ALMOST PROPER GIT-STACKS AND DISCRIMINANT AVOIDANCE

We prove that the classifying stack of an reductive group scheme over a field is very close to being proper. Using this we prove a result about isotrivial families of varieties. Fix a polarized variety with reductive automorphism group. To prove that every isotrivial family with this fibre has a rational section it suffices to prove this when the base is projective, i.e., the discriminant of the family is empty.


Introduction
Consider an algebraic stacks of the form [Spec(k)/G] where G is a geometrically reductive group scheme over the field k.It turns out that such a stack is nearly proper, see Proposition 2.4.1.
Next, consider a pair (V, L) consisting of a projective variety V over k and an invertible sheaf.Also, fix an integer d ≥ 1.We would like to know if every ddimensional family of polarized varieties X → S, N ∈ Pic(X) all of whose fibres are isomorphic to (V, L) has a rational section.For example this is true if V is a nodal plane cubic.
Theorem 1.0.1.(Theorem 2.1.3.)Assume G = Aut(V, L) is geometrically reductive.If X → S has a section whenever S is a projective variety of dimension d then there is a section no matter what (X, N )/S you start with, provided dim S ≤ d.
Loosely speaking this means that if you prove the existence of rational sections whenever the discriminant is empty then you prove it in general.For example, it implies that if you are trying to find rational sections on families of polarized homogenous varieties over surfaces then it suffices to do so in the case of families of homogenous varieties over projective nonsingular surfaces.Our proof of this theorem depends on the result on GIT-stacks mentioned above.This effort is part of our work on using the geometry moduli spaces of rational curves on fibres of families X → S to prove the existence of rational sections of X/S.See the forthcoming papers [?].In this paper we handle the special case of a family of Grassmanians over a smooth projective surface S, see Theorem 4.2.1.The method is to fibre the surface S in curves C t .For each C = C t we study the moduli space Σ of sections of some large degree of X| C → C. We prove that the MRC-quotient of Σ equals the Jacobian of the curve C. Finally we use the Graber-Harris-Starr theorem to conclude.
In the end, putting everything together we deduce the period-index theorem for Brauer classes over function fields of surfaces, compare [?].
Theorem 1.0.2.Let k be an algebraically closed field and let k(S) be the function field of a surface over S. The the period equals the index for any Brauer class over k(S).

Isotrivial families
The title of this section is a little misleading as usually one thinks of an isotrivial family as having finite monodromy.As the reader will see such families are certainly examples to which our discussion applies, but we also allow for a positive dimensional structure group.The families will be isotrivial in the sense that the fibres over a Zariski open will be all isomorphic to a fixed variety V .
2.1.Statement of the result.Let k be an algebraically closed field of any characteristic.We assume given a variety V over k and an ample invertible sheaf L over V .We let m = dim V .We introduce another integer d ≥ 1 which will be an upper bound for the dimension of the base of our families.We are going to ask the following question: Is it true that for any polarized family of varieties over a ≤ d-dimensional base whose general fibre is V , there is a rational point on the generic fibre?We make this more precise as follows.
Situation 2.1.1.Here we are given a triple (K/k, X → S, N ), with the following properties: (a) The field K is an algebraically closed field extension of k.(b) The map X → S is a morphism of projective varieties over K. (c) The dimension of S is at most d.(d) We are given an invertible sheaf N on X. (e) For a general point s ∈ S(K) we have (X s , N s ) ∼ = (V K , L K ).
The notation (V K , L K ) refers to the base change of the pair (V, L) to SpecK.Thus (e) means that there exists a Zariski open U ⊂ S such that (X s , N s ) ∼ = (V K , L K ) as pairs over K for all s ∈ U .Considering a suitable Hilbert scheme this then implies that all geometric fibres of X → S over U are isomorphic to a suitable base change of V .
Question 2.1.2.Suppose we are in Situation 2.1.1.Is there a rational point on the generic fibre of X → S? In other words: Is X(K(S)) not empty?
A natural problem that arises when studying this question is the possibility of bad fibres in the family X → S. Let us define the discriminant ∆ of a family (K/k, X → S, N ) as in Situation 2.1.1 as the Zariski closure of the set of points s ∈ S(K) such that (X s , L s ) is not isomorphic to (V K , L K ).A priori the codimension of (the closure of) ∆ is assumed ≥ 1, and typically it will be 1.In this section we show that it often suffices to answer Question 2.1.2 in cases where the codimension of ∆ is bigger, at least as long as we are answering the question for all families.
It is not surprising that the automorphism group G of the pair (V, L) is an important invariant of the situation.The group scheme G has T -valued points which are pairs (φ, α), where φ : V T → V T is an automorphism of schemes over T , and α : φ * L T → L T is an isomorphism of invertible sheaves.The group law is given by (φ, α) ).We leave it to the reader to show that G is an affine group scheme (since L is ample).In the following theorem G • red denotes the reduction of the connected component of G.Note that G • red is a smooth affine group scheme (since k is algebraically closed, and hence perfect).
Theorem 2.1.3.Fix (V, L) and d as above.Assume that G • red is reductive.If the answer to Question 2.1.2is yes whenever ∆ = ∅, then the answer to Question 2.1.2is yes in all cases.
The proof has 2 parts: deformation and specialization.The deformation argument proves the following: For every triple (K/k, X → S, N ), there is a dense open subset U ⊂ X and a deformation of (X U → U, N | X U ) to a triple (K /k, X → S , N ) with trivial discriminant.The specialization argument proves the following: Every rational point of the generic fiber of X → S specializes to a rational point of the generic fiber of X → S. Thus Question 2.1.2has a positive answer for (K/k, X → S, N ) if it has a positive answer for (K /k, X → S , N ).

2.2.
A bijective correspondence.To deform the pair (X U → U, N | X U ), it is convenient to first convert the pair into a G-torsor over U , deform the torsor, and then convert this back into a triple.This subsection describes how to convert between pairs and G-torsors.As in subsection 2.1, denote by G the automorphism group scheme of (V, L).
Let U be any k-scheme.Let (X → U, N ) be a pair where X → U is a flat proper morphism and N an invertible sheaf on X.We assume that every geometric fiber of (X, N ) over U is isomorphic to the base-change of (V, L).Consider the functor that associates to a scheme T → U over U the set of pairs (φ, α), where φ : X T → V T is an isomorphism over T and α : There is a left G-action G × T → T on T over U (by post-composing, see the definition of the group structure on G).
Proof.It suffices to prove that (X, N ) is locally in the fppf topology of U isomorphic to the constant family (V, L) × U .To prove this we need some notation.
Take N so large that L N is very ample on V and has vanishing higher cohomology groups.Let n = dim Γ(V, L N ).A choice of basis of Γ(V, L N ) determines a closed immersion i : V → P n−1 .This determines a point [i] of the Hilbert scheme Hilb = Hilb P n−1 /k .The smooth algebraic group PGL n acts on Hilb, and we denote Z the orbit of [i].Note that Z is a smooth scheme of finite type over k, and that there is a flat surjective morphism PGL n → Z. See [?, Generalitiesaboutorbits] By construction the pullback of the universal family over Z to PGL n is canonically isomorphic to V × PGL n , and the invertible sheaf O(1) pulls back to L N O.
The question is local on U so we may assume that U is affine.By our choice of N above, the invertible sheaf N N is very ample on every fibre of X over U with vanishing higher cohomology groups.Hence after possibly shrinking U we can find a closed immersion X → P n−1 U which restricts to a the embedding given by the full linear series of N N on every geometric fibre.Consider the associated moduli map m : U → Hilb.Since U is reduced, and since each pair (X s , N s ) is isomorphic to a base change of (V, L) we see that m(U ) ⊂ Z.
This implies there is some surjective flat morphism U → U such that there is a U isomorphism X ∼ = V × U with the property that N N pulls back to L N .Namely, take U = U × Z PGL n and unwind the definitions.To finish, do the same thing for N + 1 to get some U → U .Then over U = U × U U there is an isomorphism of the pullback of (X, N ) and the base change of (V, L).This proves the reult.
Conversely, given a left G-torsor T over U we will construct a flat proper family of varieties X → U and an invertible sheaf N on X such that Isom U ((X, N ), (V U , L U )) is isomoprhic to T .
The structure morphism π : T → U is a flat surjective morphism of finite type.We are going to descend the constant family V × T to U using a descend datum Before we describe the descent datum, we recall that the map ψ : G × T → T × U T , (g, t) → (gt, t) is an isomorphism.Also, let us denote m : V × G → V the map (v, g) → gv, where gv denote the natural action of g ∈ G on v ∈ V .Finally, we take To verify the cocycle condition on T × U T × U T , we can think of φ as the map (v, gt, t) → (g −1 v, gt, t).If on V ×T × U T × U T we have a point (v, g 1 g 2 t, g 2 t, t) then pr * 23 (φ)(v, g 1 g 2 t, g 2 t, t) = (g 2 v, g 1 g 2 t, g 2 t, t) and pr * 12 (φ) • pr * 23 (φ)(v, g 1 g 2 t, g 2 t, t) = (g 1 g 2 v, g 1 g 2 t, g 2 t, t) and pr * 13 (φ)(v, g 1 g 2 t, g 2 t, t) = ((g 1 g 2 )v, g 1 g 2 t, g 2 t, t).Thus pr * 13 (φ) = pr * 12 (φ) • pr * 23 (φ) as desired.Because all the maps in question lift canonically to the invertible ample sheaf L this actually defines a descent datum on the pair (V, L) for T → U .As L is ample, this descent datum is effective, cf.[Gro62,No. 190,§B.1].Thus there exists a pair (X → U, N ) over U and an isomorphism δ : The above constructions give a bijective correspondence between pairs (X → U, N ) and left G-torsors over U in case U is a reduced scheme over k.
Remark 2.2.3.The construction of the family (X, N )/U starting from the torsor T works more generally when k is more generally a ring as long as: (1) V is a flat projective scheme of finite presentation over k, (2) L is ample, and (3) the automorphism group scheme G = Aut(V, L) is flat over k.

2.3.
Deforming torsors over a Henselian DVR.Before proving Theorem 2.1.3,it is useful to say what is known without the hypothesis that G is reductive.We thank Ofer Gabber, Jean-Louis Colliot-Thélène and Max Lieblich for explaining the following proposition.
Proposition 2.3.1.Let R be a Henselian DVR with residue field K, and let G be a flat separated group scheme of finite type over Spec R. Every torsor for the closed fiber G K over Spec K is the closed fiber of a torsor for G over Spec R.
Proof.We first give a proof when G is affine which is all we will use in this paper.Choose a closed immersion G → GL n,R , see [?, Generalitiesaboutgroupschemes] Let X = GL n,R /G, and note that X → Spec R is smooth, see [?, Generalitiesaboutorbits] Since H 1 (K, GL n,K ) = {1}, any torsor for G K is the fibre of the map GL n,K → X K over a K-point of X.Since R is Henselian the map X(R) → X(K) is surjective, and hence every G K -torsor lifts.In the general case we argue as follows.By [LMB00, Prop.10.13.1], which relies upon Artin's criterion for algebraicity of a stack, the classifying stack BG is an algebraic stack over Spec R. By [LMB00, Thm.6.3], for each G K -torsor there exists an affine R-scheme X, a smooth morphism φ : X → BG, and a K-point x of X such that φ(x) corresponds to the given G K -torsor.Denote by t : Spec R → BG the 1-morphism associated to the trivial G-torsor.Since φ is smooth, the basechange pr R : Spec R × t,BG,φ X → Spec R is smooth.Since t is a surjective flat morphism, the base-change, pr X : Spec R × t,BG,φ X → X is surjective and flat.By [GD67, §6.5], it follows that X is smooth over Spec R. Since R is Henselian and X is smooth over Spec R, X(R) → X(K) is surjective; in particular there is an R-morphism Spec R → X extending the given K-point of X.The composition of this morphism with φ determines a G-torsor over Spec R whose closed fiber is isomorphic to the given G K -torsor over Spec K.
Corollary 2.3.2.Let R be a DVR with residue field k, and let G be a separated, finite type, flat group scheme over Spec R. Let U be a finite type, integral k-scheme, and let T U → U be a G k -torsor.There exists an integral, flat, quasi-projective R-scheme Y , with nonempty special fibre Y k , a G-torsor T → Y , and an open immersion j : Proof.First we show there exists an integral, flat, quasi-projective R-scheme Z and an open immersion j : Z k → U .Namely, we may replace U by an affine open, and hence we may assume that U is regular, see [?, Generalitiesaboutcommutativealgebra] In particular this implies that U → Spec k is a local complete intersection morphism, see [?, Generalitiesaboutcommutativealgebra] So after shrinking U some more we may assume that Define R to be the local ring of Z at the generic point of Z k .Then R is a Noetherian 1-dimensional local ring.Denote by π a uniformizer of R. Clearly, π maps into m R and R /πR is the function field of Z k , i.e., the function field of U .Because R is R-flat, π is a nonzerodivisor.Thus R is a DVR with residue field K = k(U ).By Proposition 2.3.1, the G k torsor over R /πR lifts to a G-torsor T h over the Henselization of R .By a standard limit argument, this lift exists over an étale extension R → R contained in the Henselization of R .Note that the residue field R /πR of R is still the function field of U .By a standard limit argument, there is an étale morphism Y → Z such that Y k → Z k is an open immersion and such that R is the local ring of Y at the generic point of Y k .After replacing Y by an open subscheme, there is a G-torsor T over Y that pulls back to T h over R .We leave it to the reader to see that, after possible shrinking Y again, this torsor satisfies the conditions of the corollary.
Corollary 2.3.2 above is a weak version of the deformation principle that we will esthablish later on.The remaining issue is whether there exists a datum (Y → Spec R, T → Y, j : Y K → U ) such that the generic fiber of Y → Spec R is projective.Presumably this is not always possible, but in case G is reductive we will show that it is.Corollary 2.3.2 can be used to lift problems in char p > 0 to characteristic 0. Suppose that R is a complete discrete valuation ring with algebraically closed residue field k.Let Ω be an algebraic closure of the fraction field of R. We have in mind the case where char(k) = p > char(L) = 0. Suppose that V R is a flat projective R scheme, and that L R is an ample invertible sheaf over V R .We assume that V Ω and V k are varieties.Let G R denote the automorphism group scheme of (V R , L R ) over R.
Corollary 2.3.3.Notations and assumptions as above.Fix d ∈ N. Assume that G R is flat over R. If the answer to Question 2.1.2is always "yes" for the pair (V Ω , L Ω ) then the answer is always "yes" for the pair (V k , L k ).
Proof.Let (K/k, X → S, N ) be a triple as in Situation 2.1.1 for the pair (V k , L k ).Let U be the open subscheme of S over which all geometric fibres of (X, N ) are isomorophic to the base change of (V k , L k ).The construction in Subsection 2.2 gives a corresponding G K -torsor T U over U .There exists an extension of complete discrete valuation rings R ⊂ R such that the induced extension of residue fields is K/k, see [ Let Ω be an algebraic closure of the field of fractions Q(R ) of R .Since R ⊂ R we may and do assume that Ω ⊂ Ω .Note that we do not know that the geometric fibre Y Ω is irreducible.However, our assumptions imply that X has a Ω (Y )-valued point for every irreducible component Y of Y Ω .To conclude we apply the lemma below.
Lemma 2.3.4.Suppose that R is a DVR with algebraically closed residue field K. Let Ω be an algebraic closure of Q(R).Let Y → Spec R be a flat, finite type morphism, X → Y a projective morphism and let ξ ∈ Y K .Assume in addition that (a) ξ is the generic point of an irreducible component C of the scheme Y K , (b) the scheme Y K is reduced at ξ, and (c) for every irreducible component Y of Y Ω there exists a Ω(Y )-valued point of X.Then X has a K(C)-valued point.
Proof.Note that right from the start we may replace R by its completion, and hence we may assume that R is excellent.This will garantee that the integral closure of R in a finite extension of Q(R) is finite over R. (Not necessary in char 0.) See [?, Generalitiesaboutalgebra] By hypothesis, and a standard limit argument, there is a section of extends to a O-valued point of X R , and in particular we obtain a κ(ξ) = K(C)valued point of (X K ) κ(ξ) = X K(C) as desired.
For example this corollary always applies to the case where (V, L) is the pair consisting of a Grassmanian and its ample generator.
2.4.Deforming torsors for a reductive group.Under the additional hypothesis that G is a geometrically reductive linear algebraic group we can prove a stronger version of Corollary 2.3.2.First we prove that BG is proper over k in some approximate sense.
Proposition 2.4.1.Let G be a geometrically reductive group scheme over the field k.For each integer c, there exists a smooth k-scheme X, a smooth morphism φ : X → BG, and an open immersion j : X → X such that (i) X is a projective k-scheme, (ii) for every infinite field K and every morphism Spec K → BG, there exists a lift Spec K → X under φ.
The proof uses geometric invariant theory to construct X ⊂ X.With more care it may be possible to remove the assumption that K be infinite from (ii).
Proof.By definition G is a linear group scheme.Let V be a finite dimensional k-vector space, and let ρ : G → GL(V ) be a closed immersion of group schemes.Consider ρ : G → SL(V ⊕ k ⊕ k) defined by ρ(g) = diag(ρ (g), det(ρ (g)) −1 , 1) (diagonal blocks).Observe that the intersection of Image(ρ) and G m Id is the trivial group scheme.Thus, without loss of generality, assume ρ is a closed embedding of G into SL(V ) such that Image(ρ) ∩ G m Id is the trivial group scheme.In other words, the induced morphism of group schemes Pρ : Denote the dimension of V by n > 1.Let W be a finite-dimensional k-vector space of dimension c.Denote by H the finite-dimensional k-vector space Hom(W, Hom(V, V )).There is a linear action σ : GL(V ) × H → H, where an element g ∈ GL(V ) acts on a linear map h : W → Hom(V, V ) by σ(g, h)(w) = g • h(w).This restricts to a linear action of G on H.
The linear action of G on H determines an action of G on the projective space PH of lines in H.It comes with a natural linearization of the invertible sheaf L := O PH (1) so that the action of G on H 0 (PH, O(1)) = Hom(H, k) is the dual of ρ.Denote by PH ss , resp.PH s (0) , the semistable, resp.properly stable, locus for the action of G on the pair (PH, L).Denote by X the uniform categorical quotient PH ss G and denote by p : PH ss → X the quotient morphism.These exist by [MFK94, Thm.1.10, App.1.A, App.1.C].By the remark on [MFK94, p. 40], X is projective.Also, some power of L is the pullback under p of an ample invertible sheaf on X.Thus (i) is satisfied for X.
For every element w ∈ W − {0}, define F w to be the homogeneous, degree n polynomial on H defined by F w (h) = det(h(w)).For every g ∈ SL(V ), Thus F w is invariant for the action of SL(V ).Thinking of F w as an element of Γ(PH, O(n)) it is invariant for the action of G. Denote by H w ⊂ H, resp.PH w ⊂ PH, the open complement of the zero locus of F w .By what was said above, PH w is contained in PH ss .The next step is to prove that PH w is contained in PH s (0) , and, in fact, the geometric quotient Let W be a subspace of W such that W = span(w) ⊕ W . Denote by H ⊂ H the subspace H = Hom(W , Hom(V, V )).There is a morphism q w : H w → GL(V ) × H defined by the rule h → (h(w), h(w) −1 h| W ). The morphism q w is GL(V )equivariant if we act on GL(V ) × H on the first factor only.There is an inverse morphism r w : GL(V ) × H → H w sending a pair (g, h ) to the unique linear map W → Hom(V, V ) such that w → g and w → gh (w ) for every w ∈ W . Thus, as a scheme with a left GL(V )-action, H w is isomorphic to GL(V ) × H .For the same reason, as a scheme with a PGL(V )-action, PH w is isomorphic to PGL(V ) × H . Thus the categorical quotient of PH w by the action of G is the induced morphism In particular, the action of G on PH w is proper and free so that PH w is contained in PH ss (0) .
Denote U = PH w , where the union is over all w ∈ W − {0}.This is a G-invariant open subscheme of PH s (0) .Therefore there exists a unique open subscheme X ⊂ X such that p −1 (X) = U .By the last paragraph, p : T → X is a G-torsor.Since U is smooth and p is flat, by [GD67, §6.5] also X is smooth.
Associated to the G-torsor U over X, there is a 1-morphism φ : X → BG.There are also morphisms of stacks Finally, the codimension of X −X is at least as large as the codimension of PH −U .Choosing a basis (w 1 , . . ., w d ) for W , PH −U is contained in the common zero locus of F w1 , . . ., F wc , which clearly has codimension c.Therefore X −X has codimension at least c in X.This proves (iii).
Corollary 2.4.2.Let the field k and the group scheme G be as in Proposition 2.4.1.Let R be a DVR containing k with residue field K. Let U be a finite type, integral K-scheme, and let T U → U be a G-torsor.There exists a triple (Y → SpecR, T → Y, j : Y K → U ) as in Corollary 2.3.2 with the additional property that the generic fiber of Y is projective.
Proof.We may assume that dim U > 0. Let c be an integer larger than dim(U ).Let (φ : X → BG, X ⊂ X) be as in Proposition 2.4.1.The torsor T U corresponds to a 1-morphism U → BG.By condition (ii), the base-change morphism Spec K(U ) → BG lifts to a morphism Spec K(U ) → X. (Note that K(U ) is infinite since dim U > 0.) After replacing U by a dense open subscheme, this comes from a morphism f : U → X lifting U → BG.Also, replace U by an open subscheme that is quasiprojective, say a nonempty open affine.Then for some positive integer N , there is a locally closed immersion of K-schemes, f : U → (X × P N k ) K such pr X • f equals f .Denote by m the codimension of f (U ) in (X × P N k ) K .
The scheme (X × P N k ) R is flat and projective over Spec R. Choose a closed immersion in P M R for some positive integer M .As in the proof of 2.3.2 we will use that the scheme U is a local complete intersection at a general point, and we will use that X is smooth over k.This implies that f (U ) is dense in a component of a complete intersection of (X × P N k ) K in P M K .More precisely, for some positive integer e, there exist homogoneous, degree e polynomials F 1 , . . ., F m on P M K such that the scheme Y K := V(F 1 , . . ., F m ) ∩ (X × P N k ) K has pure dimension dim(U ) and contains a nonempty open subscheme U that is an open subscheme of f (U ).Let F 1 , . . ., F c be homogeneous, degree e polynomials on P M R such that for every i = 1, . . ., m, ( * ) Denote by Y the zero scheme It is easy to see that the set of all possible choices of F i satisfying ( * ) forms a Zariski dense set of points in the relevant vector space of degree e polynomials over the field of fractions Q(R) of R. Thus the dimension count shows there exists a choice of Let η be a generic point of Y that specializes to the generic point of U .Replace Y by the closure of η, so that now Y is integral.(Presumably, a suitable application of Bertini's theorem could be used to replace this step.)Then Y is an integral, flat, projective R-scheme, the closed fiber contains U as an open subscheme, and the generic fiber is contained in Spec R × Spec k (X × P This is an integral, flat, quasi-projective R-scheme whose generic fiber is projective.Moreover, Y K equals U , which admits a dense, open immersion in S. Finally, the projection pr X : Y → X, and the 1-morphism φ • pr X : Y → BG determine a G-torsor T over Y .By construction, the restriction of this G-torsor to U is isomorphic to the pullback of T U by the open immersion, as desired. We remark that we did not claim that the generic fibre of Y → SP (R) is geometrically irreducible.Since X is smooth and geometrically irreducible over k, it seems that with a carefull choice of the F i and some additional arguments one can obtain this property as well maving Y Q(R) smooth over Q(R).
Next we deduce a corollary to help prove Theorem 2.1.3.Let k be an algebraically closed field, and let (V, L) be a pair of a projective k-scheme and an ample invertible sheaf.Denote by G/k the group scheme G = Aut(V, L).Let (K/k, X → S, N ) be as in Situation 2.1.1.Denote by G • red the reduced, connected component of the identity of G.
Corollary 2.4.3.Notations as above.Let R be a DVR containing k and with residue field K.If G • red is reductive, there exists an integral, flat, quasi-projective R-scheme Y , a projective, flat morphism f : X → Y , an invertible sheaf N on X, and an open immersion j : Y K → S such that: (i) every geometric fiber of ( X, N ) over Y equals the base-change of (V, L), (ii) the restriction of ( X, N ) to Y K is isomorphic to the pullback of j * (X, N ), and (iii) the generic fiber of Y → Spec R is projective.In particular, let S be an irreducible component of the geometric generic fibre of Y → Spec R. Then ( X → S , N ) over R is a triple (K /k, X → S , N ) with empty discriminant.
Proof.If G  By construction, this has empty discriminant.By hypothesis, the generic fiber of X → S has a K (S )-point.At this point we apply Lemma 2.3.4 to conclude.

Simple applications
We are going to apply this in a more serious way to Grassmanians, but let us indicate some simple applications of Theorem ?? this section.
3.1.Fermat Hypersurfaces.As a first case we take V a Fermat hypersurface of degree d in ), say over the complex numbers C. In this case the group scheme G is an extension of a finite group by G m so certainly reductive.Consider the following family with general fibre (V, L) over P 2 : We learned about this family from Tom Graber.This family does not have a rational point over k(P 2 ).Reference [?][TomGraber].The reader may enjoy finding an elementary proof of this by looking at what it means to have a polynomial solution to the above.We conclude from Theorem 3.1 that there is a smooth projective family over a projective surface with every fibre isomorphic to (V, L), without a rational section.We like this example because it is not immediately obvious how to write one down explicitly.
There is another reason why the family given by ( * ) is interesting.Tsen's theorem asserts that, if n ≥ d 2 then any degree d hypersurface X ⊂ P n F , where F is the function field of a surface has a rational point.The authors of this paper wonder what the obstruction to the existence of a rational point is in the boundary case, namely degree d in P d 2 −1 .One guess is that it is a Brauer class, i.e., an element α in the Brauer group of F such that for finite extensions F /F one has: X(F ) = ∅ ⇔ α| F = 0. (Compare future work of the authors.)However, the example above shows that this is not the case.Namely, in our example F = C(x, y) where x = X/Z and y = Y /Z.Anand Depokar pointed out that ( * ) obtains a rational point over F (ξ) where ξ is a dth root of a nonzero polynomial of the form (Just take T 0 = ξ and T i+jd = a 1/d i,j .)Let C ⊂ P 2 be an irreducible curve, not the line at infinity Z = 0. Suppose that α ramifies along C. The ramification data gives a cyclic extension There is a choice of of a i,j such that the rational function f (x, y) restricts to a rational function on C such that both f | C and g −1 f | C are not d th powers.(Left to the reader.)Thus the pullback of α to F is still ramified along the pullback of C to the surface whose function field is C(x, y)(ξ).Contradiction.Hence C does not exist.However, the only Brauer class on P 2 ramified along a single line is 0.

Projective spaces.
Another case is where we take the pair (V, L) to be (P n , O(n + 1)).Note that O(n + 1) = ω −1 P n so the families in question are canonically polarized, and we are just talking about the problem of having nontrivial families of Brauer-Severi varieties.In particular, our theorem reduces the problem of proving the nullity of the Brauer group of a curve to the problem of proving the nonexistence of Brauer-Severi varieties having no rational sections over projective nonsingular curves.As far as we know this is not really helpfull, since the proof of Tsen's theorem is pretty straigthforward anyway.However, it illustrates the idea!

The period-index problem
In this section we show how to use the ideas above to find a proof of the period index problem for Brauer classes over surfaces.We refer to the paper [?] and the lecture [?] for a more thorough discussion of the problem.4.1.Brauer classes and Grassmanians.Suppose that F is a field and that α ∈ Br(F ) is a Brauer class.The period of α is the smallest integer m > 0 such that mα = 0.The index of α is the smallest integer e > 0 such that there is a central simple algebra B of degree e representing α.Fix a central simple algebra of some degree d representing the class α, i.e., dim F A = d 2 .Note that the index of α divides d.For each integer 1 ≤ e ≤ d consider the variety X e parametrizing right ideals of rank ed in A. In other words a T -valued point of X e corresponds to a sheaf of right ideals I ⊂ O T ⊗ A, locally a direct summand of rank ed as O T -modules.Note that if X e has a rational point I ⊂ A, then the commutant of A acting on I is an algebra B of degree e representing −α.Hence the index of the Brauer class α is the smallest integer e such X e has a rational point.Geometrically the varieties X e are isomorphic to Grassmanians.Namely, suppose we choose an isomorphism ψ : In this way, using ψ, we obtain the isomorphism with the Grassmanian of e-dimensional subspaces of the standard d-dimensional vector space.Since Pic(Grass(e, d)) = Z with a canonical ample generator, we see that each X e gives rise to a Brauer class α e , namely the obstruction to descending this ample generator to an invertible sheaf over X e .
Proof.With the notation above we can write down the Galois cocycle of α.Namely, the isomorphism class of A is given by the 1-cocycle ∂ψ = {σ → σ(ψ) with exact rows and columns.
Proof.Clearly the group scheme G m × GL m acts on the pair (Grass(n, m), O(1)).
We leave it to the reader to show that one obtains the short exact sequences of the statement.(Note that if n = m/2 there is an additional automorphism of Grass(n, m) coming from taking duals.) Proof of Theorem 1.0.2.Let S be a surface over k = k, and let F = k(S).Let α ∈ Br(F ) = Br(k(S)).Lemma 4.1.1means that if we take e equal to the period of α then the variety X e carries an ample invertible sheaf which is geometrically an ample generator of Pic(Grass(e, d)).We conclude that the period-index problem for the field F is equivalent to the problem of finding rational points on forms of Grassmanians which are endowed with an invertible sheaf that is geometrically an ample generator for the Picard group.It suffices to do this in all cases where e < d/2 since we can always replace A by Mat(2 × 2, A) to artificially increase d.
Thus it suffices to prove the answer to Question 2.1.2is always yes in the following situations.Namely, k is an algebraically closed field, the integer d = 2, and the pair (V, L) is the pair (Grass(n, m), O(1)), where 1 ≤ n < m/2.Lemma 4.1.2shows that Aut(V, L) is reductive.
By Theorem 2.1.3it suffices to consider only those cases where the surface S is projective.By resolution of surface singularities we may also assume that the surface is smooth.This case is Theorem 4.2.1 below.
4.2.Families of Grassmanians over smooth projective surfaces.In this subsection let S be a smooth projective surface over the algebraically closed field k.Suppose that X → S is a projective morphism and that N is an invertible sheaf on X. Assume that 1 ≤ n < m/2 and assume that for every point s ∈ S(k) the fibre X s is isomorphic to the Grassmanian Grass(n, m) and that N restricts to the ample generator of the Picard group.
Theorem 4.2.1.With notations as above.The morphism X → S has a rational section.
The rest of this subsection is devoted to the proof of this theorem.Some of the discussion that follows applies more generally to the problem of finding a rational section of any family of varieties over a surface.Note that we do not assume that the characteristic of k is 0 so we have to be a little carefull.Let {C t , t ∈ P 1 } be a Lefschetz pencil on S. We replace S by the blow up in C 0 ∩C ∞ and we replace X by the pullback to this blow up.Thus we obtain is the fibre over t ∈ P 1 .We denote ∆ ⊂ P 1 the discriminant locus of f (a finite closed subscheme).We write X t = h −1 (t) for the fibre of X → P 1 k over t ∈ P 1 k .From now on we think of S as a family of curves over P 1 and of X as a family of families of Grassmanians over curves.After replacing N by N ⊗ g * (M) for some very ample invertible sheaf on S we may assume that N is ample on X.Let p a = p a (C t ) be the common value of the arithmetic genus of the curves C t (of course this is just the genus of C t for t not in ∆).For d ∈ N set P d (X) = dX + 1 − p a ∈ Z[X].and set Hilb d = Hilb P d X/P 1 ,N −→ P 1 .In other words, this is the projective scheme over P 1 whose fibres Hilb d,t parametrize 1-dimensional closed subschemes Z ⊂ X t whose arithmetic genus is p a and whose N -degree is d.Let us denote ι : Z d −→ X × P 1 Hilb d the universal family of closed subschemes.Note that g induces a morphism between flat projective families of 1-dimensional schemes over Hilb d as follows x x q q q q q q q q q q Hilb d We are only going to use a small piece of this Hilbert scheme.Let us denote U d ⊂ Hilb d the open subscheme whose geometric points correspond to closed subschemes Z ⊂ X t such that (1) Z is a nonsingular irreducible curve, m −1 N (τ (P 1 )) → P 1 .Theorem 4.2.4 gives information on the geometric generic fibre of m τ , namely it implies that this is a rationally connected variety.In characteristic zero this suffices, using the theorem of Graber-Harris-Starr [GHS03], to conclude there exists a lift of τ to a rational section ρ : P 1 → U of U → P 1 .In characteristic p > 0 we can use Lemma 4.2.5 below to see that we may compactify the generic fibre F η of m τ so that all the conditions of the main theorem of [dJS03] are satisfied.Hence in this case we obtain ρ as well.
By Lemma 4.2.2 the pullback ρ * Z is birational to S. Thus is the desired rational section of X → S.
Lemma 4.2.5.Let K be a field and let V be a variety over K. Suppose that V K has a compactification V K ⊂ V K which is a projective normal variety with boundary of codimension ≥ 2. Then there is a projective normal compactification V ⊂ V with boundary having codimension ≥ 2.
Proof.Let K ⊂ L ⊂ K be a finite extension of K over which the compactification is defined.Let L L be an ample invertible sheaf over V L which is the restriction of an ample invertible sheaf on VL .If we can pick L such that it is the pullback of an invertible sheaf on V then we are done.(Left to the reader.) In characteristic zero this does not pose a problem since we may assume that L/K is Galois with group G and then we can simply replace L by ⊗ σ∈G σ * L which descends.In characteristic p > 0 this reduces the question to the case where L/K is purely inseparable.However in this case the invertible sheaf L n with n = [L : K] will descend since the n-th power of an element of the structure sheaf of V L is an element of the structure sheaf of V .
Proof of 4.2.4.Let t be a geometric point of P 1 not in the discriminant locus ∆ of S → P 1 .Thus C = C t = f −1 (t) is a nonsingular geometric fibre of S → P 1 .Let Y = X t → C be the family of Grassmanians over C. Then U d,t is an open subscheme of the Hilbert scheme of Y , consisting of points that correspond to sections of the morphism Y → C, with a suitable stability condition as in Lemma 4.2.3.We claim that to prove Theorem 4.2.4 it suffices to prove the corresponding statements for U d,t → Pic d C for all geometric points t of P 1 −∆.The only somewhat subtle point is that the bounds implicit in the statement d >> 0 may depend on t.There are two ways to deal with this issue.The first is to notice that in the proof of Theorem 4.2.1 we only every used the case where t is a geometric generic point of P 1 .The second way to deal with this is to notice that once the assertions of Theorem 4.2.4 have been proven for all θ lying over a geometric generic point of P 1 , then the assertions hold for all geometric points θ lying over a fixed Zariski open of P 1 − ∆.The additional finite list of points can then be dealt with separatedly.
From now on we work with a fixed family of Grassmanians Y → C, and ample invertible sheaf N ∈ Pic(Y ) over a fixed algebraically closed field as above.As before the ample sheaf restricts to an ample generator on the fibres.The corresponding Hilbert scheme is where B ∈ Pic d (C) and where A is an invertibe sheaf of very negative degree.A general element e ∈ E corresponds to a stable rank n sheaf, and we obtain a rational map from the projective space of lines in E to the fibre of det over [B].One checks that this morphism is defined away from codimension 2 and that it is generically smooth.A general high degree rational curve in P(E) maps to the desired curve a.
[H/G] → BG and [PH/G] → BG (because BG = [Spec k/G].By construction, X is 2-equivalent to an open substack of [PH/G] as a stack over BG.The morphism [PH/G] → BG is smooth, since PH is smooth.Hence X → BG is smooth.For every field K and 1-morphism Spec K → BG, the 2-fibered product Spec K × BG [H/G] is a K-vector space, and Spec K × BG [PH/G] is the associated projective space.Thus Spec K × BG [PH/G] ∼ = P cn 2 −1 .Finally, Spec K × BG X is a nonempty open subscheme of Spec K × BG [PH/G].Since K is infinite every dense open subset of P dn 2 −1 K contains a K-point.This proves (ii).
Henselian, contains k and has residue field K. Let X → Y → Spec R and N be as in Corollary 2.4.3.Denote by Ω/k an algebraic closure of the field of fractions Q(R) of R. Let S be any irreducible component of Y Ω and let X = X| S , N = N | S .Thus (Ω/k, X → S , N ) is a triple as in Situation 2.1.1.
Hilb d = Hilb P d Y,N , which is now a scheme over the ground field, and we have open subschemes U d ⊂ U d ⊂ Hilb d as above.(In other words we have dropped the subscript t from the notation.)The Brauer group of C is trivial, hence we can write Y = Grass(n, V) for some locally free sheaf V of rank m.Let W univ ⊂ V ⊗ O Y denote the universal locally direct summand.Comparing fibre by fibre we observe that N | Y ⊗ ∧ n W univ ∼ = (Y → C) * H for some invertible sheaf H on C. Thus a section Z = σ(C) of Y → C corresponds to a locally direct summand W ⊂ V of rank n.The stability condition of Lemma 4.2.3 corresponds to the stability condition for this same W.The corresponding point in Pic d C is the point corresponding to the invertible sheaf (∧ n W) −1 ⊗ H. Hence the degree of Z with respect to N is d = − deg(W) + deg(H).Thus increasing d corresponds to decreasing d = deg W. We will see below that the integer h(t) of the statement of Theorem 4.2.4 can be taken to be h(t) = − deg(H).Let M s (C, n, d ) denote the coarse moduli scheme of rank n degree d stable locally free sheaves on C. The arguments above show that there exists a sequence of morphisms U d → M s (C, n, d ) → Pic d C → Pic d C factoring the morphism of the Theorem.The last of these morphisms is an isomorphism.The first morphism is identified with an open subscheme of a projective bundle.Namely, the fibre over the point [W] is the open subschemes inside P(Hom(W, V)) corresponding to maps W → V having maximal rank in all points of C. Since W is stable, if d >> 0, so d = deg(W) << 0 then the codimension of the complement is m − n − 1.For later use we describe this projective bundle a little better.There is a surjective étale morphism T → M s (C, n, d ) such that a tautological locally free sheaf W T over T × C exists.After refining T → M s (C, n, d ) we may assume that there exists an isomorphism γ : pr * 1 W T −→ pr * 2 W T over T × M s (C,n,d ) T × C.Over the triple fibre product T × M s T × M s T × C the combination pr * 23 γ • pr * 12 γ differs from pr * 13 γ by an invertible function on T × M s T × M s T .(This invertible function defines a Brauer class on M s which is the obstruction for the existence of a tautological bundle.)All of this means that we can consider the projective bundle P(pr 1, * (Hom(W T , pr * 2 V))) over T .The isomorphism γ induces an isomorphism of projective bundles and the cocylce condition for γ over the triple fibre product implies that this is a descent datum.The conclusion is that we get a projective bundle P → M s (C, n, d ) and an open immersion z u u u u u u u u u u u M s (C, n, d ) .The morphism det : M s (C, n, d ) → Pic d C is smooth.This fact corresponds, via deformation theory, to the fact that H 1 (Tr) : H 1 (C, End(W)) → H 1 (C, O C ) is surjective for every locally free sheaf W.There is a morphism a : P 1 → M s (C, n, d ) so that the pull back of the tangent sheaf T M s (C,n,d )/Pic d C is ample.One way to see this is to consider the Ext group E = Ext C (B ⊗ A ⊗−n+1 , A ⊕n−1 ) Generalitiesaboutalgebra]. We apply Corolloray 2.3.2 to obtain Y → Spec R , T → Y and j : Y K → U .According to Conclusion 2.2.2 and Remark 2.2.3 there exists a pair • red being reductive implies that it is a geometrically reductive group scheme over k by a result of Haboush, see [?, GIT] Note that G • red is a closed normal subgroup scheme of G and that the quotient G/G • red is a finite group scheme.A finite group scheme over k is geometrically reductive, and an extension of geometrically reductive group schemes is reductive, see [?, GeneralitiesaboutGIT] Hence G is geometrically reductive.Thus the result of the Corollora follows from Corollary 2.4.2 above by applying the bijective correspondence of Conclusion 2.2.2.Proof of Theorem 2.1.3.Let us start with an arbitrary triple ( F * ) to lift this 1-cocycle ∂ψ to a 1-cocycle for GL d , i.e., the boundery of ∂ψ for the short exact sequence G m → GL m → PGL m .The obstruction to descend the ample invertible sheaf O(1) on Grass(e, d) to F is the boundary of the 1-cocycle ∂ψ for the short exact sequence G m → G e,d → PGL d , see Lemma 4.1.2for notation.By Lemma 4.1.2below this boundary is d − e times α as desired.The automorphism group scheme of Grass(n, m) is canonically isomorphic to PGL m .(2) The automorphism group scheme G n,m of the pair (Grass(n, m), O(1)) the extension of PGL m by G m fitting into the following diagram The description of the morphism U d → M s (C, n, d ) above implies that we can lift the morphism a to a morphism b : P 1 → U d .By adding lines in fibres of U d → M s (C, n, d ) and smoothing we can make sure that the restriction of the tangent sheaf T U d /Pic d This implies that M s (C, n, d ) is a projective variety.Thus in this case the projective bundle P → M s (C, n, d ) is itself a projective variety, and the fibres of the induced morphism P → Pic d C are the desired compactifications.