzyk, and S. Mori (eds.), Recent developments in algebraic geometry. To Miles Reid for his 70th birthday, pp. 183–215, London Math. Soc. Lecture Note Ser. 478, Cambridge University Press, Cambridge, 2022 Zbl 1524.14040 MR 4480569

, 21

N. Pavic and E. Shinder, Derived categories of nodal del Pezzo threefolds. Taiwanese J. Math. 29 (2025), no. 6, 1607–1632 Zbl 08150207 MR 5003266

, 22

F. Xie, Nodal quintic del Pezzo threefolds and their derived categories. Math. Z. 304 (2023), no. 3, article no. 48, 28 pp. Zbl 1528.14022 MR 4610379

]. We follow the approach detailed in [15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

]. To illustrate this theory, let

X

be a nodal genus

0

curve consisting of curves

E ≅ Y ≅ ℙ^{1}

intersecting transversely at a point. The contraction of the branch

E

through the map

f : X \to Y

yields the semiorthogonal decomposition

𝐃^{b} (X) = ⟨ \ker 𝐑 f_{*}, 𝐋 f^{*} 𝐃^{b} (ℙ^{1}) ⟩,

where we have that

𝐋 f^{*}

is fully faithful. Absorption is then given by the category

\ker 𝐑 f_{*} = ⟨ 𝒪_{E} (- 1) ⟩,

and denoting

P := 𝒪_{E} (- 1),

one finds that

\begin{matrix} {Ext}^{•} (P, P) & ≅ ℂ [θ] \\ \ker 𝐑 f_{*} & ≃ ⟨ P ⟩ ≃ 𝐃^{b} (ℂ [r] / r^{2}), \end{matrix}

where

\deg θ = 2

and

\deg r = - 1 .

We say that

P

is a

ℙ^{\infty, 2}

object, since

{Ext}^{•} (P, P)

resembles the cohomology ring of

ℂ ℙ^{\infty} .

It was shown in [15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

] that

ℙ^{\infty, 2}

objects can be used to absorb singularities on nodal varieties of odd dimension. In contrast, in even dimensions, the absorbing category is instead given by

𝐃^{b} (ℂ [w] / w^{2}) ≃ ⟨ P ⟩

with

\deg w = 0,

and

{Ext}^{•} (P, P) = ℂ [ε]

with

\deg ε = 1 .

In this case,

P

is called a

ℙ^{\infty, 1}

object.

Perhaps the most interesting part of this story is a property of

ℙ^{\infty, 2}

objects known as deformation absorption: given a

ℙ^{\infty, 2}

object on a variety, the pushforward into a smoothing of the variety generates an admissible subcategory of the smoothing. Essentially, this tells us that the

ℙ^{\infty, 2}

object vanishes from the semiorthogonal decomposition when smoothing the variety. In the example above, this is exemplified by considering a smoothing of

X

to

ℙ^{1} .

This powerful property was used in [16

A. Kuznetsov and E. Shinder, Derived categories of Fano threefolds and degenerations. Invent. Math. 239 (2025), no. 2, 377–430 Zbl 1564.14091 MR 4850600

] to study relationships between the Kuznetsov components of smooth Fano threefolds, by first degenerating a threefold to a nodal variety and then taking a resolution.

1.1. Results and structure

The cases in the literature have so far focused on isolated singularities. Presently, we turn our attention to a complex threefold with a non-isolated compound

A_{2}

singularity, constructed as follows. We first consider the curve

ℭ := {x = s^{2} u = 0} \subset ℙ_{x : y}^{1} \times ℙ_{s : t}^{1} \times ℙ_{u : v}^{1}

and then perform a blow-up

X := {Bl}_{ℭ} {(ℙ^{1})}^{3} .

Then,

X

is locally given by the equation

{x b = s^{2} u} \subset 𝔸_{x, s, u, b}^{4},

so we can think of the singularity as a family of surface nodes compounded with a threefold node at the origin. In this article, we essentially perform a series of calculations, checking that there is a categorical absorption which is as nicely behaved as one could expect.

Theorem 1.1

(

=

Theorem 3.3). The curve $ℭ$ and threefold $X$ admit a categorical absorption, where the absorbing category is generated by an object $P$ with the property

{Ext}^{•} (P, P) ≅ ℂ [ε, θ],

where $\deg ε = 1$ and $\deg θ = 2 .$

We call an object with this self-Ext algebra a compound

ℙ^{\infty},

or

c ℙ^{\infty},

object, to reflect the idea that it is simultaneously a

ℙ^{\infty, 1}

and

ℙ^{\infty, 2}

object.

The work in [16

A. Kuznetsov and E. Shinder, Derived categories of Fano threefolds and degenerations. Invent. Math. 239 (2025), no. 2, 377–430 Zbl 1564.14091 MR 4850600

] shows us that it can be useful to consider the structure of a resolution and of a smoothing; consequently, in Section 4, we show that the semiorthogonal decomposition produced for

X

is induced by one on a specific choice of resolution of singularities

\tilde{X} .

In particular, our resolution has as an exceptional divisor a surface and a curve intersecting transversely. The key geometric input which allows us to perform computations in the derived category of

\tilde{X}

is the following proposition.

Proposition 1.2

(

=

Proposition 4.3). The space $\tilde{X}$ can be obtained from ${(ℙ^{1})}^{3}$ as a series of blow-ups along smooth rational curves.

Using this, we obtain a full exceptional collection on

\tilde{X} .

However, this collection is not compatible with the contraction

σ : \tilde{X} \to X,

in the sense that the pushforward under

σ

does not recover the absorption constructed on

X .

To overcome this, we perform a series of mutations and explicitly find an admissible category

\tilde{𝒟} \subset 𝐃^{b} (\tilde{X})

that serves as a ‘categorical resolution’ for our absorbing category.

Theorem 1.3

(

=

Theorem 4.8). Let $σ : \tilde{X} \to X$ be the resolution of singularities map. There is an admissible subcategory $\tilde{𝒟} \subset 𝐃^{b} (\tilde{X})$ such that the absorbing category $𝒟 \subset 𝐃^{b} (X)$ is equivalent to the Verdier quotient

𝒟 ≃ \tilde{𝒟} / (\ker 𝐑 σ_{*} \cap \tilde{𝒟}) .

The category

\tilde{𝒟}

is generated by an exceptional collection of length

4,

and under the pushforward

𝐑 σ_{*}

all the generating objects are mapped to the

c ℙ^{\infty}

object

P .

We investigate the algebra structure of this collection.

Theorem 1.4

(

=

Theorem 4.16). The morphisms in the exceptional sequence

\tilde{𝒟} = ⟨ ℰ_{0}^{''}, ℱ_{0}^{''}, ℰ_{1}, ℱ_{1} ⟩

are described as follows:

(1)
${Ext}^{•} (ℰ_{0}^{''}, ℱ_{0}^{''}) ≅ {Ext}^{•} (ℰ_{1}, ℱ_{1}) ≅ ℂ \oplus ℂ [- 1],$
(2)
${Ext}^{•} (ℱ_{0}^{''}, ℰ_{1}) ≅ ℂ \oplus ℂ [- 1],$
(3)
${Ext}^{•} (ℱ_{0}^{''}, ℱ_{1}) ≅ {Ext}^{•} (ℰ_{0}^{''}, ℰ_{1}) ≅ ℂ \oplus ℂ [- 1] \oplus ℂ^{\oplus 2} [- 2],$
(4)
${Ext}^{•} (ℰ_{0}^{''}, ℱ_{1}) ≅ ℂ \oplus ℂ [- 1] \oplus ℂ^{\oplus 2} [- 2] \oplus ℂ^{\oplus 2} [- 3] .$

To compute how morphisms compose, we note that there is an isomorphism of graded vector spaces between the

Ext -

spaces in the theorem above and the truncated self-Ext-spaces of the

c ℙ^{\infty}

object:

\begin{matrix} {Ext}^{•} (ℰ_{0}^{''}, ℱ_{0}^{''}) & ≅ {Ext}^{•} (ℰ_{1}, ℱ_{1}) ≅ {Ext}^{•} (ℱ_{0}^{''}, ℰ_{1}) ≅ {Ext}^{\leq 1} (P, P), \\ {Ext}^{•} (ℰ_{0}^{''}, ℰ_{1}) & ≅ {Ext}^{•} (ℱ_{0}^{''}, ℱ_{1}) ≅ {Ext}^{\leq 2} (P, P), \\ {Ext}^{•} (ℰ_{0}^{''}, ℱ_{1}) & ≅ {Ext}^{\leq 3} (P, P) . \end{matrix}

This indicates that the morphisms between objects of

\tilde{𝒟}

should be seen as lifting some map

P \to P [i] .

We prove the following.

Proposition 1.5

(

=

Proposition 4.19). All compositions in $\tilde{𝒟}$ are non-zero.

In particular, omitting compositions in non-zero degrees, we can visualise the categorical resolution as the following quiver.

Here, solid arrows indicate degree

0

maps, dashed arrows indicate degree

1

maps and dotted arrows indicate degree

2

maps.

Finally, we briefly consider the properties of our

c ℙ^{\infty}

objects with respect to smoothings of the curve

ℭ

in Section 5.

Proposition 1.6

(

=

Proposition 5.1). Let $𝔛$ be the smoothing of $ℭ$ obtained by blowing up $ℙ^{1} \times ℙ^{1}$ in a fattened point $Spec ℂ [s] / s^{2} .$ Then, the pushforward of the $c ℙ^{\infty}$ object $P \in 𝐃^{b} (ℭ)$ into $𝔛$ generates an admissible subcategory of $𝐃^{b} (𝔛),$ and hence provides a deformation absorption with respect to the smoothing $𝔛 .$

Notation and conventions.

(1)
We work over the base field $ℂ .$
(2)
For a dg-algebra $A$ with finite-dimensional cohomology, we denote by $𝐃 (A)$ the unbounded derived category of dg-modules over $A,$ and $𝐃^{b} (A) \subset 𝐃 (A)$ is the subcategory of dg-modules with finite-dimensional total cohomology.
(3)
Tensor products and pushforward and pullback functors between derived categories are implicitly derived.
(4)
Given a scheme $X$ with closed subscheme $Z \subset X$ and a birational map $Y \to X,$ we denote by $\bar{Z}$ the strict transform of $Z$ in $Y .$

2. Preliminaries

2.1. Derived categories

In this section, we recap general results in the theory of derived categories which we will use freely.

2.1.1. Mutations of semiorthogonal decompositions. Given a semiorthogonal decomposition of a triangulated category where each component is admissible

𝒯 = ⟨ 𝒜_{1}, \dots, 𝒜_{n} ⟩,

it is possible to reorder the components to obtain a new semiorthogonal decomposition. The first method is to perform a left mutation of

𝒜_{i + 1}

over

𝒜_{i}

𝒯 = ⟨ 𝒜_{1}, \dots, 𝒜_{i - 1}, 𝕃_{𝒜_{i}} 𝒜_{i + 1}, 𝒜_{i}, 𝒜_{i + 2}, \dots, 𝒜_{n} ⟩,

and the second is to perform a right mutation of

𝒜_{i}

over

𝒜_{i + 1}

𝒯 = ⟨ 𝒜_{1}, \dots, 𝒜_{i - 1}, 𝒜_{i + 1}, ℝ_{𝒜_{i + 1}} 𝒜_{i}, 𝒜_{i + 2}, \dots, 𝒜_{n} ⟩ .

In each case, the mutation is an equivalence of categories

\begin{matrix} 𝕃_{𝒜_{i}} 𝒜_{i + 1} & ≃ 𝒜_{i + 1}, \\ ℝ_{𝒜_{i + 1}} 𝒜_{i} & ≃ 𝒜_{i} . \end{matrix}

We omit the construction, but note that when

𝒜_{i}

and

𝒜_{i + 1}

are generated by exceptional objects

ℰ_{i}

and

ℰ_{i + 1},

respectively, we have formulae for the left and right mutations through the following exact triangles [4

A. I. Bondal, Representation of associative algebras and coherent sheaves (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25–44. English translation. Math. USSR-Izv. 34 (1990), no. 1, 23–42 Zbl 0692.18002 MR 0992977

]:

\begin{matrix} RHom (ℰ_{i}, ℰ_{i + 1}) \otimes ℰ_{i} & \to ℰ_{i + 1} \to 𝕃_{ℰ_{i}} (ℰ_{i + 1}), \\ ℝ_{ℰ_{i + 1}} (ℰ_{i}) & \to ℰ_{i} \to RHom {(ℰ_{i}, ℰ_{i + 1})}^{*} \otimes ℰ_{i + 1} . \end{matrix}

We also note an easy consequence of Serre duality.

Lemma 2.1.

If $X$ is Gorenstein with a semiorthogonal decomposition

𝐃^{b} (X) = ⟨ 𝒜, ℬ ⟩,

we have that

𝐃^{b} (X) = ⟨ ℬ \otimes ω_{X}, 𝒜 ⟩

is also a semiorthogonal decomposition, where $ω_{X}$ is the canonical bundle.

2.1.2. Derived nine-lemma. When working in the derived category, the ordinary nine-lemma in homological algebra no longer holds. Instead, we have the following derived nine-lemma (see [17

J. P. May, The additivity of traces in triangulated categories. Adv. Math. 163 (2001), no. 1, 34–73 Zbl 1007.18012 MR 1867203

, Lemma 2.6] for a proof).

Lemma 2.2.

Suppose that the top-left square commutes in the following diagram and the top two rows and left two columns are exact triangles.

Then, there exists an object $Z^{''}$ and maps $f, g, h, k, k^{'}, k^{''}$ such that the bottom-right square anti-commutes, all other squares commute and all rows and columns form exact triangles.

2.1.3. Derived category of a blow-up. Suppose

X

is a scheme and

Z \subset X

is a locally complete intersection of codimension

c \geq 2 .

Consider the blow-up

{Bl}_{Z} X .

Then, there is a semiorthogonal decomposition [18

D. O. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves (in Russian). Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 852–862. English translation. Russ. Acad. Sci. Izv. Math. 41 (1993), no. 1, 133–141 Zbl 0798.14007 MR 1208153

]

(2.1)

𝐃^{b} ({Bl}_{Z} X) = ⟨ Φ_{c - 1} (𝐃^{b} (Z)), \dots, Φ_{1} (𝐃^{b} (Z)), 𝐃^{b} (X) ⟩,

where

𝐃^{b} (X)

is embedded fully faithfully by pulling back, and the functors

Φ_{i}

are fully faithful and constructed as follows. The exceptional divisor is a projective bundle over

π : E \to Z,

so letting

j : E \to {Bl}_{Z} X

be the inclusion, we define

(2.2)

Φ_{i} (ℱ) = j_{*} (π^{*} ℱ \otimes 𝒪 (- i)) .

2.1.4. The cone on the pull-push counit. Let

i : E \to X

be the inclusion of a Cartier divisor in a scheme

X .

A useful computation tool will be the following description of the cone on the canonical adjunction morphism [2

A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties. 1995 arXiv:alg-geom/9506012v1

]:

(2.3)

i^{*} i_{*} ℱ \to ℱ \to ℱ \otimes 𝒩_{E / X}^{\lor} [2] .

2.1.5. Bondal–Orlov localisation. The Bondal–Orlov localisation conjecture relates the derived categories of a singular variety

X

and a resolution

π : \tilde{X} \to X .

Conjecture 2.3

([6

A. I. Efimov, Homotopy finiteness of some DG categories from algebraic geometry. J. Eur. Math. Soc. (JEMS) 22 (2020), no. 9, 2879–2942 Zbl 1478.14010 MR 4127943

, Conjecture 1.9], [3

A. Bondal and D. Orlov, Derived categories of coherent sheaves. In Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 47–56, Higher Ed. Press, Beijing, 2002 Zbl 0996.18007 MR 1957019

, Section 5]). If $π : \tilde{X} \to X$ is a resolution of rational singularities, then $π_{*}$ induces an equivalence between $𝐃^{b} (X)$ and a Verdier quotient of $𝐃^{b} (\tilde{X})$ :

{\tilde{π}}_{*} : 𝐃^{b} (\tilde{X}) / \ker π_{*} \to 𝐃^{b} (X) .

In fact, the conjecture is true in several settings (see [6

A. I. Efimov, Homotopy finiteness of some DG categories from algebraic geometry. J. Eur. Math. Soc. (JEMS) 22 (2020), no. 9, 2879–2942 Zbl 1478.14010 MR 4127943

, 20

N. Pavic and E. Shinder,

K -

theory and the singularity category of quotient singularities. Ann. K-Theory 6 (2021), no. 3, 381–424 Zbl 1490.14020 MR 4310324

]). The most relevant situation for us will be when the resolution has one-dimensional fibres, in which case the conjecture was proven in [1

A. Bondal, M. Kapranov, and V. Schechtman, Perverse schobers and birational geometry. Selecta Math. (N.S.) 24 (2018), no. 1, 85–143 Zbl 1436.14037 MR 3769727

]. An immediate consequence of this is that we may descend an appropriate semiorthogonal decomposition of

𝐃^{b} (\tilde{X})

to one for

𝐃^{b} (X)

[15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

, Proposition 4.1]. If

𝐃^{b} (\tilde{X}) = ⟨ \tilde{𝒜}, \tilde{ℬ} ⟩

is a semiorthogonal decomposition such that

\ker π_{*} \subset \tilde{𝒜},

then we obtain a semiorthogonal decomposition

𝐃^{b} (X) = ⟨ 𝒜, ℬ ⟩

with

\begin{matrix} 𝒜 & ≃ \tilde{𝒜} / (\ker π_{*} \cap \tilde{𝒜}) ≃ \tilde{𝒜} / \ker π_{*}, \\ ℬ & ≃ \tilde{ℬ} / (\ker π_{*} \cap \tilde{ℬ}) ≃ \tilde{ℬ}, \end{matrix}

where

𝒜, ℬ

must generate

𝐃^{b} (X)

because

π_{*}

is essentially surjective.

2.1.6. Spherical objects. An object

K \in 𝐃^{b} (X),

where

X

is a complex smooth and projective variety, is called spherical if [7

D. Huybrechts, Fourier–Mukai transforms in algebraic geometry. Oxford Math. Monogr., Oxford University Press, Oxford, 2006, 307 pp. Zbl 1095.14002 MR 2244106

]

(1)
$K \otimes ω_{X} ≅ K$ and
(2)
$Hom (K, K [i]) ≅ H^{i} (S^{dim X}, ℂ),$

where

S^{dim X}

is the sphere of dimension equal to

dim X .

A spherical object induces an autoequivalence of the derived category by twisting around the object. For any object

ℰ,

the twist

T_{K} (ℰ)

is defined through the exact triangle

(⨁_{i} Hom (K, ℰ [i]) \otimes K [- i]) \to ℰ \to T_{K} (ℰ) .

Being an autoequivalence, it has an inverse known as the dual twist, which is similarly described by an exact triangle.

2.2. Projective bundles and ruled surfaces

We review some theory which will be essential for computations later on. Fix a smooth variety

B,

and let

ℱ

and

ℒ

be invertible sheaves. Let

ℰ = ℒ \oplus ℱ .

Then, there are two possible conventions for the corresponding projective bundle; in our case, we take the definition to be

π : ℙ (ℰ) := \underline{Proj} (Sym (ℰ)) \to B,

so that there is a map

π^{*} ℰ \to 𝒪 (1) .

We will write

ℙ (ℰ) = ℙ (ℒ_{x_{0}} \oplus ℱ_{x_{1}})

to denote that the coordinates on the fibres are

[x_{0} : x_{1}] .

There is a section

S

corresponding to the surjection

ℰ \to ℒ;

equivalently, it is given by the vanishing of the coordinate

x_{1} .

The skyscraper sheaf of the section is defined by taking the cokernel of the composition

π^{*} ℱ \to π^{*} ℰ \to 𝒪 (1),

yielding

π^{*} ℱ \to 𝒪 (1) \to 𝒪_{S} (1) \to 0

and twisting by

𝒪 (- 1)

π^{*} ℱ \otimes 𝒪 (- 1) \to 𝒪 \to 𝒪_{S} \to 0 .

Then, the above sequence is the Koszul resolution of

𝒪_{S},

so it follows from the fact that

{𝒪 (1) |}_{S} ≅ ℒ

that the normal bundle of

S

is given by

ℱ^{\lor} \otimes ℒ .

Note that the first map is ‘multiplication by

x_{1}

’, and hence

x_{1}

is a section of the line bundle

π^{*} ℱ^{\lor} \otimes 𝒪 (1) .

We collect some results on ruled surfaces, which will arise frequently when we blow up a curve

C ≅ ℙ^{1}

in a smooth threefold

X .

In this situation, we recall that the exceptional divisor is isomorphic to

ℙ_{C} (𝒩_{C / X}^{\lor})

and has normal bundle

𝒪 (- 1) .

Consider

π : Σ = ℙ_{ℙ^{1}} (𝒪_{x} \oplus 𝒪 {(a)}_{y}) \to ℙ^{1} .

Then, the divisor corresponding to

𝒪 (1)

is given by the class of the section

H := V (x),

and we denote

(2.4)

𝒪 (a, b) := 𝒪 {(1)}^{\otimes a} \otimes π^{*} 𝒪 (b),

or, equivalently, it is the line bundle corresponding to the divisor

a H + b F,

where

F

is a fibre. Then, the canonical bundle is given by the formula

(2.5)

ω_{Σ} ≅ 𝒪 (- 2, - 2 + a) .

If

ℰ = 𝒪 \oplus 𝒪 (a),

then pushing forward the line bundle

𝒪 (d)

to the base

ℙ^{1}

gives us

(2.6)

π_{*} 𝒪 (d) = {\begin{matrix} {Sym}^{d} (ℰ) & if d \geq 0, \\ {(𝒪 (a) \otimes {Sym}^{- 2 - d} (ℰ))}^{\lor} [- 1] & if d < 0 . \end{matrix}

2.3. Categorical absorption for nodal varieties

This section gives an overview of the theory of categorical absorption for nodal varieties, with [15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

] as our primary reference. We will only concern ourselves with dimensions up to

3,

but note that the theory can be extended to arbitrary dimension.

2.3.1. Categorical ordinary double points. We recall the definitions given earlier.

Definition 2.4.

An object

P \in 𝐃^{b} (X)

for a variety

X

is called a

ℙ^{\infty, q}

object if

{Ext}^{•} (P, P) ≅ ℂ [θ] \deg θ = q .

Definition 2.5.

The categorical ordinary double point of degree

p

is

𝐃^{b} (ℂ [w] / w^{2})

with

\deg w = p .

The theory is split up by parity of

dim X

: in all examples,

q = 1

when

X

is even dimensional, and

q = 2

when

X

is odd dimensional.

For the odd-dimensional case, the key example is the nodal curve considered earlier, which we can see as the vanishing locus

X = V (s u) \subset ℙ_{s : t}^{1} \times ℙ_{u : v}^{1} .

In this case, the pushforward of a line bundle

𝒪 (a)

on either of the two branches gives a

ℙ^{\infty, 2}

object. We have seen that we can consider contracting one of the branches

f : X \to ℙ_{s : t}^{1};

it is then a direct consequence of the fact that

f_{*} 𝒪_{X} ≅ 𝒪_{ℙ^{1}}

that the pullback

f^{*}

is fully faithful. One can show that the null category

\ker f_{*}

is admissible in this case, and that there is a semiorthogonal decomposition [15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

, Proposition 6.15]

𝐃^{b} (X) = ⟨ 𝐃^{b} (ℂ [w] / w^{2}), 𝐃^{b} (ℙ^{1}) ⟩

with

\deg w = - 1 .

Here, the absorbing category

𝐃^{b} (ℂ [w] / w^{2})

is generated by the pushforward of the

𝒪 (- 1)

sheaf on the branch being contracted.

The simplest (but still useful) example of a nodal even-dimensional scheme is the fat point

Spec ℂ [w] / w^{2} .

In this case, the object

ℂ ≅ 𝒪 / w

is a

ℙ^{\infty, 1}

object which generates

𝐃^{b} (ℂ [w] / w^{2}),

where

\deg w = 0 .

More interestingly, we may consider the nodal surface

X := ℙ (1, 1, 2)

[8

J. Karmazyn, A. Kuznetsov, and E. Shinder, Derived categories of singular surfaces. J. Eur. Math. Soc. (JEMS) 24 (2022), no. 2, 461–526 Zbl 1493.14023 MR 4382477

, Example 3.17]. To construct a categorical absorption here, we consider the resolution of singularities given by

σ : 𝔽_{2} \to ℙ (1, 1, 2),

where we denote by

π : 𝔽_{2} ≅ ℙ (𝒪 \oplus 𝒪 (- 2)) \to ℙ^{1}

the second Hirzebruch surface. In this case, the exceptional divisor

E

is given by the section of the bundle which has self-intersection

(- 2),

and it can be shown that the kernel of

σ_{*}

is generated by

𝒪_{E} (- 1) .

Let

F

be the class of a fibre, so that

𝒪 (F) ≅ π^{*} 𝒪 (1) .

Then,

E

is the divisor corresponding to

𝒪 (1),

and so there is a full exceptional sequence

\begin{matrix} 𝐃^{b} (𝔽_{2}) & = ⟨ π^{*} 𝐃^{b} (ℙ^{1}) \otimes 𝒪 (- 1), π^{*} 𝐃^{b} (ℙ^{1}) ⟩ \\ = ⟨ 𝒪 (- E - 2 F), 𝒪 (- E - F), 𝒪 (- F), 𝒪 ⟩ . \end{matrix}

One sees that there is a short exact sequence

0 \to 𝒪 (- E - F) \to 𝒪 (- F) \to 𝒪_{E} (- 1) \to 0

which implies that

\ker σ_{*} \subset ⟨ 𝒪 (- E - F), 𝒪 (- F) ⟩ .

An important feature here is that the sheaf

𝒪_{E} (- 1)

is a

2 -

spherical object, and

𝒪 (- E - F)

is the dual spherical twist of

𝒪 (- F)

around

𝒪_{E} (- 1) .

We note that the pushforwards

σ_{*} 𝒪

and

σ_{*} 𝒪 (- E - 2 F)

remain exceptional in

𝐃^{b} (ℙ (1, 1, 2)),

and hence

σ_{*} ⟨ 𝒪 (- E, - F), 𝒪 (- F) ⟩

is a candidate for our categorical absorption. We can compute, using (2.6), that

{Ext}^{•} (𝒪 (- E - F), 𝒪 (- F)) ≅ ℂ \oplus ℂ [- 1] .

Since the cone on the morphism

𝒪 (- E - F) \to 𝒪 (- F)

is in

\ker σ_{*},

we find that

σ_{*} 𝒪 (- E - F) ≅ σ_{*} 𝒪 (- F) .

The idea is that this object should be a

ℙ^{\infty, 1}

object; intuitively, the degree 0 morphism becomes an isomorphism and the degree

1

morphism then gives rise to the self-extension of the

ℙ^{\infty, 1}

object.

These observations are formalised in the following theorems.

Definition 2.6

([15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

, Section 3.1]). The graded Kronecker quiver of degree

q

is the path dg-algebra of the following dg-quiver:

where there are two arrows of degrees

0

and

q,

respectively, and the differentials are zero. The graded Kronecker quiver category of degree

q,

denoted

𝒦 r_{q},

is the derived category of this algebra.

Theorem 2.7

([15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

, Corollary 3.3]). Let $⟨ ℰ, ℰ^{'} ⟩ \subset 𝐃^{b} (\tilde{X})$ be an exceptional sequence, where $\tilde{X}$ is a variety such that ${Ext}^{•} (ℰ, ℰ^{'}) ≅ ℂ \oplus ℂ [- q] .$ Then, $⟨ ℰ, ℰ^{'} ⟩ ≃ 𝒦 r_{q} .$

Theorem 2.8

([15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

, Proposition 3.7]). Let $⟨ ℰ, ℰ^{'} ⟩ ≃ 𝒦 r_{q},$ and define the object $K$ using the distinguished triangle

K \to ℰ \to ℰ^{'} .

Then, $K$ is a $1 + q -$ spherical object, and there is an equivalence

𝒦 r_{q} / ⟨ K ⟩ ≃ 𝐃^{b} (ℂ [w] / w^{2}) \deg w = 1 - q,

and the image of $ℰ$ under $σ_{*} : 𝒦 r_{q} \to 𝒦 r_{q} / ⟨ K ⟩$ is a $ℙ^{\infty, q}$ object.

In the more general setting, the theorem can be applied as follows. Suppose we have a crepant resolution

σ : \tilde{X} \to X

for a nodal surface or threefold

X,

and assume that we have

H^{> 0} (\tilde{X}, 𝒪) = 0,

so that line bundles are exceptional objects. Then, labeling the exceptional curve as

C,

we have a

d -

spherical object

K := 𝒪_{C} (- 1),

where

d = dim X .

One can show that

K

generates

\ker σ_{*} .

Now suppose

ℰ

is an exceptional object such that

dim {Ext}^{•} (ℰ, K) = 1 .

A simple way to find such an object is to look for a divisor which intersects the curve transversely at a point, and then take the associated line bundle. Now, we can simply take the spherical twist of

ℰ

around

K;

if

{Ext}^{•} (K, ℰ)

is concentrated in degree

n,

this takes the form

K [- n] \to ℰ \to ℰ^{'} .

Since a spherical twist is an autoequivalence, we see that

ℰ^{'}

is an exceptional object, so we have an admissible subcategory

⟨ ℰ, ℰ^{'} ⟩

of

𝐃^{b} (\tilde{X})

which contains

\ker σ_{*} = ⟨ K ⟩ .

It is then not hard to show, simply by taking long exact sequences for

Ext,

that

⟨ ℰ, ℰ^{'} ⟩ ≃ 𝒦 r_{q}

for

q = d - 1,

and hence taking the Verdier quotient gives us a

ℙ^{\infty, q}

object generating a categorical ordinary double point of degree

1 - q .

As a result, we are able to construct a categorical absorption for the nodal variety.

2.3.2. Deformation absorption. Finally, we discuss smoothings of nodal varieties. Suppose that

f : 𝔛 \to B

is a flat projective morphism from a smooth space

𝔛

to a smooth curve

B,

with the central fibre isomorphic to

X

and all other fibres smooth, so that

𝔛

gives a smoothing of

X .

Now suppose that the nodal variety

X

has an absorption by a

ℙ^{\infty, 2}

object

P

as follows:

𝐃^{b} (X) = ⟨ P, {}^{⟂}P ⟩ .

Then, for any smoothing

i : X \overset{}{↪} 𝔛,

the object

i_{*} P

can be shown to be exceptional [15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

, Theorem 1.8]. In particular, we have a semiorthogonal decomposition

𝐃^{b} (𝔛) = ⟨ i_{*} P, 𝒟 ⟩

which is said to be

B -

linear in the sense that there is a well-defined notion of base changing the semiorthogonal decomposition to a fibre of the map

f : 𝔛 \to B

[13

A. Kuznetsov, Base change for semiorthogonal decompositions. Compos. Math. 147 (2011), no. 3, 852–876 Zbl 1218.18009 MR 2801403

]. Performing this base change to a non-central fibre

𝔛_{b},

we find that the

i_{*} P

component vanishes since it is only supported on the central fibre, and so

𝐃^{b} (𝔛_{b}) ≃ 𝒟_{b} .

In effect, we have a family of smooth and proper categories

𝒟_{b},

and for

b = o,

we acquire a

ℙ^{\infty, 2}

object to account for the singular behaviour. We say that

ℙ^{\infty, 2}

objects provide a deformation absorption, and furthermore, since this occurs for every smoothing, we say that the deformation absorption is universal.

3. Compound $ℙ^{\infty}$ objects

We begin by generalising the example of a nodal curve by trivially thickening one of the branches. Consider

(3.1)

ℭ = {s^{2} u = 0} \subset ℙ_{s : t}^{1} \times ℙ_{u : v}^{1}

which is a copy of

ℭ_{0} := V (u) ≅ ℙ_{s : t}^{1}

intersecting the thickened branch

𝔜 := V (s^{2}) ≅ ℙ_{u : v}^{1} \times ℂ [s] / s^{2}

at a fat point. There is a contraction

f : ℭ \to ℭ_{0},

and it follows once again from the fact that

f_{*} 𝒪_{ℭ} = 𝒪_{ℭ_{0}}

that there is a semiorthogonal decomposition

𝐃^{b} (ℭ) = ⟨ \ker f_{*}, f^{*} 𝐃^{b} (ℭ_{0}) ⟩

with

f^{*}

fully faithful. Let

𝔜_{red} = V (s)

be the reduction of the fattened branch

𝔜 .

Denote by

𝒫 (a)

the pushforward of the line bundle

𝒪 (a)

on

𝔜_{red}

to

ℭ .

Lemma 3.1.

The kernel of $f_{*}$ is generated by $𝒫 (- 1) .$

Proof.

Firstly, consider the projections

q : 𝔜 \to ℂ [s] / s^{2}

and

p : 𝔜 \to ℙ^{1} .

Then, we have that

\ker q_{*} = ⟨ q^{*} 𝐤 \otimes p^{*} 𝒪 (- 1) ⟩,

where we denote by

𝐤 = (ℂ [s] / s^{2}) / s

the generator of

𝐃^{b} (ℂ [s] / s^{2}) .

The pushforward of this sheaf to

ℭ

is precisely

𝒫 (- 1) .

To reduce to this case, we show that any object in the kernel of

f_{*}

is a pushforward of an object in

𝐃^{b} (𝔜) .

Firstly, let

ℱ

be a sheaf on

ℭ

supported set-theoretically on

𝔜_{red} .

Then, we can find a short exact sequence

0 \to ℱ_{1} \to ℱ \to ℱ_{2} \to 0,

where

supp ℱ_{1}

is contained in

{u = s^{r} = 0}

for some

r \geq 1

and

supp ℱ_{2} \subset 𝔜 .

Now assume that

supp ℱ

is a thickening of

𝔜,

so that

ℱ ≇ ℱ_{2}

and hence

ℱ_{1} \neq 0 .

Since

supp ℱ_{1} \subset ℭ_{0},

we find that

σ_{*} ℱ_{1} \neq 0 .

Now, left exactness of

σ_{*}

implies that

σ_{*} ℱ \neq 0 .

Hence, any sheaf

ℱ \in \ker σ_{*}

has scheme-theoretic support contained in

𝔜 .

Finally, if

ℱ^{•} \in 𝐃^{b} (ℭ)

is a complex, we use a standard spectral sequence argument (see, e.g., [5

T. Bridgeland, Flops and derived categories. Invent. Math. 147 (2002), no. 3, 613–632 Zbl 1085.14017 MR 1893007

, Lemma 3.1]) using

E_{2}^{p, q} = R^{p} σ_{*} (H^{q} (ℱ^{•})) ⟹ R^{p + q} σ_{*} ℱ^{•} .

This degenerates on the second page, so that if

supp ℱ^{•}

is a thickening of

𝔜,

then the non-vanishing of

σ_{*} H^{q} (ℱ^{•})

for some

q

implies the non-vanishing of

σ_{*} ℱ^{•} .

Now we let

𝒪 (a, b)

denote the restriction of the line bundle on

ℙ^{1} \times ℙ^{1}

with degree

a

along

ℙ_{s : t}^{1}

and degree

b

along

ℙ_{u : v}^{1} .

Then, the object

𝒫 (- 1)

has an infinite locally free resolution

0 \leftarrow 𝒫 (- 1) \leftarrow 𝒪 (0, - 1) \overset{\cdot s}{\leftarrow} 𝒪 (- 1, - 1) \overset{\cdot s u}{\leftarrow} 𝒪 (- 2, - 2) \overset{\cdot s}{\leftarrow} 𝒪 (- 3, - 2) \leftarrow \dots,

so that

{Ext}^{i} (𝒫 (- 1), 𝒫 (- 1))

can be computed using the chain complex

0 \to 𝒪 \overset{0}{\to} 𝒪 \overset{0}{\to} 𝒪 (1) \overset{0}{\to} 𝒪 (1) \overset{0}{\to} 𝒪 (2) \overset{0}{\to} 𝒪 (2) \overset{0}{\to} \dots,

where all maps are zero, since the coordinate

s

becomes

0

when restricted to

𝔜_{red} .

Hence,

ℰ x t^{i} (𝒫 (- 1), 𝒫 (- 1))

is the pushforward of a line bundle on

𝔜_{red},

and to compute global

Ext,

we can simply take global sections as the local-to-global spectral sequence degenerates on the second page. If

n

is even, then this shows that

{Ext}^{n} (𝒫 (- 1), 𝒫 (- 1))

has dimension

n / 2 + 1,

with basis given by

{u^{i} v^{j} | i + j = n / 2} .

If

n

is odd, then

{Ext}^{n} (𝒫 (- 1), 𝒫 (- 1))

has dimension

(n + 1) / 2

with basis given by

{u^{i} v^{j} | i + j = (n - 1) / 2} .

Concretely, the dimensions of

{Ext}^{n} (𝒫 (- 1), 𝒫 (- 1))

form a sequence:

(1, 1, 2, 2, 3, 3, \dots) .

We guess from this observation that the algebra structure is that of a polynomial ring with a generator of degree

1

and a generator of degree

2;

this guess is verified in the next theorem.

Definition 3.2.

An object

P \in 𝐃^{b} (X)

for a variety

X

is called a compound

ℙ^{\infty}

object, or

c ℙ^{\infty},

if there is an isomorphism of graded algebras

{Ext}^{•} (P, P) ≅ ℂ [ε, θ],

where

\deg ε = 1

and

\deg θ = 2 .

Theorem 3.3.

The object $P = 𝒫 (- 1)$ is a $c ℙ^{\infty}$ object on $ℭ .$

Proof.

Consider the locally free resolution for

P .

The element of

{Ext}^{1} (P, P) ≅ ℂ

arises from the mapping

𝟙 : 𝒪 (- 1, - 1) \to P,

and we lift this to a morphism

ε : P \to P [1]

by defining a chain map on the locally free resolution, so that all triangles and quadrilaterals in the diagram below commute.

Next,

{Ext}^{2} (P, P) ≅ ℂ^{\oplus 2}

has a basis given by the elements

{u, v} .

We lift the element given by

v

to a chain map

θ : P \to P [2]

similarly.

One now sees that

ε \circ θ = θ \circ ε

since both compositions are given by the chain map lifting the element

v \in {Ext}^{3} (P, P) .

In general, we claim that the lift of the element

u^{i} v^{j} \in {Ext}^{n} (P, P)

is given by the chain map

ε^{2 i} θ^{j}

if

n

is even, and

ε^{2 i + 1} θ^{j}

if

n

is odd. This is verified by observing that in the

n

even case, the chain map lifting

ε^{2 i} θ^{j}

is given by

while in the odd case, the chain map lifting

ε^{2 i + 1} θ^{j}

is given by

hence concluding the proof.

Remark 3.4.

In fact, the above proof shows that the algebra

RHom (P, P)

is formal, that is, quasi-isomorphic to

{Ext}^{•} (P, P) .

The chain maps constructed above give us the quasi-isomorphism.

Remark 3.5.

There is in fact a general class of examples where one can perform the same trick, given by considering the trivially thickened curve

ℙ^{1} \times ℂ [s] / s^{n}

intersecting a smooth curve transversely. For

n > 2,

with

P

defined analogously, the algebra

RHom (P, P)

will not be formal.

Note that

ω_{ℭ} ≅ 𝒪 (0, - 1) .

Hence, by the Serre duality statement of Lemma 2.1, we also have the following semiorthogonal decomposition for

ℭ,

which we will use in the sequel.

(3.2)

𝐃^{b} (ℭ) = ⟨ 𝐃^{b} (ℭ_{0}), 𝒫 (- 1) \otimes ω_{ℭ}^{- 1} ⟩ = ⟨ 𝐃^{b} (ℭ_{0}), 𝒫 ⟩ .

Note here that

𝒫

is the skyscraper sheaf along

𝔜_{red},

and we will define

(3.3)

𝒟 := ⟨ 𝒫 ⟩ \subset 𝐃^{b} (X)

to be the absorbing subcategory of

𝐃^{b} (X) .

Remark 3.6.

Given a

c ℙ^{\infty}

object

P,

we expect that there is an equivalence given by Koszul duality

\begin{matrix} ⟨ P ⟩ ≃ 𝐃^{perf} (ℂ [ε, θ]) ≃ 𝐃^{b} (ℂ [w, r] / (w^{2}, r^{2})) & \deg ε = 1, \deg θ = 2, \\ \deg w = 0, \deg r = - 1 . \end{matrix}

This would follow by computing

{Ext}^{•} (ℂ, ℂ)

over the algebra

ℂ [w, r] / (w^{2}, r^{2}) .

After verifying that the dg-algebras involved are formal, for example, using a similar argument to the one given in Remark 3.4, the desired Koszul duality statement would then follow from [11

B. Keller, Derived categories and tilting. In L. A. Hügel, D. Happel, and H. Krause (eds.), Handbook of tilting theory, pp. 49–104, London Math. Soc. Lecture Note Ser. 332, Cambridge University Press, Cambridge, 2007 MR 2384608

, Theorem 8.5 (b)].

4. Semiorthogonal decomposition for a resolution of singularities

Let

B

be a smooth threefold which contains the curve

ℭ

considered in Section 3 as a locally complete intersection. In this section, to allow us to perform explicit calculations, we will also assume that

B

contains an open subset

(4.1)

U_{B} ≅ 𝔸_{x, s}^{2} \times ℙ_{u : v}^{1}

such that

ℭ \cap U_{B}

is just the curve

ℭ

missing a point at infinity on the reduced branch. For example, we may take

B = ℙ_{x : y}^{1} \times ℙ_{s : t}^{1} \times ℙ_{u : v}^{1}

which contains

ℭ

as the vanishing locus

V (x, s^{2} u)

and

U_{B}

as the locus

{y \neq 0, t \neq 0} .

Let

X

be the threefold constructed through the blow-up

X := {Bl}_{ℭ} B .

Then,

X

is singular with a line of surface nodes compounded with a threefold nodal singularity at one point. In particular, this singularity is non-isolated. By Orlov’s blow-up formula given in equation (2.1) combined with the absorption for

ℭ

given by equation (3.2), we immediately obtain a semiorthogonal decomposition

𝐃^{b} (X) = ⟨ 𝐃^{b} (ℭ), 𝐃^{b} (B) ⟩ = ⟨ 𝐃^{b} (ℙ_{s : t}^{1}), 𝒟, 𝐃^{b} (B) ⟩,

where

𝒟

is the absorbing category as in (3.3), generated by the

c ℙ^{\infty}

object

𝒫 .

It is worth considering how each component of the decomposition is embedded, using (2.2). Firstly, we define a line bundle on

ℭ

ℒ (a) := 𝒪 (a, 0)

which is the pullback of a line bundle

𝒪 (a)

via the contraction map

ℭ \to ℭ_{0} ≅ ℙ_{s : t}^{1} .

Note

ℒ (a)

has degree

a

along the

ℙ_{s : t}^{1}

branch and is trivial along the thickened branch.

Next,

𝐃^{b} (ℭ)

is embedded into

X

by the functor

Φ : 𝐃^{b} (ℭ) \to 𝐃^{b} (X)

which pulls back to the exceptional locus, tensors with the

𝒪 (- 1)

bundle and pushes forward into

X .

Since

X

is the blow-up of

B

along

ℭ,

we observe that the exceptional locus has two components, both of which are

ℙ^{1} -

bundles:

(4.2)

F \to ℙ_{s : t}^{1},

G \to ℙ_{u : v}^{1} \times ℂ [s] / s^{2} .

We can also pullback along

ℙ^{1} \overset{}{↪} ℙ^{1} \times ℂ [s] / s^{2}

to obtain

(4.3)

G_{red} \to ℙ_{u : v}^{1} .

Then, we can write our semiorthogonal decomposition as

𝐃^{b} (X) = ⟨ Φ (ℒ (1)), Φ (ℒ (2)), Φ (𝒫), 𝐃^{b} (B) ⟩,

where the sheaves

Φ (ℒ (a))

are supported on

F \cup G

and

Φ (𝒫)

is supported on

G_{red} .

Note that we have chosen

𝐃^{b} (ℙ_{s : t}^{1}) = ⟨ 𝒪 (1), 𝒪 (2) ⟩ .

Locally over

U_{B},

X

takes the form

U_{X} := {x b = s^{2} u a} \subset ℙ_{U_{b}} (𝒪 {(1)}_{a} \oplus 𝒪_{b})

and hence has a compound

A_{2}

singularity of the form described previously. We find that our semiorthogonal decomposition provides a categorical absorption for this singularity. The goal of this section is to show that the semiorthogonal decomposition descends from one for a resolution of

X .

Remark 4.1.

If we instead start with

ℙ^{1} \times ℂ [s] / s^{n}

intersecting a

ℙ^{1}

transversely, then we obtain a threefold with a non-isolated compound

A_{n}

singularity locally given by the equation

{x b = s^{n} u} \subset 𝔸^{4},

so that we have a line of surface

A_{n - 1}

singularities, compounded with a threefold node at the origin.

4.1. Constructing a geometric resolution

We now describe the process of constructing the resolution, and refer to Figure 1 for a visualisation. To resolve the singularities, we first blow up along the exceptional component

F

defined in (4.2) to obtain a singular threefold

\hat{X}

and the map

{\hat{f}}_{X} : \hat{X} \to X .

Since

F

intersects the singular locus only at the threefold node, we find that

\hat{X}

has a

ℙ^{1} -

family of surface node singularities. To resolve these, we blow up along the strict transform of

G_{red}

defined in (4.3) to obtain a smooth threefold

\tilde{X}

and the map

φ_{\hat{X}} : \tilde{X} \to \hat{X} .

Define

σ : \tilde{X} \to X

to be the composition

\tilde{X} \to \hat{X} \to X .

It may be helpful to refer to the surface analogue: we may consider the blow-up

{Bl}_{⟨ x^{2}, y ⟩} 𝔸_{x, y}^{2} ≅ {x^{2} a = y b} \subset 𝔸^{2} \times ℙ_{a : b}^{1},

so that the exceptional divisor

V (x^{2}, y)

is a thickened

ℙ^{1},

and the space has a nodal singularity at the point

(0, 0, [0 : 1]) .

Then, blowing up the reduced exceptional divisor

V (x, y)

gives a minimal resolution of the singularity; our second blow-up

\tilde{X} \to \hat{X}

is a family version of this process.

Figure 1.

Constructing the resolution. Here,

\bar{F}, {\bar{G}}_{red}

denote the strict transforms in

\hat{X}

and

\tilde{F}, {\tilde{G}}_{red}

denote the strict transforms in

\tilde{X} .

4.1.1. Local analysis. We illustrate this construction locally by considering the open subset

U_{B}

defined in (4.1). Recall that we obtain an open subset of

X

U_{X} := {x b = s^{2} u a} \subset ℙ_{U_{b}} (𝒪 {(1)}_{a} \oplus 𝒪_{b}) =: P_{X} .

Locally,

F \cap U_{X} ≅ V (x, u)

and

G \cap U_{X} ≅ V (s^{2}, x) .

To obtain an open subset of

\hat{X},

we blow up at the ideal

⟨ x, u ⟩

to obtain

U_{\hat{X}} := {\begin{matrix} b d = s^{2} a c \\ x c = u d \end{matrix}} \subset ℙ_{P_{X}} (𝒪_{c} \oplus π_{B}^{*} 𝒪 {(1)}_{d}) =: P_{\hat{X}},

where we denote

π_{B} : P_{X} \to U_{B} .

We can now resolve the singularities by blowing up at

⟨ d, s ⟩

:

U_{\tilde{X}} := {\begin{matrix} b f = s a c e \\ x c = u d \\ d e = s f \end{matrix}} \subset ℙ_{P_{\hat{X}}} (𝒪 {(1)}_{e} \oplus π_{B}^{*} 𝒪 {(1)}_{f}) =: P_{\tilde{X}},

where this time we are denoting

π_{B} : P_{\hat{X}} \to U_{B}

and

𝒪 (1)

is the tautological line bundle on

P_{\hat{X}} .

As a GIT quotient,

P_{\tilde{X}}

is constructed as the quotient

ℂ^{10} / {(ℂ^{*})}^{4}

with weights shown in Table 1.

$x$	$s$	$u$	$v$	$a$	$b$	$c$	$d$	$e$	$f$
$0$	$0$	$1$	$1$	$- 1$	$0$	$0$	$- 1$	$0$	$- 1$
				$1$	$1$	$0$	$0$	$0$	$0$
						$1$	$1$	$- 1$	$0$
								$1$	$1$

Table 1.

Matrix of weights for

P_{\tilde{X}},

with omitted values equal to

0 .

Note that the exceptional locus of

U_{\hat{X}} \to U_{X}

is a curve

V (x, u, s, b) ≅ ℙ_{c : d}^{1} .

We denote by

(4.4)

C = V (x, u, s, b, e) ≅ ℙ_{c : d}^{1} \subset U_{\tilde{X}}

the strict transform of this curve in

\tilde{X} .

The strict transform

{\bar{G}}_{red}

of

G_{red}

in

\hat{X}

intersects the singular locus along the section where the fibre coordinate

b

vanishes. The effect of blowing up this locus is to introduce an exceptional surface

E \subset U_{\tilde{X}} .

The exceptional locus of the composition

σ : \tilde{X} \to X

is the union

E \cup C,

and it is precisely the preimage of the reduced singular locus

V (x, b, s) \subset X .

Note that the two subvarieties

E

and

C

intersect transversely at a point.

Lemma 4.2.

Let ${\tilde{G}}_{red}$ be the strict transform of $G_{red}$ in $\tilde{X} .$ Then, we have

\begin{matrix} {\tilde{G}}_{red} & ≅ ℙ_{ℙ_{u : v}^{1}} (𝒪_{a} \oplus 𝒪 {(1)}_{b}), \\ E & ≅ ℙ_{ℙ_{u : v}^{1}} (𝒪_{e} \oplus 𝒪 {(1)}_{f}) . \end{matrix}

Proof.

In

\hat{X},

the locus

{\bar{G}}_{red}

is given by

{\bar{G}}_{red} = V (x, d, s) \subset U_{\hat{X}},

and so its strict transform is given by

{\tilde{G}}_{red} = V (x, d, s, f) \subset U_{\tilde{X}} .

The second statement follows from similarly noting that

E = V (x, d, s, b) \subset U_{\tilde{X}} .

We also see that the strict transform

\bar{F} \subset \hat{X}

of

F

intersected with

U_{\hat{X}}

is a nodal surface

{b d = s^{2} a c} \subset 𝔸_{s}^{1} \times ℙ_{a : b}^{1} \times ℙ_{c : d}^{1}

which is isomorphic to the blow-up of

𝔸_{s}^{1} \times ℙ_{a : b}^{1}

at a fattened point cut out by the ideal

⟨ s^{2} a, b ⟩ .

Denoting by

\tilde{F}

the strict transform of

F

in

\tilde{X},

we find that

\tilde{F}

is a ruled surface blown up at a point, and then blown up again at a point on the exceptional curve.

4.1.2. Alternative construction of the resolution. A key computational tool will be the following alternative construction of

\tilde{X},

where we only blow up smooth varieties along a

ℙ^{1} .

This allows us to easily obtain a concrete semiorthogonal decomposition for

𝐃^{b} (\tilde{X}) .

Proposition 4.3.

There are varieties $W, Y$ and a sequence of contractions $φ_{W}, ψ, h_{Y},$ where each map is a blow-up of a smooth rational curve such that the following diagram commutes.

(4.5)

Furthermore, letting $E_{B}, E_{Y}, E_{W}$ be the exceptional loci of $h_{Y}, ψ, φ_{W},$ respectively, and ${\tilde{E}}_{B}, {\tilde{E}}_{Y}$ the strict transforms of $E_{B}, E_{Y},$ respectively, to $\tilde{X},$ we have that

(1)
${\tilde{E}}_{Y}$ coincides with $E;$
(2)
$E_{W}$ coincides with ${\tilde{G}}_{red};$
(3)
${\tilde{E}}_{B}$ coincides with $\tilde{F} .$

Proof.

The map

h_{Y} : Y \to B

is obtained by blowing up

B

along the reduced branch

ℭ_{0} ≅ ℙ_{s : t}^{1} \subset ℭ \subset B

to obtain a variety

h_{Y} : Y \to B .

Now,

\hat{X}

can be obtained by blowing up the strict transform

\bar{𝔜}

of the non-reduced branch

𝔜 \subset ℭ \subset B .

Hence, we have a map

φ_{Y} : \tilde{X} \to Y,

and we see that the total transform of the ideal cutting out the reduced curve

\bar{𝔜_{red}}

is locally principal in

\tilde{X} .

Letting

ψ : W \to Y

be the blow-up in

\bar{𝔜_{red}},

we see that

\tilde{X} \to Y

factors as

\tilde{X} \to W \to Y

by the universal property of blow-ups.

Note that outside of the open subset

U_{B}

both constructions of

\tilde{X}

only blow up

B

at the single point at infinity on

ℭ_{0} \subset ℭ;

otherwise, all blow-ups have centres contained in some open set over

U_{B} .

Define an open subset of

Y

over

U_{B}

as

U_{Y} := {x c = u d} \subset ℙ_{U_{B}} (𝒪_{c} \oplus 𝒪 {(1)}_{d}) =: P_{Y} .

Similarly, there is an open subset of

W

found by blowing up

P_{Y}

at

⟨ d, s ⟩

and pulling back, giving

U_{W} := {\begin{matrix} x c = u d \\ d e = s f \end{matrix}} \subset ℙ_{P_{Y}} (𝒪 {(1)}_{e} \oplus π_{B}^{*} 𝒪 {(1)}_{f}) =: P_{W},

where we denote

π_{B} : P_{Y} \to U_{B} .

Finally, we blow up

P_{W}

along the ideal

⟨ f, s c e ⟩

and pull back the equations for

U_{W}

to give

U_{\tilde{X}} = {\begin{matrix} x c = u d \\ d e = s f \\ b f = (s c e) a \end{matrix}} \subset ℙ_{P_{W}} (π_{B}^{*} 𝒪 {(1)}_{a} \oplus 𝒪_{b}) =: P_{\tilde{X}}^{'},

where we denote

π_{B} : P_{W} \to U_{B} .

It is clear that

P_{\tilde{X}}^{'} ≅ P_{\tilde{X}}

(e.g., by writing out the matrix of weights for

P_{\tilde{X}}^{'}

and comparing with Table 1), and so we recover

U_{\tilde{X}} .

The centre of

ψ : W \to Y

is given by

Z_{W} = V (x, d, s) \subset U_{Y},

and the exceptional locus of

ψ : W \to Y

is given by

E_{Y} = V (x, d, s) ≅ ℙ_{ℙ_{u : v}^{1}} (𝒪_{e} \oplus 𝒪 {(1)}_{f}) \subset U_{W} .

The centre of the blow-up

\tilde{X} \to W

is given by

Z_{\tilde{X}} = V (f, d, x, s) \subset U_{W}

which is a copy of

ℙ_{u : v}^{1}

arising as a section of

E_{Y} \to ℙ_{u : v}^{1},

and we see that

E_{W} = V (f, d, s, x) \subset U_{\tilde{X}}

is the exceptional divisor of the blow-up

\tilde{X} \to W .

Now, it is clear by considering the equations cutting out each subvariety that

E_{W}

coincides with

{\tilde{G}}_{red}

and

{\tilde{E}}_{Y}

coincides with

E

in

\tilde{X} .

It is also clear that

{\tilde{E}}_{B}

coincides with

\tilde{F}

in

\tilde{X}

since both give the preimage of

ℭ_{0}

along the map

\tilde{X} \to B .

The alternative construction is visualised in Figure 2.

Figure 2.

Alternative construction of the resolution, with blow-up centres indicated in red. Here,

{\bar{E}}_{B}

denotes the strict transform of

E_{B}

to

W,

and

{\tilde{E}}_{B}, {\tilde{E}}_{Y}

denote the strict transforms of

E_{B}, E_{Y},

respectively, to

\tilde{X} .

4.1.3. Exceptional loci. Next, we consider the irreducible component

E

of the exceptional locus of

σ,

recalling the notation for line bundles on a ruled surface introduced in equation (2.4).

Lemma 4.4.

For the divisor $E \subset \tilde{X},$ we have that

\begin{matrix} 𝒩_{E / \tilde{X}} & ≅ 𝒪 (- 2, 1), \\ {ω_{\tilde{X}} |}_{E} & ≅ 𝒪 (0, - 2) . \end{matrix}

Proof.

Recall that

E ≅ ℙ_{ℙ_{u : v}^{1}} (𝒪_{e} \oplus 𝒪 {(1)}_{f})

by Lemma 4.2. The bundle

𝒩_{E / \tilde{X}}

has degree

- 2

along the fibres of the bundle

E \to ℙ_{u : v}^{1},

which we can see by considering that each fibre is the exceptional curve in the resolution of a surface node. Also note that since

E \subset \tilde{X}

is the strict transform of

E_{Y} \subset W,

E

is the blow-up of

E_{Y}

along the section given by

V (f) \subset E_{Y} .

This implies that the restriction of

𝒩_{E / \tilde{X}}

to the section

V (e)

is the same as the restriction of

𝒩_{E_{Y} / W} ≅ 𝒪 (- 1)

to

V (e) .

Hence, we find that

𝒩_{E / \tilde{X}} ≅ 𝒪 (- 2, 1),

and using the formula for

ω_{E}

given in (2.5), we compute that

{ω_{\tilde{X}} |}_{E} ≅ ω_{E} \otimes 𝒩_{E / \tilde{X}}^{\lor} ≅ 𝒪 (- 2, - 1) \otimes 𝒪 (2, - 1) ≅ 𝒪 (0, - 2) .

Finally, we consider the curve

C .

Recall the definition of a spherical object from Section 2.1.6.

Lemma 4.5.

The object $𝒪_{C} (- 1)$ is a $3 -$ spherical object in $𝐃^{b} (\tilde{X}) .$

Proof.

This follows if we can contract

C

to an ordinary double point. Starting from

B,

let

ℭ_{node}

be the union of the reduction of the branches, and consider the blow-up

Y^{'} = {Bl}_{ℭ_{node}} B

so that

Y^{'}

is a nodal variety. Now note that the total transform of the locus

ℭ_{node}

in

Y

is the union of

E_{B},

which is a Cartier divisor, and the centre

Z_{W}

of the blow-up

W \to Y .

It follows from the universal property of the blow-up that

W

can be written as a blow-up of

Y^{'}

at the strict transform of

ℭ_{0},

which is a ruled surface in

Y^{'}

intersecting the node. In particular, there is an exceptional curve

C_{1}

which contracts to give the node. The blow-up

φ_{W} : \tilde{X} \to W

has centre disjoint from

C_{1},

and we can compute by working locally over

U_{B}

that

C_{1}

is identified with the curve

C

under the blow-up. Hence,

C

can also be contracted to a threefold node.

4.2. A full exceptional sequence on the resolution

With the geometric setup established, we may now analyse

𝐃^{b} (\tilde{X})

using Orlov’s blow-up formula once again. We recall the exceptional loci in the alternative construction from Section 4.1.3, as well as the maps described in diagram (4.5). The centre of each blow-up in the alternative construction is a locally complete intersection. This gives us the following semiorthogonal decompositions, where for a

ℙ^{1} -

bundle

π : Σ \to ℙ^{1}

embedded into a variety

T

as

j : Σ \to T,

we denote by

𝒪_{Σ} (a, b)

the pushforward

j_{*} (𝒪 (a) \otimes π^{*} 𝒪 (b))

:

\begin{matrix} 𝐃^{b} (Y) & = ⟨ 𝐃^{b} (ℙ_{s : t}^{1}), 𝐃^{b} (B) ⟩ \\ = ⟨ 𝒪_{E_{B}} (- 1, - 1), 𝒪_{E_{B}} (- 1, 0), h_{Y}^{*} 𝐃^{b} (B) ⟩, \\ 𝐃^{b} (W) & = ⟨ 𝐃^{b} (ℙ_{u : v}^{1}), 𝐃^{b} (Y) ⟩ \\ = ⟨ 𝒪_{E_{Y}} (- 1, - 1), 𝒪_{E_{Y}} (- 1, 0), ψ^{*} 𝐃^{b} (Y) ⟩, \\ 𝐃^{b} (\tilde{X}) & = ⟨ 𝐃^{b} (ℙ_{u : v}^{1}), 𝐃^{b} (W) ⟩ \\ = ⟨ 𝒪_{E_{W}} (- 1, - 1), 𝒪_{E_{W}} (- 1, 0), φ_{W}^{*} 𝐃^{b} (W) ⟩ . \end{matrix}

The component in the semiorthogonal decomposition equivalent to

𝐃^{b} (B)

is immediately seen to be orthogonal to

\ker σ_{*} .

We therefore consider the following objects:

(4.6)

\begin{matrix} ℰ_{0} & := 𝒪_{E_{W}} (- 1, - 1), & ℰ_{1} & := 𝒪_{E_{W}} (- 1, 0), \\ ℱ_{0} & := φ_{W}^{*} 𝒪_{E_{Y}} (- 1, - 1), & ℱ_{1} & := φ_{W}^{*} 𝒪_{E_{Y}} (- 1, 0), \\ 𝒜_{0} & := φ_{Y}^{*} 𝒪_{E_{B}} (- 1, - 1), & 𝒜_{1} & := φ_{Y}^{*} 𝒪_{E_{B}} (- 1, 0), \end{matrix}

so that

\ker σ_{*}

lies in the exceptional sequence

\ker σ_{*} \subset ⟨ ℰ_{0}, ℰ_{1}, ℱ_{0}, ℱ_{1}, 𝒜_{0}, 𝒜_{1} ⟩ .

It is useful to obtain a geometric description for the objects

ℱ_{i}

and

𝒜_{i} .

The following lemmas are certainly specialisations of more general results on the properties of pulling back skyscraper sheaves along blow-ups, but we give the concrete calculations here for completeness, using the properties of ruled surfaces given in Section 2.2.

Lemma 4.6.

For $i \in {0, 1},$ $ℱ_{i}$ is the pushforward of the line bundle on $E \cup E_{W}$ which is

(1)
the line bundle $𝒪 (- 1, i - 1)$ on $E;$
(2)
the line bundle $𝒪 (0, i - 1)$ on $E_{W} .$

Proof.

We show the statement for

ℱ_{1},

using its description from equation (4.6), and the corresponding statement for

ℱ_{0}

follows by twisting appropriately. Firstly, taking the exact sequence on

W

0 \to 𝒪 \to 𝒪 (E_{Y}) \to 𝒪_{E_{Y}} (- 1, 0) \to 0

and applying the pullback

φ_{W}^{*}

shows that we have an exact sequence

0 \to 𝒪 \to 𝒪 (E + E_{W}) \to φ_{W}^{*} 𝒪_{E_{Y}} (- 1, 0) \to 0

on

\tilde{X} .

Note that this statement holds even though the functor

φ_{W}^{*}

is derived; in other words,

𝐋^{i} φ_{W}^{*} (𝒪_{E_{Y}} (- 1, 0))

vanishes for

i \neq 0 .

We find that the objects

ℱ_{i}

are supported on

E \cup E_{W} .

Note that

E \cap E_{W} = V (f) \subset E,

and

f

is a section of

𝒪 (1, - 1) .

Hence,

{𝒪 (E + E_{W}) |}_{E} ≅ 𝒪 (- 2, 1) \otimes 𝒪 (1, - 1) ≅ 𝒪 (- 1, 0)

and since

E \cap E_{W} = V (b) \subset E_{W},

where

b

is a section of

𝒪 (1, 0),

we have

{𝒪 (E + E_{W}) |}_{E_{W}} ≅ 𝒪 (- 1, 0) \otimes 𝒪 (1, 0) ≅ 𝒪 .

We hence find that

ℱ_{1}

is a pushforward of the line bundle on

E \cup E_{W}

which is given by

𝒪 (- 1, 0)

on the

E

component and

𝒪

on the

E_{W}

component. Note that the line bundle

𝒪 (a, b)

on

E

restricts to

𝒪 (b)

on

E \cap E_{W},

so such a line bundle exists on

E \cup E_{W} .

Recall that by Proposition 4.3,

\tilde{F} \subset \tilde{X}

coincides with the strict transform of

E_{B}

in

\tilde{X} .

Lemma 4.7.

For $i \in {0, 1},$ the objects $𝒜_{i}$ are given as

𝒜_{i} ≅ 𝒪_{\tilde{F}} (- 1, i - 1) .

Proof.

For the object

𝒜_{1},

we have a short exact sequence

0 \to 𝒪 \to 𝒪 (\tilde{F}) \to φ_{Y}^{*} 𝒜_{1} \to 0,

so that

𝒜_{1}

is the pushforward of the normal bundle of

\tilde{F} .

Recall that

\tilde{F}

is a ruled surface blown up twice, and has a chain of exceptional curves

C_{E} + C_{W},

where we denote

C_{E} := \tilde{F} \cap E

and

C_{W} = \tilde{F} \cap E_{W} .

We denote by

𝒪 (z, w) := p^{*} 𝒪 (z, w),

where

p : \tilde{F} \to E_{B}

is the blow-up map. Then, the normal bundle must be of the form

𝒩_{\tilde{F} / \tilde{X}} ≅ 𝒪 (z, w) \otimes 𝒪 (a C_{E} + b C_{W}) .

We know that

z = - 1

and

w = 0

since

𝒩_{E_{B} / Y} ≅ 𝒪 (- 1, 0)

and the restriction to the fibres of

\tilde{F} \to E_{B} \to ℙ_{u : v}^{1}

is generically unchanged. Next, one computes via the inclusion

C_{E} \overset{}{↪} E \overset{}{↪} \tilde{X}

that

𝒩_{C_{E} / \tilde{X}} ≅ 𝒪 \oplus 𝒪 (- 2) .

Since

𝒩_{C_{E} / \tilde{F}} ≅ 𝒪 (- 2),

we find that

{𝒪 (a C_{E} + b C_{W}) |}_{C_{E}} ≅ 𝒪 ⟹ - 2 a + b = 0 .

Similarly,

𝒩_{C_{W} / \tilde{F}} ≅ 𝒪 (- 1)

and

𝒩_{C_{W} / \tilde{X}} ≅ 𝒪 \oplus 𝒪 (- 1),

so we find that

{𝒪 (a C_{E} + b C_{W}) |}_{C_{W}} ≅ 𝒪 ⟹ - a + b = 0 .

Hence,

b = a = 0,

and we obtain the statement for

𝒜_{1},

with the case of

𝒜_{0}

being similar.

Observe now that naïvely pushing down this semiorthogonal decomposition via

σ_{*}

does not recover the semiorthogonal decomposition we have for

X .

Recall that we expressed our semiorthogonal decomposition as

𝐃^{b} (X) = ⟨ Φ (ℒ (1)), Φ (ℒ (2)), Φ (𝒫), 𝐃^{b} (B) ⟩,

where

Φ : 𝐃^{b} (ℭ) \to 𝐃^{b} (X)

is the embedding induced by Orlov’s blow-up formula. We recall that

ℒ (a)

denotes the line bundle

𝒪 (a, 0)

on

ℭ

which has degree

a

over the reduced branch and is trivial over the non-reduced branch, and

Φ (𝒫) =: 𝒟

is the absorbing category. We can see that

σ (E_{W}) = σ (E + E_{W}) = G_{red} \subset X,

while

σ (\tilde{F}) = F \subset X .

Hence, we immediately see that

σ_{*} ℰ_{1}

is our

c ℙ^{\infty}

object

Φ (𝒫),

but

σ_{*} ℰ_{0}

is a different

c ℙ^{\infty}

object, namely

Φ (𝒫 (- 1)) .

Meanwhile, the objects

𝒜_{i}

are supported on

\tilde{F}

and hence push down to sheaves supported only on

F .

It turns out that we can obtain a suitable exceptional sequence which recovers the semiorthogonal decomposition after pushing down simply by mutating the existing sequence appropriately.

Firstly, starting with the exceptional sequence

(4.7)

⟨ ℰ_{0}, ℰ_{1}, ℱ_{0}, ℱ_{1}, 𝒜_{0}, 𝒜_{1} ⟩,

we find that the objects

ℰ_{1}, ℱ_{0}

are orthogonal in Section 4.3.2, so that

⟨ ℰ_{0}, ℱ_{0}, ℰ_{1}, ℱ_{1}, 𝒜_{0}, 𝒜_{1} ⟩

is also an exceptional sequence. Then, we perform a series of left and right mutations, using the distinguished triangles described in Section 2.1.1.

Theorem 4.8.

Let

\begin{matrix} 𝒜_{i}^{''} & := 𝕃_{ℰ_{1}} 𝕃_{ℱ_{1}} 𝒜_{i}, \\ ℰ_{0}^{''} & := ℝ_{𝒜_{1}^{''}} ℝ_{𝒜_{0}^{''}} ℰ_{0}, \\ ℱ_{0}^{''} & := ℝ_{𝒜_{1}^{''}} ℝ_{𝒜_{0}^{''}} ℱ_{0} . \end{matrix}

Then, there is a semiorthogonal decomposition of $𝐃^{b} (\tilde{X})$ given by

𝐃^{b} (\tilde{X}) = ⟨ 𝒜_{0}^{''}, 𝒜_{1}^{''}, ℰ_{0}^{''}, ℱ_{0}^{''}, ℰ_{1}, ℱ_{1}, 𝐃^{b} (B) ⟩

such that:

(1)
$σ_{*} 𝒜_{0}^{''} ≅ Φ (ℒ (1))$ and $σ_{*} 𝒜_{1}^{''} ≅ Φ (ℒ (2));$
(2)
the admissible subcategory $\tilde{𝒟}$ defined using the exceptional sequence

$\tilde{𝒟} := ⟨ ℰ_{0}^{''}, ℱ_{0}^{''}, ℰ_{1}, ℱ_{1} ⟩$

induces an equivalence

$\tilde{𝒟} / (\ker σ_{*} \cap \tilde{𝒟}) \tilde{\to} 𝒟,$

where $𝒟 = ⟨ Φ (𝒫) ⟩ \subset 𝐃^{b} (X)$ is the absorbing category.

The mutations performed in the above theorem are as follows:

\begin{matrix} ⟨ ℰ_{0}, ℱ_{0}, ℰ_{1}, ℱ_{1}, 𝒜_{0}, 𝒜_{1} ⟩ & ⇝ ⟨ ℰ_{0}, ℱ_{0}, ℰ_{1}, 𝒜_{0}^{'}, 𝒜_{1}^{'}, ℱ_{1} ⟩ \\ ⇝ ⟨ ℰ_{0}, ℱ_{0}, 𝒜_{0}^{''}, 𝒜_{1}^{''}, ℰ_{1}, ℱ_{1} ⟩ \\ ⇝ ⟨ 𝒜_{0}^{''}, 𝒜_{1}^{''}, ℰ_{0}^{''}, ℱ_{0}^{''}, ℰ_{1}, ℱ_{1} ⟩, \end{matrix}

where we firstly mutate

⟨ 𝒜_{0}, 𝒜_{1} ⟩

to the left of

⟨ ℰ_{1}, ℱ_{1} ⟩,

and then mutate

⟨ ℰ_{0}, ℱ_{0} ⟩

to the right of

⟨ 𝒜_{0}^{''}, 𝒜_{1}^{''} ⟩ .

The theorem then states that pushforward under

σ_{*}

recovers the semiorthogonal decomposition of

𝐃^{b} (X) .

In particular, once the image of each component under

σ_{*}

is identified, the final statement follows as a consequence of Bondal–Orlov localisation, as described in Section 2.1.5. Since

\tilde{𝒟}

is an admissible component of a smooth projective variety, we suggest thinking of it as a categorical resolution for the absorbing category. Note that there are various formalisations of the notion of a categorical resolution in the literature (see, e.g., [12

A. Kuznetsov, Lefschetz decompositions and categorical resolutions of singularities. Selecta Math. (N.S.) 13 (2008), no. 4, 661–696 Zbl 1156.18006 MR 2403307

, 14

A. Kuznetsov and V. A. Lunts, Categorical resolutions of irrational singularities. Int. Math. Res. Not. IMRN 2015 (2015), no. 13, 4536–4625 Zbl 1338.14020 MR 3439086

]); here, we use the term less formally.

In the following Sections 4.3 and 4.4, we firstly obtain a complete description of the morphisms in the original exceptional sequence (4.7), as well the morphisms between the exceptional objects and to the objects in

\ker σ_{*} .

Using this, we are then able to perform the necessary mutations, and give a proof of Theorem 4.8.

4.3. Morphisms in the exceptional sequence

4.3.1. Morphisms to the null category. We firstly compute the morphisms between each object in the collection and some key objects in the kernel. The results are summarised in Tables 2 and 3.

$A$	$B$
	$𝒪_{C} (- 1)$	$𝒪_{E} (- 1, - 1)$	$𝒪_{E} (- 1, 0)$
$𝒜_{i}$	$ℂ$	0	0
$ℱ_{0}$	$ℂ [- 1]$	$ℂ$	$ℂ^{\oplus 2}$
$ℱ_{1}$	$ℂ [- 1]$	0	$ℂ$
$ℰ_{0}$	0	$ℂ [- 1]$	$ℂ^{\oplus 2} [- 1]$
$ℰ_{1}$	0	0	$ℂ [- 1]$

Table 2.

{Ext}^{•} (A, B),

where

B

is an object in the null category.

$A$	$B$
	$𝒜_{i}$	$ℱ_{0}$	$ℱ_{1}$	$ℰ_{0}$	$ℰ_{1}$
$𝒪_{C} (- 1)$	$ℂ [- 3]$	$ℂ [- 2]$	$ℂ [- 2]$	$0$	$0$
$𝒪_{E} (- 1, - 1)$	$0$	$ℂ [- 2]$	$ℂ^{\oplus 2} [- 2]$	$ℂ [- 1]$	$ℂ^{\oplus 2} [- 1]$
$𝒪_{E} (- 1, 0)$	$0$	$0$	$ℂ [- 2]$	$0$	$ℂ [- 1]$

Table 3.

{Ext}^{•} (A, B)

where

A

is an object in the null category.

4.3.1.1. Morphisms to $𝒪_{C} (- 1)$ . The objects

ℰ_{0}

and

ℰ_{1}

are orthogonal to

𝒪_{C} (- 1),

because they are supported on

E_{W}

which is disjoint from

C .

Both

ℱ_{0}

and

ℱ_{1}

are supported on

E \cup E_{W},

and we see that

E

and

C

intersect transversely, which gives us that

{Ext}^{•} (ℱ_{i}, 𝒪_{C} (- 1)) ≅ ℂ [- 1] .

Consider the objects

𝒜_{i},

and let

j : \tilde{F} \to \tilde{X}

be the inclusion. We have a splitting distinguished triangle for

j^{*} 𝒜_{i} ≅ j^{*} j_{*} 𝒪 (- 1, i - 1)

𝒪 (0, i - 1) [1] \to j^{*} 𝒜_{1} \to 𝒪 (- 1, i - 1)

which shows that

\begin{matrix} {Ext}^{•} (𝒜_{i}, 𝒪_{C} (- 1)) & ≅ {Ext}_{\tilde{F}}^{•} (j^{*} 𝒜_{i}, 𝒪_{C} (- 1)) \\ ≅ {Ext}_{\tilde{F}}^{•} (𝒪 (- 1, i - 1) \oplus 𝒪 (0, i - 1) [1], 𝒪_{C} (- 1)) \\ ≅ {Ext}_{C}^{•} (𝒪 (- 1) \oplus 𝒪 [1], 𝒪 (- 1)) ≅ ℂ . \end{matrix}

The above calculations also let us compute morphisms from

𝒪_{C} (- 1),

since it is a

3 -

spherical object by Lemma 4.5, so that

{Ext}^{i} (𝒪_{C} (- 1), A) ≅ {Ext}^{3 - i} (A, 𝒪_{C} (- 1))

for any object

A .

4.3.1.2. Morphisms between $𝒜_{i}$ and $𝒪_{E} (- 1, a) .$ The objects

𝒜_{0}, 𝒜_{1}

are completely orthogonal to these objects because

{(φ_{Y})}_{*} 𝒪_{E} (- 1, a) = 0

for any

a,

and because

{Ext}^{i} (𝒪_{E} (- 1, a), -) ≅ {Ext}^{3 - i} (-, 𝒪_{E} (- 1, a - 2))

by Lemma 4.4.

4.3.1.3. Morphisms between $ℱ_{i}$ and $𝒪_{E} (- 1, a) .$ Next we compute the morphisms from

ℱ_{0}

to

𝒪_{E} (- 1, - 1) .

The key in this computation is that

E

is the strict transform of

E_{Y} \subset W,

so

{Ext}_{\tilde{X}}^{•} (ℱ_{0}, 𝒪_{E} (- 1, - 1)) ≅ {Ext}_{W}^{•} (𝒪_{E_{Y}} (- 1, - 1), 𝒪_{E_{Y}} (- 1, - 1)) ≅ ℂ,

and similarly

{Ext}_{\tilde{X}}^{•} (ℱ_{0}, 𝒪_{E} (- 1, 0)) ≅ {Ext}_{W}^{•} (𝒪_{E_{Y}} (- 1, - 1), 𝒪_{E_{Y}} (- 1, 0)) ≅ ℂ^{\oplus 2}

because

⟨ 𝒪_{E_{Y}} (- 1, - 1), 𝒪_{E_{Y}} (- 1, 0) ⟩ ≃ 𝐃^{b} (ℙ^{1}) .

We similarly find that

\begin{matrix} {Ext}_{\tilde{X}}^{•} (ℱ_{1}, 𝒪_{E} (- 1, 0)) & ≅ ℂ, \\ {Ext}_{\tilde{X}}^{•} (ℱ_{1}, 𝒪_{E} (- 1, - 1)) & ≅ 0 . \end{matrix}

To compute morphisms in the reverse direction, we use Serre duality as before:

\begin{matrix} {Ext}^{•} (𝒪_{E} (- 1, - 1), ℱ_{0}) & ≅ ℂ [- 2], \\ {Ext}^{•} (𝒪_{E} (- 1, - 1), ℱ_{1}) & ≅ ℂ^{\oplus 2} [- 2], \\ {Ext}^{•} (𝒪_{E} (- 1, 0), ℱ_{0}) & = 0, \\ {Ext}^{•} (𝒪_{E} (- 1, 0), ℱ_{1}) & ≅ ℂ [- 2] . \end{matrix}

4.3.1.4. Morphisms between $ℰ_{i}$ and $𝒪_{E} (- 1, a) .$ Instead of using a geometric method to compute morphisms from

ℰ_{i},

we make the following observation.

Lemma 4.9.

For $i \in {0, 1},$ there are short exact sequences

(4.8)

0 \to ℰ_{i} \to ℱ_{i} \to 𝒪_{E} (- 1, i - 1) \to 0 .

Proof.

We can form the following commuting diagram, where the rows and the first two columns are exact sequences.

Applying the nine-lemma then gives us the desired short exact sequence from the rightmost column, and running the argument with the appropriate line bundle twist gives us the sequence for

ℰ_{0} .

The remaining morphisms in the tables are an immediate consequence of these distinguished triangles and the following statements:

\begin{matrix} {Ext}^{•} (𝒪_{E} (a, b), 𝒪_{E} (a, b)) & ≅ ℂ \oplus ℂ [- 2], \\ {Ext}^{•} (𝒪_{E} (a, b), 𝒪_{E} (a, b - 1)) & = 0 \end{matrix}

which follow using the usual distinguished triangle (2.3) for

{𝒪_{E} (a, b) |}_{E}

𝒪 (a + 2, b - 1) [1] \to {𝒪_{E} (a, b) |}_{E} \to 𝒪 (a, b),

where we have used that

𝒩_{E / \tilde{X}} ≅ 𝒪 (- 2, 1)

and invoked equation (2.6) in order to compute cohomology of line bundles on the ruled surface

E .

4.3.2. Morphisms between objects in the exceptional sequence. We summarise the results in Table 4.

$A$	$B$
	$ℰ_{0}$	$ℰ_{1}$	$ℱ_{0}$	$ℱ_{1}$	$𝒜_{0}$	$𝒜_{1}$
$ℰ_{0}$	$ℂ$	$ℂ^{\oplus 2}$	$ℂ \oplus ℂ [- 1]$	$ℂ^{\oplus 2} \oplus ℂ^{\oplus 2} [- 1]$	$ℂ [- 1]$	$ℂ [- 1]$
$ℰ_{1}$	$0$	$ℂ$	$0$	$ℂ \oplus ℂ [- 1]$	$ℂ [- 1]$	$ℂ [- 1]$
$ℱ_{0}$	$0$	$0$	$ℂ$	$ℂ^{\oplus 2}$	$ℂ [- 1]$	$ℂ [- 1]$
$ℱ_{1}$	$0$	$0$	$0$	$ℂ$	$ℂ [- 1]$	$ℂ [- 1]$
$𝒜_{0}$	$0$	$0$	$0$	$0$	$ℂ$	$ℂ^{\oplus 2}$
$𝒜_{1}$	$0$	$0$	$0$	$0$	$0$	$ℂ$

Table 4.

{Ext}^{•} (A, B),

where

A, B

are objects in the exceptional sequence.

4.3.2.1. Morphisms to $𝒜_{i} .$ To begin, we see that

\begin{matrix} {Ext}_{\tilde{X}}^{i} (ℱ_{1}, 𝒜_{1}) & ≅ {Ext}_{W}^{i} (𝒪_{E_{Y}} (- 1, 0), ψ^{*} 𝒪_{E_{B}} (- 1, 0)) \\ ≅ {Ext}_{W}^{3 - i} (ψ^{*} 𝒪_{E_{B}} (- 1, 0), 𝒪_{E_{Y}} (- 2, - 1)) \\ ≅ {Ext}_{W}^{3 - i} (𝒪_{E_{B}} (- 1, 0), ψ_{*} 𝒪_{E_{Y}} (- 2, - 1)) \\ ≅ {Ext}_{Y}^{3 - i} (𝒪_{E_{B}} (- 1, 0), 𝒪_{Z_{W}} (- 2) [1]) ≅ ℂ for i = 1 \end{matrix}

and that there are no morphisms in any other degree. One sees, since the map is induced by a transversal intersection at a single point, that the same result holds for

{Ext}^{•} (ℱ_{j}, 𝒜_{k})

for any choice of

j, k \in {0, 1} .

Since the objects

𝒜_{i}

are orthogonal to

𝒪_{E} (- 1, a)

for any

a,

we find that

{Ext}^{•} (ℰ_{j}, 𝒜_{k}) ≅ {Ext}^{•} (ℱ_{j}, 𝒜_{k})

for any

j, k \in {0, 1} .

4.3.2.2. Morphisms to $ℱ_{i} .$ For each

i \in {0, 1},

using that

{Ext}^{•} (ℰ_{i}, 𝒪_{E} (- 1, i - 1)) ≅ ℂ [- 1],

we obtain that

{Ext}^{•} (ℰ_{i}, ℱ_{i}) ≅ ℂ \oplus ℂ [- 1] .

Since

{Ext}^{•} (ℰ_{1}, 𝒪_{E} (- 1, - 1)) = {Ext}^{•} (ℰ_{1}, ℰ_{0}) = 0,

we must also have that

{Ext}^{•} (ℰ_{1}, ℱ_{1}) = 0 .

Finally, using that

{Ext}^{•} (ℰ_{0}, 𝒪_{E} (- 1, 0)) ≅ ℂ^{\oplus 2} [- 1],

we find that

{Ext}^{•} (ℰ_{0}, ℱ_{1}) ≅ ℂ^{\oplus 2} \oplus ℂ^{\oplus 2} [- 1] .

4.4. Mutating the semiorthogonal decomposition

We are now in a position to perform the mutations described previously. The first set is defined by the following exact triangles:

(4.9)

𝒜_{0} \to 𝒜_{0}^{'} \to ℱ_{1},

(4.10)

𝒜_{1} \to 𝒜_{1}^{'} \to ℱ_{1}

We have that the objects

𝒜_{i}^{'}

are sheaves supported on

\tilde{F} \cup E_{W} \cup E .

Now note that the orthogonality of

𝒜_{0}, 𝒜_{1}

with

𝒪_{E} (- 1, 0)

implies that there is an isomorphism

{Ext}^{•} (ℱ_{1}, 𝒜_{j}) ≅ {Ext}^{•} (ℰ_{1}, 𝒜_{j}) ≅ ℂ [- 1]

which we obtain by applying

{Ext}^{•} (-, 𝒜_{j})

to the triangle (4.8). This isomorphism is obtained by precomposing with

ℰ_{1} \to ℱ_{1},

so we obtain a commuting triangle.

Applying

{Ext}^{•} (ℰ_{1}, -)

to the triangles (4.9) and (4.10) gives us exact sequences

\begin{matrix} 0 & \to Hom (ℰ_{1}, ℱ_{1}) \tilde{\to} {Ext}^{1} (ℰ_{1}, 𝒜_{j}) \\ \overset{0}{\to} {Ext}^{1} (ℰ_{1}, 𝒜_{j}^{'}) \tilde{\to} {Ext}^{1} (ℰ_{1}, ℱ_{1}) \to 0, \end{matrix}

where the first non-zero map is an isomorphism due to the above commuting triangle. Hence, we have

{Ext}^{1} (ℰ_{1}, 𝒜_{j}^{'}) ≅ {Ext}^{1} (ℰ_{1}, ℱ_{1}) ≅ ℂ .

The next set of mutations is therefore defined by the following exact triangles:

(4.11)

\begin{matrix} 𝒜_{0}^{'} \to 𝒜_{0}^{''} \to ℰ_{1}, \\ 𝒜_{1}^{'} \to 𝒜_{1}^{''} \to ℰ_{1} . \end{matrix}

We now have the job of tidying up the exceptional sequence

⟨ ℰ_{0}, ℱ_{0}, 𝒜_{0}^{''}, 𝒜_{1}^{''}, ℰ_{1}, ℱ_{1} ⟩ ⇝ ⟨ 𝒜_{0}^{''}, 𝒜_{1}^{''}, ℰ_{0}^{''}, ℱ_{0}^{''}, ℰ_{1}, ℱ_{1} ⟩ .

Lemma 4.10.

We have that

{Ext}^{•} (ℱ_{0}, 𝒜_{0}^{''}) ≅ {Ext}^{•} (ℱ_{0}, 𝒜_{1}^{''}) ≅ ℂ,

and if $ℱ_{0}^{'} := Cone (ℱ_{0} \to 𝒜_{0}^{''}),$ then we have that

{Ext}^{•} (ℱ_{0}^{'}, 𝒜_{1}^{''}) ≅ ℂ .

Proof.

Applying

{Ext}^{•} (ℱ_{0}, -)

to (4.11) gives an isomorphism

{Ext}^{•} (ℱ_{0}, 𝒜_{0}^{'}) ≅ {Ext}^{•} (ℱ_{0}, 𝒜_{0}^{''})

since

{Ext}^{•} (ℱ_{0}, ℰ_{1}) = 0 .

Similarly, applying

{Ext}^{•} (ℱ_{0}, -)

to (4.9) gives a long exact sequence of

Ext

groups

\begin{matrix} 0 & \to Hom (ℱ_{0}, 𝒜_{0}) \to Hom (ℱ_{0}, 𝒜_{0}^{'}) \to Hom (ℱ_{0}, ℱ_{1}) \\ \to {Ext}^{1} (ℱ_{0}, 𝒜_{0}) \to {Ext}^{1} (ℱ_{0}, 𝒜_{0}^{'}) \to {Ext}^{1} (ℱ_{0}, ℱ_{1}) \to 0 \end{matrix}

which reads

\begin{matrix} 0 & \to 0 \to Hom (ℱ_{0}, 𝒜_{0}^{'}) \to ℂ^{\oplus 2} \\ \to ℂ \to {Ext}^{1} (ℱ_{0}, 𝒜_{0}^{'}) \to 0 . \end{matrix}

Then, the map

ℂ^{\oplus 2} ≅ Hom (ℱ_{0}, ℱ_{1}) \to {Ext}^{1} (ℱ_{0}, 𝒜_{0}) ≅ ℂ

is non-zero, due to the fact that the extension of

ℱ_{0}

by

𝒜_{0}

is induced by the intersection of

E_{W} \cup E

with

\tilde{F},

and there is a map

ℱ_{0} \to ℱ_{1}

which is non-vanishing in a neighbourhood of the intersection locus, which is a reducible curve.

The second part of the lemma amounts to showing that the map

ℂ^{\oplus 2} ≅ Hom (𝒜_{0}^{''}, 𝒜_{1}^{''}) \to Hom (ℱ_{0}, 𝒜_{1}^{''}) ≅ ℂ

induced by precomposing with

ℱ_{0} \to 𝒜_{0}^{''}

is non-zero. By mutating over

ℰ_{1}

and then

ℱ_{1},

we see that this is equivalent to showing that the map

ℂ^{\oplus 2} ≅ Hom (𝒜_{0}, 𝒜_{1}) \to {Ext}^{1} (ℱ_{2}, 𝒜_{1}) ≅ ℂ

is non-zero, where

ℱ_{2} = φ_{W}^{*} 𝒪_{E_{Y}} (- 1, 1)

and the morphism is given by shifting and then precomposing with

ℱ_{2} \to 𝒜_{1} [1] .

The argument to show that this map is non-zero is analogous to the one in the previous paragraph.

We use the above lemma to create the following distinguished triangles defining an object

ℱ_{0}^{''}

:

(4.12)

ℱ_{0} \to 𝒜_{0}^{''} \to ℱ_{0}^{'},

(4.13)

ℱ_{0}^{'} \to 𝒜_{1}^{''} \to ℱ_{0}^{''} .

This object has a more explicit description.

Lemma 4.11.

Let $𝒦$ be the object defined by the extension

(4.14)

0 \to 𝒪_{C} (- 1) \to 𝒦 \to 𝒪_{E} (- 1, 0) \to 0 .

Then, $ℱ_{0}^{''}$ is given by the extension

(4.15)

0 \to 𝒦 \to ℱ_{0}^{''} \to ℰ_{1} \to 0 .

Proof.

Since

ℰ_{1}

is orthogonal to

𝒪_{C} (- 1),

we have

{Ext}^{•} (ℰ_{1}, 𝒦) ≅ ℂ [- 1]

giving us an extension

0 \to 𝒦 \to 𝒢 \to ℰ_{1} \to 0 .

It is clear by applying

{Ext}^{•} (ℰ_{1}, -)

to this short exact sequence that

{Ext}^{•} (ℰ_{1}, 𝒢) = 0 .

Next, we apply

{Ext}^{•} (ℱ_{1}, -)

to the triangle (4.14) defining

𝒦 .

This gives an exact sequence

\begin{matrix} 0 & \to Hom (ℱ_{1}, 𝒦) \to Hom (ℱ_{1}, 𝒪_{E} (- 1, 0)) ≅ ℂ \\ \to ℂ ≅ {Ext}^{1} (ℱ_{1}, 𝒪_{C} (- 1)) \to {Ext}^{1} (ℱ_{1}, 𝒦) \to 0 . \end{matrix}

The claim is that the middle map is an isomorphism, so that

{Ext}^{•} (ℱ_{1}, 𝒦) = 0 .

But this is clear geometrically, since the extensions

ℱ_{1} \to 𝒪_{C} (- 1) [1]

and

𝒪_{E} (- 1, 0) \to 𝒪_{C} (- 1) [1]

are induced by the intersection of

C

and

E,

and in a neighbourhood of the intersection point

ℱ_{1} \to 𝒪_{E} (- 1, 0)

is precisely the identity. Hence,

{Ext}^{•} (ℱ_{1}, 𝒦) = {Ext}^{•} (ℱ_{1}, ℰ_{1}) = 0,

so we get that

{Ext}^{•} (ℱ_{1}, 𝒢) = 0 .

Since there are no morphisms from

𝒪_{C} (- 1)

or

𝒪_{E} (- 1, 0)

to

ℰ_{0},

we see that

{Ext}^{•} (𝒢, ℰ_{0}^{''}) = 0 .

Hence,

𝒢 \in ⟨ ℱ_{0}^{''} ⟩ .

It is clear by the mutations relating

ℱ_{0}^{''}

to

ℱ_{0}

that

{Ext}^{•} (ℱ_{0}, ℱ_{0}^{''}) ≅ ℂ .

Hence, it suffices to show that

{Ext}^{•} (ℱ_{0}, 𝒢) ≅ ℂ .

We know that

{Ext}^{•} (ℱ_{0}, 𝒪_{E} (- 1, 0)) ≅ ℂ^{\oplus 2},

and by similar reasoning as we did to show that the map

Hom (ℱ_{1}, 𝒪_{E} (- 1, 0)) \to {Ext}^{1} (ℱ_{1}, 𝒪_{C} (- 1))

is non-zero, we also find that the map

ℂ^{\oplus 2} ≅ Hom (ℱ_{0}, 𝒪_{E} (- 1, 0)) \to {Ext}^{1} (ℱ_{0}, 𝒪_{C} (- 1)) ≅ ℂ

is non-zero. Hence,

{Ext}^{•} (ℱ_{0}, 𝒦) ≅ ℂ,

and since

{Ext}^{•} (ℱ_{0}, ℰ_{1}) = 0,

it follows immediately that

{Ext}^{•} (ℱ_{0}, 𝒢) ≅ ℂ .

Next we mutate the object

ℰ_{0} .

Lemma 4.12.

We have that

{Ext}^{•} (ℰ_{0}, 𝒜_{0}^{''}) ≅ {Ext}^{•} (ℰ_{0}, 𝒜_{1}^{''}) ≅ ℂ,

and if $ℰ_{0}^{'} := Cone (ℰ_{0} \to 𝒜_{0}^{''}),$ then

{Ext}^{•} (ℰ_{0}^{'}, 𝒜_{1}^{''}) ≅ ℂ .

Proof.

The first statement holds because of the exact triangle

ℰ_{0} \to ℱ_{0} \to 𝒪_{E} (- 1, - 1)

and the fact that

𝒜_{0}^{''}, 𝒜_{1}^{''}

are orthogonal to

𝒪_{E} (- 1, - 1) .

For the second statement, we use the nine-lemma for derived categories, Lemma 2.2, and find that there is a distinguished triangle

ℰ_{0}^{'} \to ℱ_{0}^{'} \to 𝒪_{E} (- 1, - 1) [1] .

We similarly define the object

ℰ_{0}^{''}

(4.16)

ℰ_{0} \to 𝒜_{0}^{''} \to ℰ_{0}^{'},

(4.17)

ℰ_{0}^{'} \to 𝒜_{1}^{''} \to ℰ_{0}^{''} .

The cone on the morphism

ℰ_{0}^{''} \to ℱ_{0}^{''}

is seen to be

𝒪_{E} (- 1, - 1) [2]

by another application of the derived nine-lemma. In particular,

ℰ_{0}^{''}

is not a sheaf.

Recall that we know that

σ_{*} ℰ_{1} ≅ Φ (𝒫),

and an easy corollary of Lemma 4.9 is that

σ_{*} ℱ_{1} ≅ σ_{*} ℰ_{1} .

The next lemmas will allow us to describe what happens if we pushforward the objects

𝒜_{i}^{''} .

Lemma 4.13.

For each $j \in {0, 1},$ the objects $𝒜_{j}$ and $𝒜_{j}^{''}$ are related by the exact triangle

𝒜_{j} \to 𝒜_{j}^{''} \to 𝒴,

where $𝒴$ is given by the non-trivial extension

ℱ_{1} \to 𝒴 \to ℰ_{1} .

Proof.

Consider the exact triangles

\begin{matrix} 𝒜_{0} \to 𝒜_{0}^{'} \to ℱ_{1}, \\ 𝒜_{0}^{'} \to 𝒜_{0}^{''} \to ℰ_{1}, \\ 𝒜_{0} \to 𝒜_{0}^{''} \to 𝒴 . \end{matrix}

Here, we use that

{Ext}^{•} (𝒜_{0}, 𝒜_{0}^{''}) ≅ {Ext}^{•} (𝒜_{0}, 𝒜_{0}^{'}) ≅ {Ext}^{•} (𝒜_{0}, 𝒜_{0}) ≅ ℂ,

where the first isomorphism is induced by postcomposing with

𝒜_{0}^{'} \to 𝒜_{0}^{''}

and the second isomorphism is induced by postcomposing with

𝒜_{0} \to 𝒜_{0}^{'} .

Therefore, the map

𝒜_{0} \to 𝒜_{0}^{''}

is a composition of the maps

𝒜_{0} \to 𝒜_{0}^{'}

and

𝒜_{0}^{'} \to 𝒜_{0}^{''},

and we can use the octahedral axiom to obtain an exact triangle

ℱ_{1} \to 𝒴 \to ℰ_{1} .

The triangle is non-split since the morphism

ℰ_{1} \to ℱ_{1} [1]

in the triangle is the composition

ℰ_{1} \to 𝒜_{0}^{'} [1] \to ℱ_{1} [1],

and we know that composing with

𝒜_{0}^{'} \to ℱ_{1}

induces an isomorphism

{Ext}^{1} (ℰ_{1}, 𝒜_{1}^{'}) ≅ {Ext}^{1} (ℰ_{1}, ℱ_{1}) .

It is geometrically easier to see what happens if we first pushforward to

\hat{X};

the next lemma shows that the kernel of this contraction is simply generated by

𝒪_{E} (- 1, - 1)

and

𝒪_{E} (- 1, 0) .

We can see this as a direct consequence of the fact that for a surface node, the kernel of the contraction is generated by the pushforward of the

𝒪 (- 1)

bundle on the exceptional curve.

Lemma 4.14.

The kernel of the contraction $\tilde{X} \to \hat{X}$ is generated by

\ker φ_{\hat{X}, *} = ⟨ 𝒪_{E} (- 1, - 1), 𝒪_{E} (- 1, 0) ⟩ .

Proof.

Let

Z \subset \hat{X}

be the reduced singular locus. Since the objects

𝒪 (- 1, - 1), 𝒪 (- 1, 0)

generate the kernel of the contraction

E \to Z,

we want to show that the kernel of

\tilde{X} \to \hat{X}

is generated by the pushforwards of these objects. We use [15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

, Theorem 5.2] to show this, which requires us to show that

φ_{\hat{X}, *} 𝒪_{\tilde{X}} (- m E) ≅ 𝒥_{Z}^{m},

where

𝒥_{Z}

is the ideal sheaf of

Z .

We can work in a neighbourhood of

Z,

but locally near

Z

we find that

\hat{X}

is just

V \times ℙ^{1}

for a nodal surface

V,

and that

\tilde{X}

is, locally around

E,

the pullback of the resolution

\tilde{V}

of

V,

that is, we have a pullback square.

Let

π : \tilde{V} \to V

be the contraction,

F

the exceptional curve and

Z_{V}

the nodal point. The desired result follows from pulling back the corresponding equivalence for

V,

which was shown in [15

A. Kuznetsov and E. Shinder, Categorical absorptions of singularities and degenerations. Épijournal Géom. Algébrique (2024), article no. 12, 42 pp., (Special volume in honour of Claire Voisin) Zbl 1545.14026 MR 4698898

, Corollary 5.6]

π_{*} 𝒪_{\tilde{V}} (- m F) ≅ 𝒥_{Z_{V}}^{m}

and using flat base change

f^{*} π_{*} ≃ π_{*} f^{*}

as needed.

Consider the extension on

𝐃^{b} (ℭ)

0 \to 𝒫 \to 𝒬 \to 𝒫 \to 0,

so that

𝒬

is the skyscraper sheaf along the non-reduced branch

ℙ_{u : v}^{1} \times ℂ [s] / s^{2} .

Lemma 4.15.

We have that

σ_{*} 𝒴 ≅ Φ (𝒬) .

Proof.

Firstly, we see that

𝒴

is orthogonal to the objects

𝒪_{E} (- 1, a)

for all

a \in ℤ .

Since

{Ext}^{•} (ℰ_{1}, 𝒪_{E} (- 1, - 1)) = {Ext}^{•} (ℱ_{1}, 𝒪_{E} (- 1, - 1)) = 0,

we immediately have

{Ext}^{•} (𝒴, 𝒪_{E} (- 1, - 1)) = 0 .

We also know that

{Ext}^{1} (ℰ_{1}, ℱ_{1}) ≅ {Ext}^{1} (ℰ_{1}, 𝒪_{E} (- 1, 0)),

that is, the map

ℰ_{1} \to 𝒪_{E} (- 1, 0) [1]

factors through

ℰ_{1} \to ℱ_{1} [1] .

Then, applying the functor

{Ext}^{•} (-, 𝒪_{E} (- 1, 0))

to the triangle

ℱ_{1} \to 𝒴 \to ℰ_{1}

gives us an exact sequence

\begin{matrix} 0 & \to Hom (𝒴, 𝒪_{E} (- 1, 0)) \to Hom (ℱ_{1}, 𝒪_{E} (- 1, 0)) \\ \tilde{\to} {Ext}^{1} (ℰ_{1}, 𝒪_{E} (- 1, 0)) \to {Ext}^{1} (𝒴, 𝒪_{E} (- 1, 0)) \to 0 \end{matrix}

which gives us that

{Ext}^{•} (𝒴, 𝒪_{E} (- 1, 0)) = 0 .

It follows then that

{Ext}^{•} (𝒴, 𝒪_{E} (- 1, a)) = 0

for all integers

a,

and hence by Serre duality that

{Ext}^{•} (𝒪_{E} (- 1, a), 𝒴) = 0

also.

Now, we pushforward to

\hat{X} .

Here, we have a distinguished triangle

φ_{\hat{X}, *} ℱ_{1} \to φ_{\hat{X}, *} 𝒴 \to φ_{\hat{X}, *} ℰ_{1} .

One sees that

φ_{\hat{X}, *} ℱ_{1} ≅ φ_{\hat{X}, *} ℰ_{1}

is the

𝒪 (- 1, 0)

bundle on the strict transform of

G_{red} .

The extension is non-split, since we showed that

𝒴

is orthogonal to

\ker φ_{\hat{X}, *},

so for example we know that

φ_{\hat{X}, *} 𝒴

has no endomorphisms in non-zero degrees. Hence,

φ_{\hat{X}, *} 𝒴

is the

𝒪 (- 1, 0)

bundle on the strict transform of

G .

Now we can see that

σ_{*} 𝒴

is the

𝒪 (- 1, 0)

bundle on

G,

or equivalently,

Φ (𝒬) .

Proof of Theorem 4.8.

To prove that

\begin{matrix} σ_{*} 𝒜_{0}^{''} & ≅ Φ (ℒ (1)), \\ σ_{*} 𝒜_{1}^{''} & ≅ Φ (ℒ (2)), \end{matrix}

recall the defining triangle for

𝒜_{1}^{'}

(4.10)

𝒜_{1} \to 𝒜_{1}^{'} \to ℱ_{1} .

In fact, we note that the transverse intersection of

E \cup E_{W}

with

\tilde{F}

along

C_{E} \cup C_{W} \subset \tilde{F}

induces an extension between

ℱ_{1}

and

𝒜_{1} .

By uniqueness, we find that

𝒜_{1}^{'}

is the pushforward of a line bundle on

E \cup E_{W} \cup \tilde{F}

which is

𝒪 (- 1, 0) \otimes 𝒪 (C_{E} + C_{W})

on

\tilde{F},

𝒪 (- 1, 0)

on

E

and

𝒪

on

E_{W} .

One verifies that this is a valid line bundle on the union since all restrictions to intersections agree. There is a line bundle on

ℙ_{s : t}^{1} \cup ℙ_{u : v}^{1} \subset C

(i.e., the union of the reduced branches) which restricts to

𝒪 (1)

on

ℙ_{s : t}^{1}

and

𝒪

on

ℙ_{u : v}^{1} .

We denote by

ℳ

the pushforward of this into

ℭ .

Then, one sees that

σ_{*} 𝒜_{1}^{'} ≅ Φ (ℳ) .

Let

𝒪_{ℭ_{0}}

be the skyscraper sheaf along the reduced branch. Then,

σ_{*} 𝒜_{1} = Φ (𝒪_{ℭ_{0}}) .

Next, there is a unique extension in

𝐃^{b} (ℭ)

between

𝒫

and

ℳ

(4.18)

ℳ \to ℒ (2) \to 𝒫 .

We can consider the following triangle, obtained by pushing forward:

𝒜_{1}^{'} \to 𝒜_{1}^{''} \to ℰ_{1} .

If this pushforward is non-split, it corresponds to the triangle obtained by embedding (4.18) using

Φ .

To show that it does not split, we pushforward the triangle

𝒜_{1} \to 𝒜_{1}^{''} \to 𝒴

which must correspond to an extension in

𝐃^{b} (ℭ)

between

𝒬

and

𝒪_{ℭ_{0}}

by Lemma 4.15. Then, one computes that there is indeed an extension induced by intersection

𝒪_{ℭ_{0}} \to ℒ (2) \to 𝒬,

so that

σ_{*} 𝒜_{1}^{''}

can only be

Φ (ℒ (2)) .

A similar argument shows

σ_{*} 𝒜_{0}^{''} ≅ Φ (ℒ (1)) .

Now let

\tilde{𝒟} = ⟨ ℰ_{0}^{''}, ℱ_{0}^{''}, ℰ_{1}, ℱ_{1} ⟩ .

The statement

\tilde{𝒟} / (\ker σ_{*} \cap \tilde{𝒟}) \tilde{\to} 𝒟

follows from the fact that the cones on the morphisms

ℱ_{0}^{''} \to ℰ_{1},

ℰ_{0}^{''} \to ℱ_{0}^{''}

and

ℰ_{1} \to ℱ_{1}

are in the kernel of

σ_{*},

so the image of

\tilde{𝒟}

under

σ_{*}

is contained within our absorbing category

𝒟 .

Combined with the fact that

𝒜_{0}^{''}, 𝒜_{1}^{''}

pushforward to

Φ (ℒ (1))

and

Φ (ℒ (2)),

respectively, we find that the image of

\tilde{𝒟}

under

σ_{*}

is in fact the entire absorbing category

𝒟,

and the desired equivalence follows from Bondal–Orlov localisation (see Section 2.1.5).

4.5. Structure of the categorical resolution

Having found the subcategory

\tilde{𝒟} \subset 𝐃^{b} (\tilde{X}),

we now endeavour to describe the morphisms between the objects and how they behave under composition. Since the subcategory is found by mutating an exceptional sequence, the resulting objects also form an exceptional sequence

\tilde{𝒟} = ⟨ ℰ_{0}^{''}, ℱ_{0}^{''}, ℰ_{1}, ℱ_{1} ⟩ .

We recall that the objects

ℰ_{1}, ℱ_{1}

are sheaves supported on the exceptional loci

E_{W}, {\tilde{E}}_{Y}

as defined in equation (4.6), while

ℰ_{0}^{''}

and

ℱ_{0}^{''}

are defined by the triangles induced by mutation given in equations (4.16), (4.17) and equations (4.12), (4.13), respectively.

4.5.1. Morphisms in the categorical resolution.

Theorem 4.16.

The morphisms in $\tilde{𝒟}$ are described as follows:

(1)
${Ext}^{•} (ℰ_{0}^{''}, ℱ_{0}^{''}) ≅ {Ext}^{•} (ℰ_{1}, ℱ_{1}) ≅ ℂ \oplus ℂ [- 1]$ ;
(2)
${Ext}^{•} (ℱ_{0}^{''}, ℰ_{1}) ≅ ℂ \oplus ℂ [- 1]$ ;
(3)
${Ext}^{•} (ℱ_{0}^{''}, ℱ_{1}) ≅ {Ext}^{•} (ℰ_{0}^{''}, ℰ_{1}) ≅ ℂ \oplus ℂ [- 1] \oplus ℂ^{\oplus 2} [- 2]$ ;
(4)
${Ext}^{•} (ℰ_{0}^{''}, ℱ_{1}) ≅ ℂ \oplus ℂ [- 1] \oplus ℂ^{\oplus 2} [- 2] \oplus ℂ^{\oplus 2} [- 3]$ .

The first claim of the theorem has already been proven in Section 4.3.2, and we prove the rest in a series of lemmas.

Lemma 4.17.

We have that ${Ext}^{•} (ℱ_{0}^{''}, ℰ_{1}) ≅ ℂ \oplus ℂ [- 1]$ and ${Ext}^{•} (ℱ_{0}^{''}, ℱ_{1}) ≅ ℂ \oplus ℂ [- 1] \oplus ℂ^{\oplus 2} [- 2] .$

Proof.

Apply

{Ext}^{•} (-, ℰ_{1})

to (4.15). Since

{Ext}^{•} (𝒪_{C} (- 1), ℰ_{1}) = 0,

we find that

{Ext}^{•} (𝒦, ℰ_{1}) ≅ {Ext}^{•} (𝒪_{E} (- 1, 0), ℰ_{1}) ≅ ℂ [- 1],

and this gives us that

{Ext}^{•} (ℱ_{0}^{''}, ℰ_{1}) ≅ ℂ \oplus ℂ [- 1] .

Applying

{Ext}^{•} (-, ℱ_{1})

to the triangle (4.14) defining

𝒦,

we find that

{Ext}^{•} (𝒦, ℱ_{1}) ≅ ℂ^{\oplus 2} [- 2]

because we have

{Ext}^{•} (𝒪_{C} (- 1), ℱ_{1}) ≅ {Ext}^{•} (𝒪_{E} (- 1, 0), ℱ_{1}) ≅ ℂ [- 2] .

Then, applying

{Ext}^{•} (-, ℱ_{1})

to the triangle (4.15) gives us that

ℂ ≅ {Ext}^{i} (ℰ_{1}, ℱ_{1}) ≅ {Ext}^{i} (ℱ_{0}^{''}, ℱ_{1})

for

i = 0, 1,

and

{Ext}^{2} (ℱ_{0}^{''}, ℱ_{1}) ≅ {Ext}^{2} (𝒦, ℱ_{1}) ≅ ℂ^{\oplus 2} .

Lemma 4.18.

We have that

\begin{matrix} {Ext}^{•} (ℰ_{0}^{''}, ℰ_{1}) & ≅ ℂ \oplus ℂ [- 1] \oplus ℂ^{\oplus 2} [- 1], \\ {Ext}^{•} (ℰ_{0}^{''}, ℱ_{1}) & ≅ ℂ \oplus ℂ [- 1] \oplus ℂ^{\oplus 2} [- 2] \oplus ℂ^{\oplus 2} [- 3] . \end{matrix}

Proof.

Applying

{Ext}^{•} (-, ℰ_{1})

to the exact triangle

ℰ_{0}^{''} \to ℱ_{0}^{''} \to 𝒪_{E} (- 1, - 1) [2]

and using the fact that

{Ext}^{•} (𝒪_{E} (- 1, - 1) [2], ℰ_{1}) ≅ ℂ^{\oplus 2} [- 3],

we get the first statement. The second statement is proven by instead applying

{Ext}^{•} (-, ℱ_{1}),

but now using that

{Ext}^{•} (𝒪_{E} (- 1, - 1) [2], ℱ_{1}) ≅ ℂ^{\oplus 2} [- 4] .

4.5.2. Algebra structure of the categorical resolution. We now investigate how the compositions behave in the categorical resolution. Notice that for a

c ℙ^{\infty}

object

P,

we have

{Ext}^{•} (P, P) ≅ ℂ \oplus ℂ [- 1] \oplus ℂ^{\oplus 2} [- 2] \oplus ℂ^{\oplus 2} [- 3] \oplus \dots,

and that we can recover the vector spaces of morphisms between objects in our categorical resolution

\tilde{𝒟}

as truncations of this algebra:

\begin{matrix} {Ext}^{•} (ℰ_{0}^{''}, ℱ_{0}^{''}) & ≅ {Ext}^{•} (ℰ_{1}, ℱ_{1}) ≅ {Ext}^{•} (ℱ_{0}^{''}, ℰ_{1}) ≅ {Ext}^{\leq 1} (P, P) \\ ≅ ℂ \oplus ℂ [- 1], \\ {Ext}^{•} (ℰ_{0}^{''}, ℰ_{1}) & ≅ {Ext}^{•} (ℱ_{0}^{''}, ℱ_{1}) ≅ {Ext}^{\leq 2} (P, P) \\ ≅ ℂ \oplus ℂ [- 1] \oplus ℂ^{\oplus 2} [- 2], \\ {Ext}^{•} (ℰ_{0}^{''}, ℱ_{1}) & ≅ {Ext}^{\leq 3} (P, P) \\ ≅ ℂ \oplus ℂ [- 1] \oplus ℂ^{\oplus 2} [- 2] \oplus ℂ^{\oplus 2} [- 3] . \end{matrix}

Subsequently, for any two objects

A, B

in the exceptional sequence, we think of an element of

{Ext}^{i} (A, B)

as lifting some element of

{Ext}^{i} (P, P) .

Since

{Ext}^{•} (P, P)

is a polynomial ring with no relations, this suggests the following.

Proposition 4.19.

All compositions in $\tilde{𝒟}$ are non-zero.

Before proving the proposition, we illustrate some consequences. Let

t_{0}, t_{1}, t_{2}, s_{0}, s_{1}

be non-zero degree 0 maps and

ε_{0}, ε_{1}, ε_{2}, δ_{0}, δ_{1}

be non-zero degree

1

maps as described by the following quiver.

Letting

a b

denote the composition

b \circ a,

a corollary of Proposition 4.19 shows that up to rescaling, we can obtain the following relations:

\begin{matrix} t_{0} t_{1} & = s_{0}, & t_{1} t_{2} & = s_{1}, \\ t_{0} ε_{1} & = ε_{0} t_{1} = δ_{0}, & t_{1} ε_{2} & = ε_{1} t_{2} = δ_{1} . \end{matrix}

Defining

ρ_{0} := ε_{0} ε_{1}

and

ρ_{1} := ε_{1} ε_{2},

we also see that

ρ_{0} t_{2} = t_{0} ρ_{1} ρ_{0} ε_{2} = ε_{0} ρ_{1} .

We can then extend to find bases

{ρ_{0}, θ_{0}}

of

{Ext}^{2} (ℰ_{0}^{''}, ℰ_{1})

and

{ρ_{1}, θ_{1}}

of

{Ext}^{2} (ℱ_{0}^{''}, ℱ_{1}),

respectively.¹

Note that we can choose

θ_{0}, θ_{1}

such that either of the relations

t_{0} θ_{1} = θ_{0} t_{2}

or

ε_{0} θ_{1} = θ_{0} ε_{2}

hold. However, it is not clear whether we can choose

θ_{0}, θ_{1}

such that both commutativity relations simultaneously hold.

We can then visualise the category

\tilde{𝒟}

through the following quiver, where we omit compositions in non-zero degrees, the solid arrows are maps of degree

0,

the dashed arrows are maps in degree

1

and the dotted arrows are maps in degree

2 .

In particular, we highlight how our resolving category is essentially built out of the graded Kronecker quivers (see Definition 2.6) that resolve

ℙ^{\infty}

objects.

The proof of Proposition 4.19 comprises the remainder of the section through a series of lemmas.

4.5.2.1. Compositions with source $ℱ_{0}^{''} .$

Lemma 4.20.

The compositions

\begin{matrix} t_{1} t_{2} : ℱ_{0}^{''} & \to ℰ_{1} \to ℱ_{1}, \\ t_{1} ε_{2} : ℱ_{0}^{''} & \to ℰ_{1} \to ℱ_{1} [1], \end{matrix}

are non-zero.

Proof.

The first statement follows from the proof of Lemma 4.17, which shows that the degree 0 map

ℱ_{0}^{''} \to ℱ_{1}

factors as

ℱ_{0}^{''} \to ℰ_{1} \to ℱ_{1} .

The degree

1

map

ℱ_{0}^{''} \to ℱ_{1} [1]

similarly factors as

ℱ_{0}^{''} \to ℰ_{1} \to ℱ_{1} [1] .

Lemma 4.21.

The composition $ε_{1} t_{2} : ℱ_{0}^{''} \to ℱ_{1}$ is non-zero.

Proof.

We deduce that

{Ext}^{•} (ℱ_{0}^{''}, 𝒪_{E} (- 1, 0)) ≅ ℂ^{\oplus 2} [- 2],

so we can immediately see that the composition

ℱ_{0}^{''} \to ℰ_{1} [1] \to ℱ_{1} [1]

is non-zero.

Lemma 4.22.

The composition $ε_{1} ε_{2} : ℱ_{0}^{''} \to ℰ_{1} [1] \to ℱ_{1} [2]$ is non-zero.

Proof.

Consider the following diagram.

(4.19)

There is a unique degree

1

morphism (up to scaling)

𝒦 \to ℰ_{1} [1],

and one can quickly show that this factors through both

𝒦 \to ℱ_{0}^{''}

and

𝒦 \to 𝒪_{E} (- 1, 0),

by applying

{Ext}^{•} (-, ℰ_{1})

to the triangles defining

𝒦

and

ℱ_{0}^{''},

respectively. Hence, by the derived nine-lemma, we can complete along the dashed arrows to obtain exact triangles. The bottom-left square must then commute, so it follows that the map

ℰ_{1} \to ℱ_{1} [1]

is non-zero and hence we find that

𝒴^{''} ≅ 𝒴 [1] .

Then, we have an exact triangle

𝒪_{C} (- 1) \to 𝒴^{'} \to 𝒴 .

We see that

Hom (𝒴^{'}, ℱ_{1}) ≅ Hom (𝒴, ℱ_{1}) ≅ ℂ

and

{Ext}^{2} (𝒴^{'}, ℱ_{1}) ≅ {Ext}^{2} (𝒪_{C} (- 1), ℱ_{1}) ≅ ℂ,

and there are no morphisms in any other degrees. Then, applying

{Ext}^{•} (-, ℱ_{1})

to the triangle

ℰ_{1} \to 𝒴^{'} \to ℱ_{0}^{''},

we see that since

{Ext}^{1} (𝒴^{'}, ℱ_{1}) = 0

that the map

{Ext}^{1} (ℰ_{1}, ℱ_{1}) \to {Ext}^{2} (ℱ_{0}^{''}, ℱ_{1})

is an injection and hence the composition

ℱ_{0}^{''} \to ℰ_{1} [1] \to ℱ_{1} [2]

is non-zero.

4.5.2.2. Compositions with source $ℰ_{0}^{''}$ and target $ℰ_{1} .$

Lemma 4.23.

The compositions

\begin{matrix} t_{0} t_{1} : ℰ_{0}^{''} & \to ℱ_{0}^{''} \to ℰ_{1}, \\ ε_{0} t_{1} : ℰ_{0}^{''} & \to ℱ_{0}^{''} [1] \to ℰ_{1} [1], \\ t_{0} ε_{1} : ℰ_{0}^{''} & \to ℱ_{0}^{''} \to ℰ_{1} [1] \end{matrix}

are non-zero.

Proof.

Consider the exact triangle

𝒦 \to ℱ_{0}^{''} \to ℰ_{1}

and apply the

{Ext}^{•} (ℰ_{0}^{''}, -)

functor. Using the fact that

ℰ_{0}^{''}

is orthogonal to

𝒪_{C} (- 1)

and the defining exact triangle for

𝒦

𝒪_{C} (- 1) \to 𝒦 \to 𝒪_{E} (- 1, 0),

we find that

{Ext}^{•} (ℰ_{0}^{''}, 𝒦) ≅ {Ext}^{•} (ℰ_{0}^{''}, 𝒪_{E} (- 1, 0)) ≅ ℂ^{\oplus 2} [- 3] .

It follows that there is an isomorphism

Hom (ℰ_{0}^{''}, ℱ_{0}^{''}) ≅ Hom (ℰ_{0}^{''}, ℰ_{1}) ≅ ℂ,

so that the composition

ℰ_{0}^{''} \to ℱ_{0}^{''} \to ℰ_{1}

is non-zero, and similarly

{Ext}^{1} (ℰ_{0}^{''}, ℱ_{0}^{''}) ≅ {Ext}^{1} (ℰ_{0}^{''}, ℰ_{1}) ≅ ℂ,

so the composition

ℰ_{0}^{''} \to ℱ_{0}^{''} [1] \to ℰ_{1} [1]

is non-zero. We can also consider the exact triangle

ℰ_{0}^{''} \to ℱ_{0}^{''} \to 𝒪_{E} (- 1, - 1) [2]

and apply the

{Ext}^{•} (-, ℰ_{1})

functor. Using that

{Ext}^{•} (𝒪_{E} (- 1, - 1) [2], ℰ_{1}) ≅ ℂ^{\oplus 2} [- 3],

we find an isomorphism

{Ext}^{1} (ℰ_{0}^{''}, ℰ_{1}) ≅ {Ext}^{1} (ℱ_{0}^{''}, ℰ_{1}) ≅ ℂ

which implies that the composition

ℰ_{0}^{''} \to ℱ_{0}^{''} \to ℰ_{1} [1]

is non-zero.

Lemma 4.24.

The composition $ε_{0} ε_{1} : ℰ_{0}^{''} \to ℱ_{0}^{''} [1] \to ℰ_{1} [2]$ is non-zero.

Proof.

Suppose for a contradiction that the composition is zero. Applying

{Ext}^{•} (-, ℱ_{0}^{''})

to the triangle

ℰ_{0}^{''} \to ℱ_{0}^{''} \to 𝒪_{E} (- 1, - 1) [2]

gives us an exact sequence

\begin{matrix} 0 & = {Ext}^{1} (ℱ_{0}^{''}, ℱ_{0}^{''}) \to {Ext}^{1} (ℰ_{0}^{''}, ℱ_{0}^{''}) \\ \to {Ext}^{2} (𝒪_{E} (- 1, - 1) [2], ℱ_{0}^{''}) \to {Ext}^{2} (ℱ_{0}^{''}, ℱ_{0}^{''}) = 0, \end{matrix}

so that we have an isomorphism

ℂ ≅ {Ext}^{1} (ℰ_{0}^{''}, ℱ_{0}^{''}) \to {Ext}^{2} (𝒪_{E} (- 1, - 1) [2], ℱ_{0}^{''}) ≅ ℂ .

Hence, the left triangle commutes in the following diagram.

Since the right triangle commutes by assumption, it suffices to show that the composition

(4.20)

𝒪_{E} (- 1, - 1) \to ℱ_{0}^{''} \to ℰ_{1} [1]

is non-zero, since then we obtain a contradiction. For this, we consider the non-split extension

ℰ_{1} \to 𝒴^{'} \to ℱ_{0}^{''},

and we only need to show that

Hom (𝒪_{E} (- 1, - 1), 𝒴^{'}) = 0,

since then the map

Hom (𝒪_{E} (- 1, - 1), ℱ_{0}^{''}) \to {Ext}^{1} (𝒪_{E} (- 1, - 1), ℰ_{1})

is an injection, and hence the composition (4.20) is non-zero. Recall that diagram (4.19) gives us an exact triangle

𝒪_{C} (- 1) \to 𝒴^{'} \to 𝒴 .

There are no morphisms from

𝒪_{E} (- 1, - 1)

to

ℰ_{1}

and

ℱ_{1}

in degree

0 .

Hence,

Hom (𝒪_{E} (- 1, - 1), 𝒴) = 0,

and since

Hom (𝒪_{E} (- 1, - 1), 𝒪_{C} (- 1)) = 0,

we have our desired result.

4.5.2.3. Compositions with source $ℰ_{0}^{''}$ and target $ℱ_{1} .$

Lemma 4.25.

For all $i \in {0, 1, 2},$ precomposing with $t_{0} : ℰ_{0}^{''} \to ℱ_{0}^{''}$ induces isomorphisms

{Ext}^{i} (ℰ_{0}^{''}, ℱ_{1}) ≅ {Ext}^{i} (ℱ_{0}^{''}, ℱ_{1})

and postcomposing with $t_{2} : ℰ_{1} \to ℱ_{1}$ induces isomorphisms

{Ext}^{i} (ℰ_{0}^{''}, ℱ_{1}) ≅ {Ext}^{i} (ℰ_{0}^{''}, ℰ_{1}) .

Proof.

Firstly, applying

{Ext}^{•} (-, ℱ_{1})

to the triangle

ℰ_{0}^{''} \to ℱ_{0}^{''} \to 𝒪_{E} (- 1, - 1) [2]

induces isomorphisms

{Ext}^{i} (ℱ_{0}^{''}, ℱ_{1}) ≅ {Ext}^{i} (ℰ_{0}^{''}, ℱ_{1})

by precomposing with

ℰ_{0}^{''} \to ℱ_{0}^{''}

for

i \in {0, 1, 2} .

Similarly, using that

{Ext}^{•} (ℰ_{0}^{''}, 𝒪_{E} (- 1, 0)) ≅ ℂ^{\oplus 2} [- 3]

and then applying

{Ext}^{•} (ℰ_{0}^{''}, -)

to the triangle (4.8)

ℰ_{1} \to ℱ_{1} \to 𝒪_{E} (- 1, 0)

gives us isomorphisms

{Ext}^{i} (ℰ_{0}^{''}, ℰ_{1}) ≅ {Ext}^{i} (ℰ_{0}^{''}, ℱ_{1})

for

i \in {0, 1, 2}

by composing with

ℰ_{1} \to ℱ_{1} .

Lemma 4.26.

Precomposing with $ε_{0} : ℰ_{0}^{''} \to ℱ_{0}^{''} [1]$ induces an isomorphism

{Ext}^{3} (ℰ_{0}^{''}, ℱ_{1}) ≅ {Ext}^{2} (ℱ_{0}^{''}, ℱ_{1})

and postcomposing with $ε_{2} : ℰ_{1} \to ℱ_{1} [1]$ induces an isomorphism

{Ext}^{3} (ℰ_{0}^{''}, ℱ_{1}) ≅ {Ext}^{2} (ℰ_{0}^{''}, ℰ_{1}) .

Proof.

By applying

{Ext}^{•} (ℱ_{0}^{''}, -)

to

(4.21)

ℱ_{1} \to 𝒴 \to ℰ_{1}

and using the results above, we find

{Ext}^{•} (ℱ_{0}^{''}, 𝒴) ≅ ℂ \oplus ℂ [- 2] .

Then, we apply the functor

{Ext}^{•} (𝒪_{E} (- 1, - 1) [2], -)

to the same triangle and see that

{Ext}^{•} (𝒪_{E} (- 1, - 1) [2], 𝒴)

can only be non-zero in degrees

0 \leq i \leq 3 .

Hence, applying

{Ext}^{•} (-, 𝒴)

to the triangle

ℰ_{0}^{''} \to ℱ_{0}^{''} \to 𝒪_{E} (- 1, - 1) [2],

we find that

{Ext}^{3} (ℰ_{0}^{''}, 𝒴) = 0 .

By applying

{Ext}^{•} (ℰ_{0}^{''}, -)

to (4.21), we obtain an isomorphism

ℂ^{\oplus 2} ≅ {Ext}^{2} (ℰ_{0}^{''}, ℰ_{1}) \tilde{\to} {Ext}^{3} (ℰ_{0}^{''}, ℱ_{1}) ≅ ℂ^{\oplus 2} .

This shows the first statement.

We use a similar argument for the second statement. Consider the non-split extension

ℱ_{0}^{''} \to 𝒵_{0} \to ℰ_{0}^{''}

and apply

{Ext}^{•} (-, ℱ_{1}) .

We see that if

{Ext}^{3} (𝒵_{0}, ℱ_{1}) = 0,

then there is an isomorphism

ℂ^{\oplus 2} ≅ {Ext}^{2} (ℱ_{0}^{''}, ℱ_{1}) \to {Ext}^{3} (ℰ_{0}^{''}, ℱ_{1}) ≅ ℂ^{\oplus 2} .

We use the following diagram where the top-left square commutes, and so we complete the dashed arrows to exact triangles using the derived nine-lemma.

One quickly sees that

{Ext}^{•} (𝒪_{E} (- 1, - 1) [2], ℱ_{1})

and

{Ext}^{•} (𝒦 [2], ℱ_{1})

are only non-zero in degree

3,

and hence

{Ext}^{i} (𝒵_{1}, ℱ_{1}) = 0

for

i \geq 4 .

From the previous calculations, we have that

{Ext}^{•} (𝒴^{'}, ℱ_{1}) ≅ ℂ \oplus ℂ [- 2] .

Hence, we find that

{Ext}^{3} (𝒵_{0}, ℱ_{1}) = 0,

by using the triangle

𝒵_{0} \to 𝒴^{'} \to 𝒵_{1} .

5. Deformation absorption through $c ℙ^{\infty}$ objects

In this section, we return to the singular curve

ℭ

defined in equation (3.1) and briefly discuss an interesting example of how

c ℙ^{\infty}

objects behave when smoothing the curve. Our goal is to investigate whether these objects provide deformation absorptions, as described in Section 2.3.2. For this section, we recall the semiorthogonal decomposition (3.2)

𝐃^{b} (ℭ) = ⟨ 𝒫 (- 1), 𝐃^{b} (ℙ_{s : t}^{1}) ⟩ .

We will denote

P := 𝒫 (- 1)

for this section. For our purposes, we will define a smoothing

𝔛

of

ℭ

to be a flat family of curves

φ : 𝔛 \to (B, o),

where

(B, o)

is a smooth pointed curve, the central fibre is

φ^{- 1} (o) ≅ ℭ

and all other fibres are smooth. Note that we do not require the total space

𝔛

to be smooth.

Consider the surface

𝔛 = {Bl}_{⟨ x, s^{2} ⟩} ℙ_{x : y}^{1} \times ℙ_{s : t}^{1}

with the natural map

𝔛 \to ℙ_{x : y}^{1} .

We can embed into a projective bundle

𝔛 ≅ {x v = s^{2} u} \subset ℙ_{ℙ^{1} \times ℙ^{1}} (𝒪 {(0, 2)}_{u} \oplus 𝒪 {(1, 0)}_{v}),

so that the space

𝔛

is nodal, and the fibre over

{x = 0}

is given by

{s^{2} u = 0} \subset ℙ_{ℙ_{s : t}^{1}} (𝒪 {(2)}_{u} \oplus 𝒪_{v})

which is isomorphic to the curve

ℭ .

Note that the generic fibre is a copy of

ℙ_{s : t}^{1},

so that

𝔛

is a smoothing of

ℭ .

Proposition 5.1.

The object $P \in 𝐃^{b} (ℭ)$ provides a deformation absorption with respect to the smoothing $𝔛 .$

Proof.

The blow-up formula shows that there is a semiorthogonal decomposition

𝐃^{b} (𝔛) = ⟨ 𝐃^{b} (ℂ [s] / s^{2}), 𝐃^{b} (ℙ^{1} \times ℙ^{1}) ⟩ .

If

i : ℭ \overset{}{↪} 𝔛

is the inclusion, then

i_{*} P

is precisely the

ℙ^{\infty, 1}

object generating the admissible subcategory

𝐃^{b} (ℂ [s] / s^{2})

of

𝐃^{b} (𝔛) .

It is not clear whether the

c ℙ^{\infty}

object provides a universal deformation absorption, that is, whether the pushforward of

P

generates an admissible category for all smoothings of

ℭ .

We note that in our example, the pushforward of the

c ℙ^{\infty}

object

P

gives a

ℙ^{\infty, 1}

object on

𝔛,

which absorbs the singularities of the total space of the smoothing. One observes that the total space of a smoothing of

ℭ

cannot itself be smooth, for purely homological reasons. Indeed, consider the object

i_{*} P .

The triviality of the normal bundle of

ℭ

in

𝔛

gives us the distinguished triangle using equation (2.3)

i^{*} i_{*} P \to P \to P [2] .

In the case that

P \to P [2]

is the zero map, it is clear that

i_{*} P

is not in

𝐃^{perf} (𝔛)

since

i^{*} i_{*} P

is not in

𝐃^{perf} (ℭ) .

Otherwise, applying

{Ext}^{•} (-, P)

to the triangle shows that

{Ext}^{j} (i_{*} P, i_{*} P) ≅ {Ext}^{j} (i^{*} i_{*} P, P) ≅ ℂ

for all

j \geq 0,

which implies

i_{*} P

is not perfect. Hence, for another smoothing, we may similarly require that

i_{*} P

absorbs the singularities of the total space.

Acknowledgements

The author would like to thank his PhD supervisor Ed Segal for invaluable guidance throughout the project, and the anonymous reviewer for helpful comments which improved the paper.

Footnotes

¹
Note that we can choose $θ_{0}, θ_{1}$ such that either of the relations $t_{0} θ_{1} = θ_{0} t_{2}$ or $ε_{0} θ_{1} = θ_{0} ε_{2}$ hold. However, it is not clear whether we can choose $θ_{0}, θ_{1}$ such that both commutativity relations simultaneously hold.

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Cite this article

Aporva Varshney, Categorical absorption of a non-isolated singularity. J. Noncommut. Geom. 20 (2026), no. 4, pp. 1419–1464

DOI 10.4171/JNCG/684

Abstract

1. Introduction

1.1. Results and structure

Theorem 1.1

Proposition 1.2

Theorem 1.3

Theorem 1.4

Proposition 1.5

Proposition 1.6

2. Preliminaries

2.1. Derived categories

Lemma 2.1.

Lemma 2.2.

Conjecture 2.3

2.2. Projective bundles and ruled surfaces

2.3. Categorical absorption for nodal varieties

Definition 2.4.

Definition 2.5.

Definition 2.6

Theorem 2.7

Theorem 2.8

3. Compound ℙ∞ objects

Lemma 3.1.

Proof.

Definition 3.2.

Theorem 3.3.

Proof.

Remark 3.4.

Remark 3.5.

Remark 3.6.

4. Semiorthogonal decomposition for a resolution of singularities

Remark 4.1.

4.1. Constructing a geometric resolution

Lemma 4.2.

Proof.

Proposition 4.3.

Proof.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

4.2. A full exceptional sequence on the resolution

Lemma 4.6.

Proof.

Lemma 4.7.

Proof.

Theorem 4.8.

4.3. Morphisms in the exceptional sequence

Lemma 4.9.

Proof.

4.4. Mutating the semiorthogonal decomposition

Lemma 4.10.

Proof.

Lemma 4.11.

Proof.

Lemma 4.12.

Proof.

Lemma 4.13.

Proof.

Lemma 4.14.

Proof.

Lemma 4.15.

Proof.

Proof of Theorem 4.8.

4.5. Structure of the categorical resolution

Theorem 4.16.

Lemma 4.17.

Proof.

Lemma 4.18.

Proof.

Proposition 4.19.

Lemma 4.20.

Proof.

Lemma 4.21.

Proof.

Lemma 4.22.

Proof.

Lemma 4.23.

Proof.

Lemma 4.24.

3. Compound $ℙ^{\infty}$ objects

5. Deformation absorption through $c ℙ^{\infty}$ objects