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.
Mathematics Subject Classification 202014F08 (primary)Keywordsabsorption of singularitiesderived categorysemiorthogonal decompositionIntroduction
Studying the bounded derived category of coherent sheaves 𝐃b(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 𝐃b(X) is by considering how it can be broken down into smaller pieces: a semiorthogonal decomposition
𝐃b(X)=⟨𝒜1,…,𝒜n⟩
is an ordered collection of full triangulated subcategories 𝒜1,…,𝒜n generating 𝐃b(X) such that ‘there are no morphisms from right to left’. Precisely, for B∈𝒜j and A∈𝒜i with j>i i\)]]>, Hom(B,A)=0.
Recently, there has been interest in studying 𝐃b(X) when X is a singular variety. One approach we may take is to consider the full triangulated subcategory of perfect complexes, 𝐃perf(X), that is, the complexes quasi-isomorphic to a bounded complex of locally free sheaves. When X is smooth, 𝐃perf(X) coincides with 𝐃b(X), since in this setting all coherent sheaves have a finite resolution by locally free sheaves, and so we can think of 𝐃perf(X) as capturing smoothness. It then makes sense to define the singularity category [19]
𝐃sg(X):=𝐃b(X)/𝐃perf(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
𝐃b(X)=⟨𝒜,ℬ⟩,
where ℬ⊂𝐃perf(X), so that 𝒜 is, in some sense, responsible for the singularity. Following [15], 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
𝐃b(X)=⟨𝐃sg(X),𝐃perf(X)⟩,
but we can try to approximate it by asking that ℬ⊂𝐃perf(X) is as large as possible. Examples studied in the literature also have the property that 𝒜 is equivalent to 𝐃b(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], 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
where degzi=0. For surfaces with type An singularities, it is shown that the singularity is often absorbed by the algebra ℂ[w]/wn+1 with degw=0, with a simple example being the blow-up of a smooth surface S in a non-reduced point Specℂ[w]/wn+1⊂S. A semiorthogonal decomposition is produced for the threefold ℙ(1,1,1,3) in [9]:
𝐃b(ℙ(1,1,1,3))=⟨𝐃b(R),𝐃b(ℂ),𝐃b(ℂ)⟩,
where this time R is an algebra with dimR=45.
The most relevant case for us will be categorical absorption of nodal varieties. Nodal threefolds are considered in various incarnations in [10, 21, 22]. We follow the approach detailed in [15]. To illustrate this theory, let X be a nodal genus 0 curve consisting of curves E≅Y≅ℙ1 intersecting transversely at a point. The contraction of the branch E through the map f:X→Y yields the semiorthogonal decomposition
𝐃b(X)=⟨ker𝐑f∗,𝐋f∗𝐃b(ℙ1)⟩,
where we have that 𝐋f∗ is fully faithful. Absorption is then given by the category ker𝐑f∗=⟨𝒪E(−1)⟩, and denoting P:=𝒪E(−1), one finds that
Ext•(P,P)≅ℂ[θ]ker𝐑f∗≃⟨P⟩≃𝐃b(ℂ[r]/r2),
where degθ=2 and degr=−1. We say that P is a ℙ∞,2 object, since Ext•(P,P) resembles the cohomology ring of ℂℙ∞. It was shown in [15] that ℙ∞,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 𝐃b(ℂ[w]/w2)≃⟨P⟩ with degw=0, and Ext•(P,P)=ℂ[ε] with degε=1. In this case, P is called a ℙ∞,1 object.
Perhaps the most interesting part of this story is a property of ℙ∞,2 objects known as deformation absorption: given a ℙ∞,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∞,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 ℙ1. This powerful property was used in [16] 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.
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 A2 singularity, constructed as follows. We first consider the curve
ℭ:={x=s2u=0}⊂ℙx:y1×ℙs:t1×ℙu:v1
and then perform a blow-up
X:=Blℭ(ℙ1)3.
Then, X is locally given by the equation {xb=s2u}⊂𝔸x,s,u,b4, 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 degε=1 and degθ=2.
We call an object with this self-Ext algebra a compound ℙ∞, or cℙ∞, object, to reflect the idea that it is simultaneously a ℙ∞,1 and ℙ∞,2 object.
The work in [16] 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 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 X~ is the following proposition.
Proposition 1.2
(= Proposition 4.3). The space X~ can be obtained from (ℙ1)3 as a series of blow-ups along smooth rational curves.
Using this, we obtain a full exceptional collection on X~. However, this collection is not compatible with the contraction σ:X~→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 𝒟~⊂𝐃b(X~) that serves as a ‘categorical resolution’ for our absorbing category.
Theorem 1.3
(= Theorem 4.8). Let σ:X~→X be the resolution of singularities map. There is an admissible subcategory 𝒟~⊂𝐃b(X~) such that the absorbing category 𝒟⊂𝐃b(X) is equivalent to the Verdier quotient
𝒟≃𝒟~/(ker𝐑σ∗∩𝒟~).
The category 𝒟~ is generated by an exceptional collection of length 4, and under the pushforward 𝐑σ∗ all the generating objects are mapped to the cℙ∞ object P. We investigate the algebra structure of this collection.
Theorem 1.4
(= Theorem 4.16). The morphisms in the exceptional sequence
𝒟~=⟨ℰ0′′,ℱ0′′,ℰ1,ℱ1⟩
are described as follows:
Ext•(ℰ0′′,ℱ0′′)≅Ext•(ℰ1,ℱ1)≅ℂ⊕ℂ[−1],
Ext•(ℱ0′′,ℰ1)≅ℂ⊕ℂ[−1],
Ext•(ℱ0′′,ℱ1)≅Ext•(ℰ0′′,ℰ1)≅ℂ⊕ℂ[−1]⊕ℂ⊕2[−2],
Ext•(ℰ0′′,ℱ1)≅ℂ⊕ℂ[−1]⊕ℂ⊕2[−2]⊕ℂ⊕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 cℙ∞ object:
Finally, we briefly consider the properties of our cℙ∞ 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 ℙ1×ℙ1 in a fattened point Specℂ[s]/s2. Then, the pushforward of the cℙ∞ object P∈𝐃b(ℭ) into 𝔛 generates an admissible subcategory of 𝐃b(𝔛), and hence provides a deformation absorption with respect to the smoothing 𝔛.
Notation and conventions.
We work over the base field ℂ.
For a dg-algebra A with finite-dimensional cohomology, we denote by 𝐃(A) the unbounded derived category of dg-modules over A, and 𝐃b(A)⊂𝐃(A) is the subcategory of dg-modules with finite-dimensional total cohomology.
Tensor products and pushforward and pullback functors between derived categories are implicitly derived.
Given a scheme X with closed subscheme Z⊂X and a birational map Y→X, we denote by Z¯ the strict transform of Z in Y.
PreliminariesDerived 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=⟨𝒜1,…,𝒜n⟩,
it is possible to reorder the components to obtain a new semiorthogonal decomposition. The first method is to perform a left mutation of 𝒜i+1 over 𝒜i
𝒯=⟨𝒜1,…,𝒜i−1,𝕃𝒜i𝒜i+1,𝒜i,𝒜i+2,…,𝒜n⟩,
and the second is to perform a right mutation of 𝒜i over 𝒜i+1
𝒯=⟨𝒜1,…,𝒜i−1,𝒜i+1,ℝ𝒜i+1𝒜i,𝒜i+2,…,𝒜n⟩.
In each case, the mutation is an equivalence of categories
𝕃𝒜i𝒜i+1≃𝒜i+1,ℝ𝒜i+1𝒜i≃𝒜i.
We omit the construction, but note that when 𝒜i and 𝒜i+1 are generated by exceptional objects ℰi and ℰi+1, respectively, we have formulae for the left and right mutations through the following exact triangles [4]:
We also note an easy consequence of Serre duality.
Lemma 2.1.
If X is Gorenstein with a semiorthogonal decomposition
𝐃b(X)=⟨𝒜,ℬ⟩,
we have that
𝐃b(X)=⟨ℬ⊗ωX,𝒜⟩
is also a semiorthogonal decomposition, where ωX 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, 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⊂X is a locally complete intersection of codimension c≥2. Consider the blow-up BlZX. Then, there is a semiorthogonal decomposition [18]
𝐃b(BlZX)=⟨Φc−1(𝐃b(Z)),…,Φ1(𝐃b(Z)),𝐃b(X)⟩,
where 𝐃b(X) is embedded fully faithfully by pulling back, and the functors Φi are fully faithful and constructed as follows. The exceptional divisor is a projective bundle over π:E→Z, so letting j:E→BlZX be the inclusion, we define
Φi(ℱ)=j∗(π∗ℱ⊗𝒪(−i)).
2.1.4. The cone on the pull-push counit. Let i:E→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]:
i∗i∗ℱ→ℱ→ℱ⊗𝒩E/X∨[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:X~→X.
Conjecture 2.3
([6, Conjecture 1.9], [3, Section 5]). If π:X~→X is a resolution of rational singularities, then π∗ induces an equivalence between 𝐃b(X) and a Verdier quotient of 𝐃b(X~):
π~∗:𝐃b(X~)/kerπ∗→𝐃b(X).
In fact, the conjecture is true in several settings (see [6, 20]). The most relevant situation for us will be when the resolution has one-dimensional fibres, in which case the conjecture was proven in [1]. An immediate consequence of this is that we may descend an appropriate semiorthogonal decomposition of 𝐃b(X~) to one for 𝐃b(X) [15, Proposition 4.1]. If
𝐃b(X~)=⟨𝒜~,ℬ~⟩
is a semiorthogonal decomposition such that kerπ∗⊂𝒜~, then we obtain a semiorthogonal decomposition
𝐃b(X)=⟨𝒜,ℬ⟩
with
𝒜≃𝒜~/(kerπ∗∩𝒜~)≃𝒜~/kerπ∗,ℬ≃ℬ~/(kerπ∗∩ℬ~)≃ℬ~,
where 𝒜,ℬ must generate 𝐃b(X) because π∗ is essentially surjective.
2.1.6. Spherical objects. An object K∈𝐃b(X), where X is a complex smooth and projective variety, is called spherical if [7]
K⊗ωX≅K and
Hom(K,K[i])≅Hi(SdimX,ℂ),
where SdimX is the sphere of dimension equal to dimX. A spherical object induces an autoequivalence of the derived category by twisting around the object. For any object ℰ, the twist TK(ℰ) is defined through the exact triangle
(⨁iHom(K,ℰ[i])⊗K[−i])→ℰ→TK(ℰ).
Being an autoequivalence, it has an inverse known as the dual twist, which is similarly described by an exact triangle.
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⊕ℱ. Then, there are two possible conventions for the corresponding projective bundle; in our case, we take the definition to be
π:ℙ(ℰ):=Proj¯(Sym(ℰ))→B,
so that there is a map π∗ℰ→𝒪(1). We will write ℙ(ℰ)=ℙ(ℒx0⊕ℱx1) to denote that the coordinates on the fibres are [x0:x1]. There is a section S corresponding to the surjection ℰ→ℒ; equivalently, it is given by the vanishing of the coordinate x1. The skyscraper sheaf of the section is defined by taking the cokernel of the composition π∗ℱ→π∗ℰ→𝒪(1), yielding
π∗ℱ→𝒪(1)→𝒪S(1)→0
and twisting by 𝒪(−1)
π∗ℱ⊗𝒪(−1)→𝒪→𝒪S→0.
Then, the above sequence is the Koszul resolution of 𝒪S, so it follows from the fact that 𝒪(1)|S≅ℒ that the normal bundle of S is given by ℱ∨⊗ℒ. Note that the first map is ‘multiplication by x1’, and hence x1 is a section of the line bundle π∗ℱ∨⊗𝒪(1).
We collect some results on ruled surfaces, which will arise frequently when we blow up a curve C≅ℙ1 in a smooth threefold X. In this situation, we recall that the exceptional divisor is isomorphic to ℙC(𝒩C/X∨) and has normal bundle 𝒪(−1). Consider
π:Σ=ℙℙ1(𝒪x⊕𝒪(a)y)→ℙ1.
Then, the divisor corresponding to 𝒪(1) is given by the class of the section H:=V(x), and we denote
𝒪(a,b):=𝒪(1)⊗a⊗π∗𝒪(b),
or, equivalently, it is the line bundle corresponding to the divisor aH+bF, where F is a fibre. Then, the canonical bundle is given by the formula
ωΣ≅𝒪(−2,−2+a).
If ℰ=𝒪⊕𝒪(a), then pushing forward the line bundle 𝒪(d) to the base ℙ1 gives us
This section gives an overview of the theory of categorical absorption for nodal varieties, with [15] 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∈𝐃b(X) for a variety X is called a ℙ∞,q object if
Ext•(P,P)≅ℂ[θ]degθ=q.
Definition 2.5.
The categorical ordinary double point of degree p is 𝐃b(ℂ[w]/w2) with degw=p.
The theory is split up by parity of dimX: 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(su)⊂ℙs:t1×ℙu:v1.
In this case, the pushforward of a line bundle 𝒪(a) on either of the two branches gives a ℙ∞,2 object. We have seen that we can consider contracting one of the branches f:X→ℙs:t1; it is then a direct consequence of the fact that f∗𝒪X≅𝒪ℙ1 that the pullback f∗ is fully faithful. One can show that the null category kerf∗ is admissible in this case, and that there is a semiorthogonal decomposition [15, Proposition 6.15]
𝐃b(X)=⟨𝐃b(ℂ[w]/w2),𝐃b(ℙ1)⟩
with degw=−1. Here, the absorbing category 𝐃b(ℂ[w]/w2) 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 Specℂ[w]/w2. In this case, the object ℂ≅𝒪/w is a ℙ∞,1 object which generates 𝐃b(ℂ[w]/w2), where degw=0. More interestingly, we may consider the nodal surface X:=ℙ(1,1,2) [8, Example 3.17]. To construct a categorical absorption here, we consider the resolution of singularities given by
σ:𝔽2→ℙ(1,1,2),
where we denote by π:𝔽2≅ℙ(𝒪⊕𝒪(−2))→ℙ1 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 𝒪E(−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
which implies that kerσ∗⊂⟨𝒪(−E−F),𝒪(−F)⟩. An important feature here is that the sheaf 𝒪E(−1) is a 2-spherical object, and 𝒪(−E−F) is the dual spherical twist of 𝒪(−F) around 𝒪E(−1). We note that the pushforwards σ∗𝒪 and σ∗𝒪(−E−2F) remain exceptional in 𝐃b(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⊕ℂ[−1].
Since the cone on the morphism 𝒪(−E−F)→𝒪(−F) is in kerσ∗, we find that
σ∗𝒪(−E−F)≅σ∗𝒪(−F).
The idea is that this object should be a ℙ∞,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∞,1 object.
These observations are formalised in the following theorems.
Definition 2.6
([15, 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 𝒦rq, is the derived category of this algebra.
Theorem 2.7
([15, Corollary 3.3]). Let ⟨ℰ,ℰ′⟩⊂𝐃b(X~) be an exceptional sequence, where X~ is a variety such that Ext•(ℰ,ℰ′)≅ℂ⊕ℂ[−q]. Then, ⟨ℰ,ℰ′⟩≃𝒦rq.
Theorem 2.8
([15, Proposition 3.7]). Let ⟨ℰ,ℰ′⟩≃𝒦rq, and define the object K using the distinguished triangle
K→ℰ→ℰ′.
Then, K is a 1+q-spherical object, and there is an equivalence
𝒦rq/⟨K⟩≃𝐃b(ℂ[w]/w2)degw=1−q,
and the image of ℰ under σ∗:𝒦rq→𝒦rq/⟨K⟩ is a ℙ∞,q object.
In the more general setting, the theorem can be applied as follows. Suppose we have a crepant resolution σ:X~→X for a nodal surface or threefold X, and assume that we have H>0(X~,𝒪)=0 0}(\tilde{X},\mathcal{O})=0\)]]>, so that line bundles are exceptional objects. Then, labeling the exceptional curve as C, we have a d-spherical object K:=𝒪C(−1), where d=dimX. One can show that K generates kerσ∗. Now suppose ℰ is an exceptional object such that dimExt•(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]→ℰ→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 𝐃b(X~) which contains kerσ∗=⟨K⟩. It is then not hard to show, simply by taking long exact sequences for Ext, that ⟨ℰ,ℰ′⟩≃𝒦rq for q=d−1, and hence taking the Verdier quotient gives us a ℙ∞,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→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∞,2 object P as follows:
𝐃b(X)=⟨P,P⟂⟩.
Then, for any smoothing i:X↪𝔛, the object i∗P can be shown to be exceptional [15, Theorem 1.8]. In particular, we have a semiorthogonal decomposition
𝐃b(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→B [13]. Performing this base change to a non-central fibre 𝔛b, we find that the i∗P component vanishes since it is only supported on the central fibre, and so
𝐃b(𝔛b)≃𝒟b.
In effect, we have a family of smooth and proper categories 𝒟b, and for b=o, we acquire a ℙ∞,2 object to account for the singular behaviour. We say that ℙ∞,2 objects provide a deformation absorption, and furthermore, since this occurs for every smoothing, we say that the deformation absorption is universal.
We begin by generalising the example of a nodal curve by trivially thickening one of the branches. Consider
ℭ={s2u=0}⊂ℙs:t1×ℙu:v1
which is a copy of ℭ0:=V(u)≅ℙs:t1 intersecting the thickened branch
𝔜:=V(s2)≅ℙu:v1×ℂ[s]/s2
at a fat point. There is a contraction f:ℭ→ℭ0, and it follows once again from the fact that f∗𝒪ℭ=𝒪ℭ0 that there is a semiorthogonal decomposition
𝐃b(ℭ)=⟨kerf∗,f∗𝐃b(ℭ0)⟩
with f∗ fully faithful. Let 𝔜red=V(s) be the reduction of the fattened branch 𝔜. Denote by 𝒫(a) the pushforward of the line bundle 𝒪(a) on 𝔜red to ℭ.
Lemma 3.1.
The kernel of f∗ is generated by 𝒫(−1).
Proof.
Firstly, consider the projections q:𝔜→ℂ[s]/s2 and p:𝔜→ℙ1. Then, we have that
kerq∗=⟨q∗𝐤⊗p∗𝒪(−1)⟩,
where we denote by 𝐤=(ℂ[s]/s2)/s the generator of 𝐃b(ℂ[s]/s2). 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 𝐃b(𝔜). Firstly, let ℱ be a sheaf on ℭ supported set-theoretically on 𝔜red. Then, we can find a short exact sequence
0→ℱ1→ℱ→ℱ2→0,
where suppℱ1 is contained in {u=sr=0} for some r≥1 and suppℱ2⊂𝔜. Now assume that suppℱ is a thickening of 𝔜, so that ℱ≇ℱ2 and hence ℱ1≠0. Since suppℱ1⊂ℭ0, we find that σ∗ℱ1≠0. Now, left exactness of σ∗ implies that σ∗ℱ≠0. Hence, any sheaf ℱ∈kerσ∗ has scheme-theoretic support contained in 𝔜.
Finally, if ℱ•∈𝐃b(ℭ) is a complex, we use a standard spectral sequence argument (see, e.g., [5, Lemma 3.1]) using
E2p,q=Rpσ∗(Hq(ℱ•))⟹Rp+qσ∗ℱ•.
This degenerates on the second page, so that if suppℱ• is a thickening of 𝔜, then the non-vanishing of σ∗Hq(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 ℙ1×ℙ1 with degree a along ℙs:t1 and degree b along ℙu:v1. Then, the object 𝒫(−1) has an infinite locally free resolution
so that Exti(𝒫(−1),𝒫(−1)) can be computed using the chain complex
0→𝒪→0𝒪→0𝒪(1)→0𝒪(1)→0𝒪(2)→0𝒪(2)→0⋯,
where all maps are zero, since the coordinate s becomes 0 when restricted to 𝔜red. Hence, ℰxti(𝒫(−1),𝒫(−1)) is the pushforward of a line bundle on 𝔜red, 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 Extn(𝒫(−1),𝒫(−1)) has dimension n/2+1, with basis given by {uivj|i+j=n/2}. If n is odd, then Extn(𝒫(−1),𝒫(−1)) has dimension (n+1)/2 with basis given by {uivj|i+j=(n−1)/2}. Concretely, the dimensions of Extn(𝒫(−1),𝒫(−1)) form a sequence: (1,1,2,2,3,3,…). 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∈𝐃b(X) for a variety X is called a compound ℙ∞ object, or cℙ∞, if there is an isomorphism of graded algebras
Ext•(P,P)≅ℂ[ε,θ],
where degε=1 and degθ=2.
Theorem 3.3.
The object P=𝒫(−1) is a cℙ∞ object on ℭ.
Proof.
Consider the locally free resolution for P. The element of Ext1(P,P)≅ℂ arises from the mapping
𝟙:𝒪(−1,−1)→P,
and we lift this to a morphism ε:P→P[1] by defining a chain map on the locally free resolution, so that all triangles and quadrilaterals in the diagram below commute.
Next, Ext2(P,P)≅ℂ⊕2 has a basis given by the elements {u,v}. We lift the element given by v to a chain map θ:P→P[2] similarly.
One now sees that ε∘θ=θ∘ε since both compositions are given by the chain map lifting the element v∈Ext3(P,P).
In general, we claim that the lift of the element uivj∈Extn(P,P) is given by the chain map ε2iθj if n is even, and ε2i+1θj if n is odd. This is verified by observing that in the n even case, the chain map lifting ε2iθj is given by
while in the odd case, the chain map lifting ε2i+1θj 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 ℙ1×ℂ[s]/sn intersecting a smooth curve transversely. For n>2 2\)]]>, with P defined analogously, the algebra RHom(P,P) will not be formal.
Note that ωℭ≅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.
𝐃b(ℭ)=⟨𝐃b(ℭ0),𝒫(−1)⊗ωℭ−1⟩=⟨𝐃b(ℭ0),𝒫⟩.
Note here that 𝒫 is the skyscraper sheaf along 𝔜red, and we will define
𝒟:=⟨𝒫⟩⊂𝐃b(X)
to be the absorbing subcategory of 𝐃b(X).
Remark 3.6.
Given a cℙ∞ object P, we expect that there is an equivalence given by Koszul duality
This would follow by computing Ext•(ℂ,ℂ) over the algebra ℂ[w,r]/(w2,r2). 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, Theorem 8.5 (b)].
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
UB≅𝔸x,s2×ℙu:v1
such that ℭ∩UB is just the curve ℭ missing a point at infinity on the reduced branch. For example, we may take
B=ℙx:y1×ℙs:t1×ℙu:v1
which contains ℭ as the vanishing locus V(x,s2u) and UB as the locus {y≠0,t≠0}. Let X be the threefold constructed through the blow-up
X:=BlℭB.
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
𝐃b(X)=⟨𝐃b(ℭ),𝐃b(B)⟩=⟨𝐃b(ℙs:t1),𝒟,𝐃b(B)⟩,
where 𝒟 is the absorbing category as in (3.3), generated by the cℙ∞ 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→ℭ0≅ℙs:t1. Note ℒ(a) has degree a along the ℙs:t1 branch and is trivial along the thickened branch.
Next, 𝐃b(ℭ) is embedded into X by the functor Φ:𝐃b(ℭ)→𝐃b(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 ℙ1-bundles:
F→ℙs:t1,
G→ℙu:v1×ℂ[s]/s2.
We can also pullback along ℙ1↪ℙ1×ℂ[s]/s2 to obtain
Gred→ℙu:v1.
Then, we can write our semiorthogonal decomposition as
𝐃b(X)=⟨Φ(ℒ(1)),Φ(ℒ(2)),Φ(𝒫),𝐃b(B)⟩,
where the sheaves Φ(ℒ(a)) are supported on F∪G and Φ(𝒫) is supported on Gred. Note that we have chosen 𝐃b(ℙs:t1)=⟨𝒪(1),𝒪(2)⟩.
Locally over UB, X takes the form
UX:={xb=s2ua}⊂ℙUb(𝒪(1)a⊕𝒪b)
and hence has a compound A2 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 ℙ1×ℂ[s]/sn intersecting a ℙ1 transversely, then we obtain a threefold with a non-isolated compound An singularity locally given by the equation
{xb=snu}⊂𝔸4,
so that we have a line of surface An−1 singularities, compounded with a threefold node at the origin.
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 X^ and the map
f^X:X^→X.
Since F intersects the singular locus only at the threefold node, we find that X^ has a ℙ1-family of surface node singularities. To resolve these, we blow up along the strict transform of Gred defined in (4.3) to obtain a smooth threefold X~ and the map
φX^:X~→X^.
Define
σ:X~→X
to be the composition X~→X^→X.
It may be helpful to refer to the surface analogue: we may consider the blow-up
Bl⟨x2,y⟩𝔸x,y2≅{x2a=yb}⊂𝔸2×ℙa:b1,
so that the exceptional divisor V(x2,y) is a thickened ℙ1, 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 X~→X^ is a family version of this process.
Constructing the resolution. Here, F¯,G¯red denote the strict transforms in X^ and F~,G~red denote the strict transforms in X~.
4.1.1. Local analysis. We illustrate this construction locally by considering the open subset UB defined in (4.1). Recall that we obtain an open subset of X
UX:={xb=s2ua}⊂ℙUb(𝒪(1)a⊕𝒪b)=:PX.
Locally, F∩UX≅V(x,u) and G∩UX≅V(s2,x). To obtain an open subset of X^, we blow up at the ideal ⟨x,u⟩ to obtain
UX^:={bd=s2acxc=ud}⊂ℙPX(𝒪c⊕πB∗𝒪(1)d)=:PX^,
where we denote πB:PX→UB. We can now resolve the singularities by blowing up at ⟨d,s⟩:
where this time we are denoting πB:PX^→UB and 𝒪(1) is the tautological line bundle on PX^. As a GIT quotient, PX~ is constructed as the quotient ℂ10/(ℂ∗)4 with weights shown in Table 1.
Matrix of weights for PX~, with omitted values equal to 0.
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
Note that the exceptional locus of UX^→UX is a curve V(x,u,s,b)≅ℙc:d1. We denote by
C=V(x,u,s,b,e)≅ℙc:d1⊂UX~
the strict transform of this curve in X~. The strict transform G¯red of Gred in 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⊂UX~. The exceptional locus of the composition σ:X~→X is the union E∪C, and it is precisely the preimage of the reduced singular locus V(x,b,s)⊂X. Note that the two subvarieties E and C intersect transversely at a point.
Lemma 4.2.
Let G~red be the strict transform of Gred in X~. Then, we have
G~red≅ℙℙu:v1(𝒪a⊕𝒪(1)b),E≅ℙℙu:v1(𝒪e⊕𝒪(1)f).
Proof.
In X^, the locus G¯red is given by G¯red=V(x,d,s)⊂UX^, and so its strict transform is given by G~red=V(x,d,s,f)⊂UX~. The second statement follows from similarly noting that E=V(x,d,s,b)⊂UX~.∎
We also see that the strict transform F¯⊂X^ of F intersected with UX^ is a nodal surface
{bd=s2ac}⊂𝔸s1×ℙa:b1×ℙc:d1
which is isomorphic to the blow-up of 𝔸s1×ℙa:b1 at a fattened point cut out by the ideal ⟨s2a,b⟩. Denoting by F~ the strict transform of F in X~, we find that 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 X~, where we only blow up smooth varieties along a ℙ1. This allows us to easily obtain a concrete semiorthogonal decomposition for 𝐃b(X~).
Proposition 4.3.
There are varieties W,Y and a sequence of contractions φW,ψ,hY, where each map is a blow-up of a smooth rational curve such that the following diagram commutes.
Furthermore, letting EB,EY,EW be the exceptional loci of hY,ψ,φW, respectively, and E~B,E~Y the strict transforms of EB,EY, respectively, to X~, we have that
E~Y coincides with E;
EW coincides with G~red;
E~B coincides with F~.
Proof.
The map hY:Y→B is obtained by blowing up B along the reduced branch
ℭ0≅ℙs:t1⊂ℭ⊂B
to obtain a variety hY:Y→B. Now, X^ can be obtained by blowing up the strict transform 𝔜¯ of the non-reduced branch 𝔜⊂ℭ⊂B. Hence, we have a map φY:X~→Y, and we see that the total transform of the ideal cutting out the reduced curve 𝔜red¯ is locally principal in X~. Letting ψ:W→Y be the blow-up in 𝔜red¯, we see that X~→Y factors as X~→W→Y by the universal property of blow-ups.
Note that outside of the open subset UB both constructions of X~ only blow up B at the single point at infinity on ℭ0⊂ℭ; otherwise, all blow-ups have centres contained in some open set over UB. Define an open subset of Y over UB as
UY:={xc=ud}⊂ℙUB(𝒪c⊕𝒪(1)d)=:PY.
Similarly, there is an open subset of W found by blowing up PY at ⟨d,s⟩ and pulling back, giving
UW:={xc=udde=sf}⊂ℙPY(𝒪(1)e⊕πB∗𝒪(1)f)=:PW,
where we denote πB:PY→UB. Finally, we blow up PW along the ideal ⟨f,sce⟩ and pull back the equations for UW to give
UX~={xc=udde=sfbf=(sce)a}⊂ℙPW(πB∗𝒪(1)a⊕𝒪b)=:PX~′,
where we denote πB:PW→UB. It is clear that PX~′≅PX~ (e.g., by writing out the matrix of weights for PX~′ and comparing with Table 1), and so we recover UX~.
The centre of ψ:W→Y is given by ZW=V(x,d,s)⊂UY, and the exceptional locus of ψ:W→Y is given by
EY=V(x,d,s)≅ℙℙu:v1(𝒪e⊕𝒪(1)f)⊂UW.
The centre of the blow-up X~→W is given by ZX~=V(f,d,x,s)⊂UW which is a copy of ℙu:v1 arising as a section of EY→ℙu:v1, and we see that
EW=V(f,d,s,x)⊂UX~
is the exceptional divisor of the blow-up X~→W. Now, it is clear by considering the equations cutting out each subvariety that EW coincides with G~red and E~Y coincides with E in X~. It is also clear that E~B coincides with F~ in X~ since both give the preimage of ℭ0 along the map X~→B.∎
The alternative construction is visualised in Figure 2.
Alternative construction of the resolution, with blow-up centres indicated in red. Here, E¯B denotes the strict transform of EB to W, and E~B,E~Y denote the strict transforms of EB,EY, respectively, to 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⊂X~, we have that
𝒩E/X~≅𝒪(−2,1),ωX~|E≅𝒪(0,−2).
Proof.
Recall that E≅ℙℙu:v1(𝒪e⊕𝒪(1)f) by Lemma 4.2. The bundle 𝒩E/X~ has degree −2 along the fibres of the bundle E→ℙu:v1, 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⊂X~ is the strict transform of EY⊂W, E is the blow-up of EY along the section given by V(f)⊂EY. This implies that the restriction of 𝒩E/X~ to the section V(e) is the same as the restriction of 𝒩EY/W≅𝒪(−1) to V(e). Hence, we find that 𝒩E/X~≅𝒪(−2,1), and using the formula for ωE given in (2.5), we compute that
ωX~|E≅ωE⊗𝒩E/X~∨≅𝒪(−2,−1)⊗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 𝒪C(−1) is a 3-spherical object in 𝐃b(X~).
Proof.
This follows if we can contract C to an ordinary double point. Starting from B, let ℭnode be the union of the reduction of the branches, and consider the blow-up
Y′=BlℭnodeB
so that Y′ is a nodal variety. Now note that the total transform of the locus ℭnode in Y is the union of EB, which is a Cartier divisor, and the centre ZW of the blow-up W→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 ℭ0, which is a ruled surface in Y′ intersecting the node. In particular, there is an exceptional curve C1 which contracts to give the node. The blow-up φW:X~→W has centre disjoint from C1, and we can compute by working locally over UB that C1 is identified with the curve C under the blow-up. Hence, C can also be contracted to a threefold node.∎
A full exceptional sequence on the resolution
With the geometric setup established, we may now analyse 𝐃b(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 ℙ1-bundle π:Σ→ℙ1 embedded into a variety T as j:Σ→T, we denote by 𝒪Σ(a,b) the pushforward j∗(𝒪(a)⊗π∗𝒪(b)):
The component in the semiorthogonal decomposition equivalent to 𝐃b(B) is immediately seen to be orthogonal to kerσ∗. We therefore consider the following objects:
It is useful to obtain a geometric description for the objects ℱi and 𝒜i. 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∈{0,1}, ℱi is the pushforward of the line bundle on E∪EW which is
the line bundle 𝒪(−1,i−1) on E;
the line bundle 𝒪(0,i−1) on EW.
Proof.
We show the statement for ℱ1, using its description from equation (4.6), and the corresponding statement for ℱ0 follows by twisting appropriately. Firstly, taking the exact sequence on W
0→𝒪→𝒪(EY)→𝒪EY(−1,0)→0
and applying the pullback φW∗ shows that we have an exact sequence
0→𝒪→𝒪(E+EW)→φW∗𝒪EY(−1,0)→0
on X~. Note that this statement holds even though the functor φW∗ is derived; in other words, 𝐋iφW∗(𝒪EY(−1,0)) vanishes for i≠0. We find that the objects ℱi are supported on E∪EW. Note that E∩EW=V(f)⊂E, and f is a section of 𝒪(1,−1). Hence,
𝒪(E+EW)|E≅𝒪(−2,1)⊗𝒪(1,−1)≅𝒪(−1,0)
and since E∩EW=V(b)⊂EW, where b is a section of 𝒪(1,0), we have
𝒪(E+EW)|EW≅𝒪(−1,0)⊗𝒪(1,0)≅𝒪.
We hence find that ℱ1 is a pushforward of the line bundle on E∪EW which is given by 𝒪(−1,0) on the E component and 𝒪 on the EW component. Note that the line bundle 𝒪(a,b) on E restricts to 𝒪(b) on E∩EW, so such a line bundle exists on E∪EW.∎
Recall that by Proposition 4.3, F~⊂X~ coincides with the strict transform of EB in X~.
Lemma 4.7.
For i∈{0,1}, the objects 𝒜i are given as
𝒜i≅𝒪F~(−1,i−1).
Proof.
For the object 𝒜1, we have a short exact sequence
0→𝒪→𝒪(F~)→φY∗𝒜1→0,
so that 𝒜1 is the pushforward of the normal bundle of F~. Recall that F~ is a ruled surface blown up twice, and has a chain of exceptional curves CE+CW, where we denote CE:=F~∩E and CW=F~∩EW. We denote by 𝒪(z,w):=p∗𝒪(z,w), where p:F~→EB is the blow-up map. Then, the normal bundle must be of the form
𝒩F~/X~≅𝒪(z,w)⊗𝒪(aCE+bCW).
We know that z=−1 and w=0 since 𝒩EB/Y≅𝒪(−1,0) and the restriction to the fibres of F~→EB→ℙu:v1 is generically unchanged. Next, one computes via the inclusion CE↪E↪X~ that 𝒩CE/X~≅𝒪⊕𝒪(−2). Since 𝒩CE/F~≅𝒪(−2), we find that
𝒪(aCE+bCW)|CE≅𝒪⟹−2a+b=0.
Similarly, 𝒩CW/F~≅𝒪(−1) and 𝒩CW/X~≅𝒪⊕𝒪(−1), so we find that
𝒪(aCE+bCW)|CW≅𝒪⟹−a+b=0.
Hence, b=a=0, and we obtain the statement for 𝒜1, with the case of 𝒜0 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
𝐃b(X)=⟨Φ(ℒ(1)),Φ(ℒ(2)),Φ(𝒫),𝐃b(B)⟩,
where Φ:𝐃b(ℭ)→𝐃b(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(EW)=σ(E+EW)=Gred⊂X, while σ(F~)=F⊂X. Hence, we immediately see that σ∗ℰ1 is our cℙ∞ object Φ(𝒫), but σ∗ℰ0 is a different cℙ∞ object, namely Φ(𝒫(−1)). Meanwhile, the objects 𝒜i are supported on 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
⟨ℰ0,ℰ1,ℱ0,ℱ1,𝒜0,𝒜1⟩,
we find that the objects ℰ1,ℱ0 are orthogonal in Section 4.3.2, so that
⟨ℰ0,ℱ0,ℰ1,ℱ1,𝒜0,𝒜1⟩
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.
where we firstly mutate ⟨𝒜0,𝒜1⟩ to the left of ⟨ℰ1,ℱ1⟩, and then mutate ⟨ℰ0,ℱ0⟩ to the right of ⟨𝒜0′′,𝒜1′′⟩. The theorem then states that pushforward under σ∗ recovers the semiorthogonal decomposition of 𝐃b(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 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, 14]); 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 kerσ∗. Using this, we are then able to perform the necessary mutations, and give a proof of Theorem 4.8.
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.
Ext•(A,B), where B is an object in the null category.
A
B
𝒪C(−1)
𝒪E(−1,−1)
𝒪E(−1,0)
𝒜i
ℂ
0
0
ℱ0
ℂ[−1]
ℂ
ℂ⊕2
ℱ1
ℂ[−1]
0
ℂ
ℰ0
0
ℂ[−1]
ℂ⊕2[−1]
ℰ1
0
0
ℂ[−1]
Ext•(A,B) where A is an object in the null category.
A
B
𝒜i
ℱ0
ℱ1
ℰ0
ℰ1
𝒪C(−1)
ℂ[−3]
ℂ[−2]
ℂ[−2]
0
0
𝒪E(−1,−1)
0
ℂ[−2]
ℂ⊕2[−2]
ℂ[−1]
ℂ⊕2[−1]
𝒪E(−1,0)
0
0
ℂ[−2]
0
ℂ[−1]
4.3.1.1. Morphisms to 𝒪C(−1). The objects ℰ0 and ℰ1 are orthogonal to 𝒪C(−1), because they are supported on EW which is disjoint from C. Both ℱ0 and ℱ1 are supported on E∪EW, and we see that E and C intersect transversely, which gives us that
Ext•(ℱi,𝒪C(−1))≅ℂ[−1].
Consider the objects 𝒜i, and let j:F~→X~ be the inclusion. We have a splitting distinguished triangle for j∗𝒜i≅j∗j∗𝒪(−1,i−1)
The above calculations also let us compute morphisms from 𝒪C(−1), since it is a 3-spherical object by Lemma 4.5, so that Exti(𝒪C(−1),A)≅Ext3−i(A,𝒪C(−1)) for any object A.
4.3.1.2. Morphisms between 𝒜i and 𝒪E(−1,a). The objects 𝒜0,𝒜1 are completely orthogonal to these objects because
(φY)∗𝒪E(−1,a)=0
for any a, and because
Exti(𝒪E(−1,a),−)≅Ext3−i(−,𝒪E(−1,a−2))
by Lemma 4.4.
4.3.1.3. Morphisms between ℱi and 𝒪E(−1,a). Next we compute the morphisms from ℱ0 to 𝒪E(−1,−1). The key in this computation is that E is the strict transform of EY⊂W, so
4.3.1.4. Morphisms between ℰi and 𝒪E(−1,a). Instead of using a geometric method to compute morphisms from ℰi, we make the following observation.
Lemma 4.9.
For i∈{0,1}, there are short exact sequences
0→ℰi→ℱi→𝒪E(−1,i−1)→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 ℰ0.∎
The remaining morphisms in the tables are an immediate consequence of these distinguished triangles and the following statements:
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•(ℱj,𝒜k) for any choice of j,k∈{0,1}. Since the objects 𝒜i are orthogonal to 𝒪E(−1,a) for any a, we find that Ext•(ℰj,𝒜k)≅Ext•(ℱj,𝒜k) for any j,k∈{0,1}.
4.3.2.2. Morphisms to ℱi. For each i∈{0,1}, using that Ext•(ℰi,𝒪E(−1,i−1))≅ℂ[−1], we obtain that
Ext•(ℰi,ℱi)≅ℂ⊕ℂ[−1].
Since Ext•(ℰ1,𝒪E(−1,−1))=Ext•(ℰ1,ℰ0)=0, we must also have that Ext•(ℰ1,ℱ1)=0. Finally, using that Ext•(ℰ0,𝒪E(−1,0))≅ℂ⊕2[−1], we find that
Ext•(ℰ0,ℱ1)≅ℂ⊕2⊕ℂ⊕2[−1].
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:
𝒜0→𝒜0′→ℱ1,
𝒜1→𝒜1′→ℱ1
We have that the objects 𝒜i′ are sheaves supported on F~∪EW∪E. Now note that the orthogonality of 𝒜0,𝒜1 with 𝒪E(−1,0) implies that there is an isomorphism
Ext•(ℱ1,𝒜j)≅Ext•(ℰ1,𝒜j)≅ℂ[−1]
which we obtain by applying Ext•(−,𝒜j) to the triangle (4.8). This isomorphism is obtained by precomposing with ℰ1→ℱ1, so we obtain a commuting triangle.
Applying Ext•(ℰ1,−) to the triangles (4.9) and (4.10) gives us exact sequences
where the first non-zero map is an isomorphism due to the above commuting triangle. Hence, we have Ext1(ℰ1,𝒜j′)≅Ext1(ℰ1,ℱ1)≅ℂ. The next set of mutations is therefore defined by the following exact triangles:
𝒜0′→𝒜0′′→ℰ1,𝒜1′→𝒜1′′→ℰ1.
We now have the job of tidying up the exceptional sequence
Then, the map ℂ⊕2≅Hom(ℱ0,ℱ1)→Ext1(ℱ0,𝒜0)≅ℂ is non-zero, due to the fact that the extension of ℱ0 by 𝒜0 is induced by the intersection of EW∪E with F~, and there is a map ℱ0→ℱ1 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⊕2≅Hom(𝒜0′′,𝒜1′′)→Hom(ℱ0,𝒜1′′)≅ℂ
induced by precomposing with ℱ0→𝒜0′′ is non-zero. By mutating over ℰ1 and then ℱ1, we see that this is equivalent to showing that the map
ℂ⊕2≅Hom(𝒜0,𝒜1)→Ext1(ℱ2,𝒜1)≅ℂ
is non-zero, where ℱ2=φW∗𝒪EY(−1,1) and the morphism is given by shifting and then precomposing with ℱ2→𝒜1[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 ℱ0′′:
ℱ0→𝒜0′′→ℱ0′,
ℱ0′→𝒜1′′→ℱ0′′.
This object has a more explicit description.
Lemma 4.11.
Let 𝒦 be the object defined by the extension
0→𝒪C(−1)→𝒦→𝒪E(−1,0)→0.
Then, ℱ0′′ is given by the extension
0→𝒦→ℱ0′′→ℰ1→0.
Proof.
Since ℰ1 is orthogonal to 𝒪C(−1), we have Ext•(ℰ1,𝒦)≅ℂ[−1] giving us an extension
0→𝒦→𝒢→ℰ1→0.
It is clear by applying Ext•(ℰ1,−) to this short exact sequence that Ext•(ℰ1,𝒢)=0. Next, we apply Ext•(ℱ1,−) to the triangle (4.14) defining 𝒦. This gives an exact sequence
The claim is that the middle map is an isomorphism, so that Ext•(ℱ1,𝒦)=0. But this is clear geometrically, since the extensions ℱ1→𝒪C(−1)[1] and 𝒪E(−1,0)→𝒪C(−1)[1] are induced by the intersection of C and E, and in a neighbourhood of the intersection point ℱ1→𝒪E(−1,0) is precisely the identity. Hence, Ext•(ℱ1,𝒦)=Ext•(ℱ1,ℰ1)=0, so we get that Ext•(ℱ1,𝒢)=0.
Since there are no morphisms from 𝒪C(−1) or 𝒪E(−1,0) to ℰ0, we see that
Ext•(𝒢,ℰ0′′)=0.
Hence, 𝒢∈⟨ℱ0′′⟩. It is clear by the mutations relating ℱ0′′ to ℱ0 that Ext•(ℱ0,ℱ0′′)≅ℂ. Hence, it suffices to show that Ext•(ℱ0,𝒢)≅ℂ. We know that
Ext•(ℱ0,𝒪E(−1,0))≅ℂ⊕2,
and by similar reasoning as we did to show that the map Hom(ℱ1,𝒪E(−1,0))→Ext1(ℱ1,𝒪C(−1)) is non-zero, we also find that the map
ℂ⊕2≅Hom(ℱ0,𝒪E(−1,0))→Ext1(ℱ0,𝒪C(−1))≅ℂ
is non-zero. Hence, Ext•(ℱ0,𝒦)≅ℂ, and since Ext•(ℱ0,ℰ1)=0, it follows immediately that Ext•(ℱ0,𝒢)≅ℂ.∎
Next we mutate the object ℰ0.
Lemma 4.12.
We have that
Ext•(ℰ0,𝒜0′′)≅Ext•(ℰ0,𝒜1′′)≅ℂ,
and if ℰ0′:=Cone(ℰ0→𝒜0′′), then
Ext•(ℰ0′,𝒜1′′)≅ℂ.
Proof.
The first statement holds because of the exact triangle
ℰ0→ℱ0→𝒪E(−1,−1)
and the fact that 𝒜0′′,𝒜1′′ are orthogonal to 𝒪E(−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
ℰ0′→ℱ0′→𝒪E(−1,−1)[1].∎
We similarly define the object ℰ0′′
ℰ0→𝒜0′′→ℰ0′,
ℰ0′→𝒜1′′→ℰ0′′.
The cone on the morphism ℰ0′′→ℱ0′′ is seen to be 𝒪E(−1,−1)[2] by another application of the derived nine-lemma. In particular, ℰ0′′ is not a sheaf.
Recall that we know that σ∗ℰ1≅Φ(𝒫), and an easy corollary of Lemma 4.9 is that σ∗ℱ1≅σ∗ℰ1. The next lemmas will allow us to describe what happens if we pushforward the objects 𝒜i′′.
Lemma 4.13.
For each j∈{0,1}, the objects 𝒜j and 𝒜j′′ are related by the exact triangle
𝒜j→𝒜j′′→𝒴,
where 𝒴 is given by the non-trivial extension
ℱ1→𝒴→ℰ1.
Proof.
Consider the exact triangles
𝒜0→𝒜0′→ℱ1,𝒜0′→𝒜0′′→ℰ1,𝒜0→𝒜0′′→𝒴.
Here, we use that Ext•(𝒜0,𝒜0′′)≅Ext•(𝒜0,𝒜0′)≅Ext•(𝒜0,𝒜0)≅ℂ, where the first isomorphism is induced by postcomposing with 𝒜0′→𝒜0′′ and the second isomorphism is induced by postcomposing with 𝒜0→𝒜0′. Therefore, the map 𝒜0→𝒜0′′ is a composition of the maps 𝒜0→𝒜0′ and 𝒜0′→𝒜0′′, and we can use the octahedral axiom to obtain an exact triangle
ℱ1→𝒴→ℰ1.
The triangle is non-split since the morphism ℰ1→ℱ1[1] in the triangle is the composition ℰ1→𝒜0′[1]→ℱ1[1], and we know that composing with 𝒜0′→ℱ1 induces an isomorphism Ext1(ℰ1,𝒜1′)≅Ext1(ℰ1,ℱ1).∎
It is geometrically easier to see what happens if we first pushforward to X^; the next lemma shows that the kernel of this contraction is simply generated by 𝒪E(−1,−1) and 𝒪E(−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 X~→X^ is generated by
kerφX^,∗=⟨𝒪E(−1,−1),𝒪E(−1,0)⟩.
Proof.
Let Z⊂X^ be the reduced singular locus. Since the objects 𝒪(−1,−1),𝒪(−1,0) generate the kernel of the contraction E→Z, we want to show that the kernel of X~→X^ is generated by the pushforwards of these objects. We use [15, Theorem 5.2] to show this, which requires us to show that
φX^,∗𝒪X~(−mE)≅𝒥Zm,
where 𝒥Z is the ideal sheaf of Z. We can work in a neighbourhood of Z, but locally near Z we find that X^ is just V×ℙ1 for a nodal surface V, and that X~ is, locally around E, the pullback of the resolution V~ of V, that is, we have a pullback square.
Let π:V~→V be the contraction, F the exceptional curve and ZV the nodal point. The desired result follows from pulling back the corresponding equivalence for V, which was shown in [15, Corollary 5.6]
π∗𝒪V~(−mF)≅𝒥ZVm
and using flat base change
f∗π∗≃π∗f∗
as needed.∎
Consider the extension on 𝐃b(ℭ)
0→𝒫→𝒬→𝒫→0,
so that 𝒬 is the skyscraper sheaf along the non-reduced branch ℙu:v1×ℂ[s]/s2.
Lemma 4.15.
We have that
σ∗𝒴≅Φ(𝒬).
Proof.
Firstly, we see that 𝒴 is orthogonal to the objects 𝒪E(−1,a) for all a∈ℤ. Since
Ext•(ℰ1,𝒪E(−1,−1))=Ext•(ℱ1,𝒪E(−1,−1))=0,
we immediately have Ext•(𝒴,𝒪E(−1,−1))=0. We also know that
Ext1(ℰ1,ℱ1)≅Ext1(ℰ1,𝒪E(−1,0)),
that is, the map ℰ1→𝒪E(−1,0)[1] factors through ℰ1→ℱ1[1]. Then, applying the functor Ext•(−,𝒪E(−1,0)) to the triangle ℱ1→𝒴→ℰ1 gives us an exact sequence
which gives us that Ext•(𝒴,𝒪E(−1,0))=0. It follows then that Ext•(𝒴,𝒪E(−1,a))=0 for all integers a, and hence by Serre duality that Ext•(𝒪E(−1,a),𝒴)=0 also.
Now, we pushforward to X^. Here, we have a distinguished triangle
φX^,∗ℱ1→φX^,∗𝒴→φX^,∗ℰ1.
One sees that φX^,∗ℱ1≅φX^,∗ℰ1 is the 𝒪(−1,0) bundle on the strict transform of Gred. The extension is non-split, since we showed that 𝒴 is orthogonal to kerφX^,∗, so for example we know that φX^,∗𝒴 has no endomorphisms in non-zero degrees. Hence, φX^,∗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 <xref rid="j_JNCG684_stat_036" ref-type="statement">4.8</xref>.
To prove that
σ∗𝒜0′′≅Φ(ℒ(1)),σ∗𝒜1′′≅Φ(ℒ(2)),
recall the defining triangle for 𝒜1′ (4.10)
𝒜1→𝒜1′→ℱ1.
In fact, we note that the transverse intersection of E∪EW with F~ along CE∪CW⊂F~ induces an extension between ℱ1 and 𝒜1. By uniqueness, we find that 𝒜1′ is the pushforward of a line bundle on E∪EW∪F~ which is 𝒪(−1,0)⊗𝒪(CE+CW) on F~, 𝒪(−1,0) on E and 𝒪 on EW. One verifies that this is a valid line bundle on the union since all restrictions to intersections agree. There is a line bundle on ℙs:t1∪ℙu:v1⊂C (i.e., the union of the reduced branches) which restricts to 𝒪(1) on ℙs:t1 and 𝒪 on ℙu:v1. We denote by ℳ the pushforward of this into ℭ. Then, one sees that σ∗𝒜1′≅Φ(ℳ). Let 𝒪ℭ0 be the skyscraper sheaf along the reduced branch. Then, σ∗𝒜1=Φ(𝒪ℭ0).
Next, there is a unique extension in 𝐃b(ℭ) between 𝒫 and ℳ
ℳ→ℒ(2)→𝒫.
We can consider the following triangle, obtained by pushing forward:
𝒜1′→𝒜1′′→ℰ1.
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
𝒜1→𝒜1′′→𝒴
which must correspond to an extension in 𝐃b(ℭ) between 𝒬 and 𝒪ℭ0 by Lemma 4.15. Then, one computes that there is indeed an extension induced by intersection
𝒪ℭ0→ℒ(2)→𝒬,
so that σ∗𝒜1′′ can only be Φ(ℒ(2)). A similar argument shows σ∗𝒜0′′≅Φ(ℒ(1)).
Now let
𝒟~=⟨ℰ0′′,ℱ0′′,ℰ1,ℱ1⟩.
The statement
𝒟~/(kerσ∗∩𝒟~)→∼𝒟
follows from the fact that the cones on the morphisms ℱ0′′→ℰ1, ℰ0′′→ℱ0′′ and ℰ1→ℱ1 are in the kernel of σ∗, so the image of 𝒟~ under σ∗ is contained within our absorbing category 𝒟. Combined with the fact that 𝒜0′′,𝒜1′′ pushforward to Φ(ℒ(1)) and Φ(ℒ(2)), respectively, we find that the image of 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).∎
Structure of the categorical resolution
Having found the subcategory 𝒟~⊂𝐃b(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
𝒟~=⟨ℰ0′′,ℱ0′′,ℰ1,ℱ1⟩.
We recall that the objects ℰ1,ℱ1 are sheaves supported on the exceptional loci EW,E~Y as defined in equation (4.6), while ℰ0′′ and ℱ0′′ 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 𝒟~ are described as follows:
Ext•(ℰ0′′,ℱ0′′)≅Ext•(ℰ1,ℱ1)≅ℂ⊕ℂ[−1];
Ext•(ℱ0′′,ℰ1)≅ℂ⊕ℂ[−1];
Ext•(ℱ0′′,ℱ1)≅Ext•(ℰ0′′,ℰ1)≅ℂ⊕ℂ[−1]⊕ℂ⊕2[−2];
Ext•(ℰ0′′,ℱ1)≅ℂ⊕ℂ[−1]⊕ℂ⊕2[−2]⊕ℂ⊕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•(ℱ0′′,ℰ1)≅ℂ⊕ℂ[−1] and Ext•(ℱ0′′,ℱ1)≅ℂ⊕ℂ[−1]⊕ℂ⊕2[−2].
Proof.
Apply Ext•(−,ℰ1) to (4.15). Since Ext•(𝒪C(−1),ℰ1)=0, we find that
Ext•(𝒦,ℰ1)≅Ext•(𝒪E(−1,0),ℰ1)≅ℂ[−1],
and this gives us that Ext•(ℱ0′′,ℰ1)≅ℂ⊕ℂ[−1].
Applying Ext•(−,ℱ1) to the triangle (4.14) defining 𝒦, we find that Ext•(𝒦,ℱ1)≅ℂ⊕2[−2] because we have
Ext•(𝒪C(−1),ℱ1)≅Ext•(𝒪E(−1,0),ℱ1)≅ℂ[−2].
Then, applying Ext•(−,ℱ1) to the triangle (4.15) gives us that
and using the fact that Ext•(𝒪E(−1,−1)[2],ℰ1)≅ℂ⊕2[−3], we get the first statement. The second statement is proven by instead applying Ext•(−,ℱ1), but now using that
Ext•(𝒪E(−1,−1)[2],ℱ1)≅ℂ⊕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 cℙ∞ object P, we have
Ext•(P,P)≅ℂ⊕ℂ[−1]⊕ℂ⊕2[−2]⊕ℂ⊕2[−3]⊕⋯,
and that we can recover the vector spaces of morphisms between objects in our categorical resolution 𝒟~ as truncations of this algebra:
Subsequently, for any two objects A,B in the exceptional sequence, we think of an element of Exti(A,B) as lifting some element of Exti(P,P). Since Ext•(P,P) is a polynomial ring with no relations, this suggests the following.
Proposition 4.19.
All compositions in 𝒟~ are non-zero.
Before proving the proposition, we illustrate some consequences. Let t0,t1,t2,s0,s1 be non-zero degree 0 maps and ε0,ε1,ε2,δ0,δ1 be non-zero degree 1 maps as described by the following quiver.
Letting ab denote the composition b∘a, a corollary of Proposition 4.19 shows that up to rescaling, we can obtain the following relations:
t0t1=s0, t1t2=s1,t0ε1=ε0t1=δ0, t1ε2=ε1t2=δ1.
Defining ρ0:=ε0ε1 and ρ1:=ε1ε2, we also see that
ρ0t2=t0ρ1ρ0ε2=ε0ρ1.
We can then extend to find bases {ρ0,θ0} of Ext2(ℰ0′′,ℰ1) and {ρ1,θ1} of Ext2(ℱ0′′,ℱ1), respectively.
Note that we can choose θ0,θ1 such that either of the relations t0θ1=θ0t2 or ε0θ1=θ0ε2 hold. However, it is not clear whether we can choose θ0,θ1 such that both commutativity relations simultaneously hold.
We can then visualise the category 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∞ 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 ℱ0′′.
Lemma 4.20.
The compositions
t1t2:ℱ0′′→ℰ1→ℱ1,t1ε2:ℱ0′′→ℰ1→ℱ1[1],
are non-zero.
Proof.
The first statement follows from the proof of Lemma 4.17, which shows that the degree 0 map ℱ0′′→ℱ1 factors as ℱ0′′→ℰ1→ℱ1. The degree 1 map ℱ0′′→ℱ1[1] similarly factors as ℱ0′′→ℰ1→ℱ1[1].∎
Lemma 4.21.
The composition ε1t2:ℱ0′′→ℱ1 is non-zero.
Proof.
We deduce that Ext•(ℱ0′′,𝒪E(−1,0))≅ℂ⊕2[−2], so we can immediately see that the composition ℱ0′′→ℰ1[1]→ℱ1[1] is non-zero.∎
Lemma 4.22.
The composition ε1ε2:ℱ0′′→ℰ1[1]→ℱ1[2] is non-zero.
Proof.
Consider the following diagram.
There is a unique degree 1 morphism (up to scaling) 𝒦→ℰ1[1], and one can quickly show that this factors through both 𝒦→ℱ0′′ and 𝒦→𝒪E(−1,0), by applying Ext•(−,ℰ1) to the triangles defining 𝒦 and ℱ0′′, 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 ℰ1→ℱ1[1] is non-zero and hence we find that 𝒴′′≅𝒴[1]. Then, we have an exact triangle
𝒪C(−1)→𝒴′→𝒴.
We see that
Hom(𝒴′,ℱ1)≅Hom(𝒴,ℱ1)≅ℂ
and
Ext2(𝒴′,ℱ1)≅Ext2(𝒪C(−1),ℱ1)≅ℂ,
and there are no morphisms in any other degrees. Then, applying Ext•(−,ℱ1) to the triangle
ℰ1→𝒴′→ℱ0′′,
we see that since Ext1(𝒴′,ℱ1)=0 that the map Ext1(ℰ1,ℱ1)→Ext2(ℱ0′′,ℱ1) is an injection and hence the composition ℱ0′′→ℰ1[1]→ℱ1[2] is non-zero.∎
4.5.2.2. Compositions with source ℰ0′′ and target ℰ1.
so that we have an isomorphism ℂ≅Ext1(ℰ0′′,ℱ0′′)→Ext2(𝒪E(−1,−1)[2],ℱ0′′)≅ℂ. Hence, the left triangle commutes in the following diagram.
Since the right triangle commutes by assumption, it suffices to show that the composition
𝒪E(−1,−1)→ℱ0′′→ℰ1[1]
is non-zero, since then we obtain a contradiction. For this, we consider the non-split extension
ℰ1→𝒴′→ℱ0′′,
and we only need to show that Hom(𝒪E(−1,−1),𝒴′)=0, since then the map
Hom(𝒪E(−1,−1),ℱ0′′)→Ext1(𝒪E(−1,−1),ℰ1)
is an injection, and hence the composition (4.20) is non-zero. Recall that diagram (4.19) gives us an exact triangle
𝒪C(−1)→𝒴′→𝒴.
There are no morphisms from 𝒪E(−1,−1) to ℰ1 and ℱ1 in degree 0. Hence,
Hom(𝒪E(−1,−1),𝒴)=0,
and since Hom(𝒪E(−1,−1),𝒪C(−1))=0, we have our desired result.∎
4.5.2.3. Compositions with source ℰ0′′ and target ℱ1.
Lemma 4.25.
For all i∈{0,1,2}, precomposing with t0:ℰ0′′→ℱ0′′ induces isomorphisms
Exti(ℰ0′′,ℱ1)≅Exti(ℱ0′′,ℱ1)
and postcomposing with t2:ℰ1→ℱ1 induces isomorphisms
Exti(ℰ0′′,ℱ1)≅Exti(ℰ0′′,ℰ1).
Proof.
Firstly, applying Ext•(−,ℱ1) to the triangle
ℰ0′′→ℱ0′′→𝒪E(−1,−1)[2]
induces isomorphisms Exti(ℱ0′′,ℱ1)≅Exti(ℰ0′′,ℱ1) by precomposing with ℰ0′′→ℱ0′′ for i∈{0,1,2}. Similarly, using that Ext•(ℰ0′′,𝒪E(−1,0))≅ℂ⊕2[−3] and then applying Ext•(ℰ0′′,−) to the triangle (4.8)
ℰ1→ℱ1→𝒪E(−1,0)
gives us isomorphisms
Exti(ℰ0′′,ℰ1)≅Exti(ℰ0′′,ℱ1)
for i∈{0,1,2} by composing with ℰ1→ℱ1.∎
Lemma 4.26.
Precomposing with ε0:ℰ0′′→ℱ0′′[1] induces an isomorphism
Ext3(ℰ0′′,ℱ1)≅Ext2(ℱ0′′,ℱ1)
and postcomposing with ε2:ℰ1→ℱ1[1] induces an isomorphism
Ext3(ℰ0′′,ℱ1)≅Ext2(ℰ0′′,ℰ1).
Proof.
By applying Ext•(ℱ0′′,−) to
ℱ1→𝒴→ℰ1
and using the results above, we find Ext•(ℱ0′′,𝒴)≅ℂ⊕ℂ[−2]. Then, we apply the functor Ext•(𝒪E(−1,−1)[2],−) to the same triangle and see that Ext•(𝒪E(−1,−1)[2],𝒴) can only be non-zero in degrees 0≤i≤3. Hence, applying Ext•(−,𝒴) to the triangle
ℰ0′′→ℱ0′′→𝒪E(−1,−1)[2],
we find that Ext3(ℰ0′′,𝒴)=0. By applying Ext•(ℰ0′′,−) to (4.21), we obtain an isomorphism
ℂ⊕2≅Ext2(ℰ0′′,ℰ1)→∼Ext3(ℰ0′′,ℱ1)≅ℂ⊕2.
This shows the first statement.
We use a similar argument for the second statement. Consider the non-split extension
ℱ0′′→𝒵0→ℰ0′′
and apply Ext•(−,ℱ1). We see that if Ext3(𝒵0,ℱ1)=0, then there is an isomorphism
ℂ⊕2≅Ext2(ℱ0′′,ℱ1)→Ext3(ℰ0′′,ℱ1)≅ℂ⊕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•(𝒪E(−1,−1)[2],ℱ1) and Ext•(𝒦[2],ℱ1) are only non-zero in degree 3, and hence Exti(𝒵1,ℱ1)=0 for i≥4. From the previous calculations, we have that Ext•(𝒴′,ℱ1)≅ℂ⊕ℂ[−2]. Hence, we find that Ext3(𝒵0,ℱ1)=0, by using the triangle
𝒵0→𝒴′→𝒵1.∎
Deformation absorption through <inline-formula><alternatives><mml:math display="inline"><mml:mi>c</mml:mi><mml:msup><mml:mi>ℙ</mml:mi><mml:mi mathvariant="normal">∞</mml:mi></mml:msup></mml:math><tex-math><![CDATA[\(c\mathbb{P}^{\infty}\)]]></tex-math></alternatives></inline-formula> objects
In this section, we return to the singular curve ℭ defined in equation (3.1) and briefly discuss an interesting example of how cℙ∞ 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)
𝐃b(ℭ)=⟨𝒫(−1),𝐃b(ℙs:t1)⟩.
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→(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,s2⟩ℙx:y1×ℙs:t1
with the natural map 𝔛→ℙx:y1. We can embed into a projective bundle
𝔛≅{xv=s2u}⊂ℙℙ1×ℙ1(𝒪(0,2)u⊕𝒪(1,0)v),
so that the space 𝔛 is nodal, and the fibre over {x=0} is given by
{s2u=0}⊂ℙℙs:t1(𝒪(2)u⊕𝒪v)
which is isomorphic to the curve ℭ. Note that the generic fibre is a copy of ℙs:t1, so that 𝔛 is a smoothing of ℭ.
Proposition 5.1.
The object P∈𝐃b(ℭ) provides a deformation absorption with respect to the smoothing 𝔛.
Proof.
The blow-up formula shows that there is a semiorthogonal decomposition
𝐃b(𝔛)=⟨𝐃b(ℂ[s]/s2),𝐃b(ℙ1×ℙ1)⟩.
If i:ℭ↪𝔛 is the inclusion, then i∗P is precisely the ℙ∞,1 object generating the admissible subcategory 𝐃b(ℂ[s]/s2) of 𝐃b(𝔛).∎
It is not clear whether the cℙ∞ 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 cℙ∞ object P gives a ℙ∞,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→P→P[2].
In the case that P→P[2] is the zero map, it is clear that i∗P is not in 𝐃perf(𝔛) since i∗i∗P is not in 𝐃perf(ℭ). Otherwise, applying Ext•(−,P) to the triangle shows that
Extj(i∗P,i∗P)≅Extj(i∗i∗P,P)≅ℂ
for all j≥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.
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