The Picard Scheme and the Dual Abelian Variety

The Picard Scheme and the Dual Abelian Variety Gabriel Dorfsman-Hopkins May 3, 2015 Contents 1 Introduction 2 1.1 Representable Functors and their Applications to Moduli Problems............... 2 1.2 Conditions for Representability................................... 3 1.3 Motivating Examples......................................... 4 2 The Picard Functors 5 3 The Picard Scheme 7 3.1 Relative Effective Divisors...................................... 7 3.2 The Module Q............................................ 9 3.3 Construction of the Picard Scheme................................. 9 4 Basic facts about the Picard Scheme 13 5 The Dual of an Abelian Variety 15 5.1 The Connected Component of the Identity of the Picard Scheme................ 16 5.2 Basic Theory of Abelian Varieties.................................. 17 5.3 Translation Invariant Line Bundles................................. 19 5.4 The Dual is a Dual.......................................... 21 1

1 Introduction 1.1 Representable Functors and their Applications to Moduli Problems Recall from [Har77, (II.2.7)] that a point x on a scheme X is equivalent to a map Spec k(x) X whose image is x (where k(x) is the residue field of x). Thus it is reasonable to think of the points of a scheme to be the maps to it from the spectra of fields. In fact, more generally, for a scheme T, we call a map T X, an T -point of X. This turns out to be a very useful convention, and is made rigorous in the following way. Let C be the site of schemes over a fixed scheme S, in the Zariski, étale, or fppf topology. For every scheme X C, we assign a functor to the category of sets as follows: h X : T Hom C (T, X). This functor is called the functor of points of X, which agrees with the convention of considering the points of X to be the maps into it. Yoneda s Lemma tells us that the association X h X is a fully faithful functor the the category of presheaves on C (i.e, the category of functors C op Set whose morphisms are natural transformations). Thus we can view C as a full subcategory of its category of presheaves, associating each scheme X with its functor of points. In order to lighten notation, it is conventional to use X to mean both the scheme and the functor, so that X(T ) = Hom(T, X) for all schemes T. A natural question is how to tell which presheaves on C actually come from schemes in this way, i.e., if F : C op Set, when is F = h X for some X C? If F is the functor of points of X, then X is said to represent F, or more loosely, that F is a scheme X. This is often referred to as Grothendieck s existence problem. First let us see an example of why we would care about this. Loosely speaking, it is often easier to describe a scheme in terms of its functor of points, so that we can control the properties we would like it to have. This is particularly clear when attempting to construct moduli spaces, as we can see in the following intentionally vague example. Example 1.1 Define a functor F : C op Set by T { stuff we like on X S T } (where stuff we like could mean line bundles, subschemes, effective divisors, quasicohoherent sheaves, etc...). Furthermore, suppose that this functor is represented by a scheme Z. Then we notice that maps T Z parametrize the objects we like on X S T. In particular, the sections of Z S parametrize the objects we like on X. Say for instance that S is the spectrum of an algebraically closed field. Then the closed points of Z parametrize the things we like on X. Thus Z is a parameter space for the objects we hope to study. This already shows the utility of defining schemes in this way, but it gets even better by utilizing the raw power of Yoneda s Lemma. Notice that F (Z) parametrizes maps Z Z, so in particular there is a thing we like W F (Z) corresponding to id Z. This is often call the universal thing we like for the following reason. Fix any V F (T ). This is equivalent to a map f : T Z. We consider the following diagram. Hom(Z, Z) F (Z) f F f Hom(T, Z) F (T ), Chasing id Z both ways, we see that V = F f(w ). We summarize this by saying that for all things we like V F (T ), there is a unique map f : T Z such that F fw = V. Thus W is indeed universal, in the sense that it has a unique way to transform into any thing we like. Often F f is pullback, or scheme theoretic preimage, or some other explicit thing we can compute. Thus, this universal object gives an explicit correspondence between points of Z and things we like. To get a bit more specific, suppose things we like are closed subschemes flat over the base. Then Z is the Hilbert Scheme. Suppose things we like are invertible sheaves. Then Z is the Picard scheme, 2

which is the subject of this paper. Suppose things we like are degree 0 line bundles on a smooth curve over an algebraically closed field Then Z is the Jacobian variety. And so on. 1.2 Conditions for Representability A first necessary condition discovered by Grothendieck, and detailed by Mumford in [Mum67, (p. 115)], is that a representable functor must be compatible with faithfully flat descent. In particular, this means that it must be a sheaf in the fppf topology, which in turn implies it is a sheaf in the étale and Zariski topologies as well. This is helpful for eliminating certain functors from consideration (i.e., presheaves that are not sheaves), but does not help in constructing sheaves. Grothendieck in [GD71, Ch 0., Section 4.5, pp 102-107] details how functors can be patched together in order to build schemes. We highlight the relevant results here. Recall that we are working in a fixed site C of schemes over S. Definition 1.2. Let F : C op Set. A subfunctor F : C op Set is a functor equipped with a natural transformation F F which is a monomorphism in the category of presheaves. Equivalently, F (T ) F (T ) is an injection of sets for all T C. F is called an open subfunctor if for all schemes T (viewed in the usual way as its functor of points in the presheaf category), and all maps T F, the fibered product T F F exists and is represented by a scheme T, and furthermore the projection T T is an open immersion. A collection {F i F } of open subfunctors is called an open cover of F if for all T F the induced open immersions {T F F i = T i T } form an open cover of T. Theorem 1.3 Suppose F : C op Set. Let {F i F } be an open cover of F such that each F i is representable. Then F is representable. Proof [GD71, (0, 4.5.4)]. This last theorem is instrumental in the construction of the Picard Scheme, as we can cover the functor we are trying to represent by functors which we can glue together. It also implies the following result. Proposition 1.4 Representability of a functor F : C op Set is Zariski local on S. Proof [GD71, (0, 4.5.5)]. A natural question to ask is whether a quotient of schemes in the presheaf category is also a scheme. First we must make sense of what we mean by a quotient. Definition 1.5. A diagram A B in a category is an equivalence relation if it is an equivalence relation of sets on all T -points. The coequilizer of this diagram is called the quotient of the equivalence relation. We can ask the following question. If F is a presheaf, and Z a scheme with a map Z F such that Z F Z Z is an equivalence relation. Then if Z F Z Z F is a coequilizer in the presheaf category, must F be a scheme too? This is unfortunately too much to hope for, even if we impose that Z F Z be a scheme. But if, also insist that F be an étale sheaf, it turns out that a flat and proper equivalence relation on a quasiprojective scheme is effective, as stated in the following proposition. 3

Proposition 1.6 Let α : Z F be a surjective map of étale sheaves. Assume Z is representable by a quasi-projective S-scheme, Z F Z is representable by an S-scheme, and the projection Z F Z is a smooth, flat, and proper map. Then F is representable by a quasi-projective S-scheme and α is a smooth map. Proof [AK80, (2.9)]. Remark 1.7 Finally, we will make a trivial remark. Let F : C op Set be a functor in the presheaf category of C. A section λ F (T ) is equivalent to a morphism T F in the presheaf category. This is really just a restatement of Yoneda s Lemma, but it turns out to be an incredibly useful way of viewing sections of sheaves, and one we will use repeatedly. The proof is quite simple. Given a map T F, we can let λ be the image of the identity under the induced map T (T ) F (T ). Conversely, given λ F (T ), we can map T F on T points by mapping g : T T to F g(λ) F (T ). It is easy to check that these are inverses. 1.3 Motivating Examples The exponential function f(x) = e 2πix gives a short exact sequence of abelian groups 0 Z C f C 1, (1) where C is the unit group of C, namely C \ {0} under multiplication. Let X be a reduced projective variety, and Z, O X, O X the sheaves of holomorphic functions with values in Z, C and C respectively. This induces the following short exact sequence of sheaves on X, 0 Z O X O X 1. Since the global holomorphic functions on a projective variety are the constant functions, taking global sections of this sequence is produces Sequence 1. Thus we can consider the long exact sequence on cohomology starting at H 1, 0 H 1 (X, Z) H 1 (X, O X ) H 1 (X, O X) H 2 (X, Z) (2) It is well known that H 1 (X, O X ) = Pic(X), the group of line bundles on X. Furthermore, suppose that X is a nonsingular curve of genus g. Then H 2 (X, Z) = Z and the map Pic X Z is the degree map, whose kernel is Pic 0 X, the group of degree 0 line bundles on X. Therefore we see that Pic 0 X = H1 (X, O X ) H 1 (X, Z) In such a way, Pic 0 X can be given the natural structure of a complex torus. 1 = C g /Z 2g. (3) The natural question to ask is whether the resulting analytic torus is in fact an algebraic variety; a question whose answer is not immediately apparent here. As it turns out, there is a completely algebraic formulation of this construction, which allows for the an interpretation of Pic X and Pic 0 X as schemes in an impressive amount of generality. This construction also sheds light on a duality theory for commutative group schemes using the space of degree 0 line bundles. As a motivation for this duality theory, let X be an abelian variety over C. Then in particular, X is a complex torus, of the form C n /Λ for n = dim X, and Λ a lattice of rank 2n. Then we can construct a dual space ˆX = Hom(X, S 1 ) = Hom(C n /Λ, R/Z) = Hom(Cn, R) Hom(Λ, Z), 1 In fact, one needs to show that the embedding H 1 (X, Z) H 1 (X, O X ) is a lattice embedding. See [Har77, (B.5)] or [Mum74, (Lecture 3)] for a more rigorous treatment. 4

which again is a complex vector space of dimension n modulo a lattice of dimension 2n, and hence has the structure of a complex torus. We will make this duality precise and algebraic, and relate it to the space of degree 0 line bundles constructed above. 2 The Picard Functors Recall that the Picard group Pic X of a scheme X is the group of isomorphism classes invertible sheaves (equivalently locally free sheaves of rank 1, or line bundles), under tensor product, with the structure sheaf serving as an identity element. Futhermore, if f : X Y, then the pullback map f : Pic Y Pic X is a group homomorphism. It is well known [Har77, (III.4.5)] that Pic X = H 1 (X, OX ). We hope to build a parameter space of line bundles on a scheme X. From here on we work in a fixed site C of schemes over a fixed scheme S, in the Zariski, étale, or fppf topologies. Let f : X S be a fixed map, separated and of finite type. To lighten notation, for all S-schemes T, and we will let X T denote X S T and let f T : X T T be the projection (equivalently, the pullback of f along T X). Our goal is to build a parameter space of line bundles on X. By Example 1.1, a natural way to do so is to make the following definition. Definition 2.1. The absolute Picard functor Pic X : C op Ab to abelian groups is defined by Pic X (T ) := Pic X T. (4) Note that this is indeed a functor, in fact, a map T T, induces X T X T and so pullback of sheaves defines a map Pic X (T ) Pic X (T ). If it were representable, we would have our parameter space just as in Example 1.1. Unfortunately, the absolute Picard functor is almost never representable, in fact, it is almost never even a sheaf! To see this, fix an S-scheme T, and suppose L Pic T were a line bundle such that ft L Pic X(T ) were not trivial. Since L is invertible, there is an open cover T T such that L T = O T. But then X T X T is also a cover, and (ft L ) XT = f T L T = O XT. Thus f T L is a nontrivial section of Pic X(T ) which becomes trivial on a cover, so Pic X cannot be a sheaf. Indeed, if we were to sheafify the absolute Picard functor, the pullback of every sheaf on T to X T would become trivial. This motivates the following definition. Definition 2.2. The relative Picard fucntor Pic X/S : C op Ab is defined by Pic X/S (T ) := Pic X T ft. (5) Pic T Although it is true that the relative Picard functor is not a priori a sheaf, it is surprising that in practice it is often representable. The discussion above shows that if + represents the sheafification in the Zarski, étale or fppf topology, the diagram + Pic X (Pic X ) + Pic X/S commutes. Thus, in particular, the absolute and relative picard functors sheafify to the same things. Let Pic X/S(zar), Pic X/S(ét), and Pic X/S(fppf) denote the sheafifications in each Grothendieck topology. Notice that each Picard functor maps naturally to the next in the presheaf category, and that each one is the sheaf associated to each of its predecessors in the indicated topology. In fact, it is easy to check that the natural maps are the sheafification maps. 5

Remark 2.3 Notice that sections of the sheaves given above are locally line bundles. That is, suppose λ Pic X/S(ét) (T ). Then we can represent λ by a cover T T and a line bundle L on X T, and the restriction of λ to T is L. Here I use the word is loosely, meaning that λ T is the image of L under the sheafification map. Another way to state this fact is in terms of Yoneda s Lemma. An element λ Pic X/S(ét) is a map T Pic X/S (ét) in the presheaf category. Thus we can restate the previous paragraph in the following way. There is an étale cover T T such that the composition T T Pic X/S(ét) factors through Pic X. Plainly, similar considerations are true in the Zariski and fppf topologies. In fact, in certain cases the loose use of the word is in the previous remark can be made rigorous, i.e., we can actually make an association between being in the image under the sheafification map, and being a class of invertible sheaves, i.e., a section of the relative Picard functor, due to the following theorem. Theorem 2.4 Assume O S f O X holds universally, that is, for all S-schemes T, the comorphism induced by f T : O T f T O XT is an isomorphism. Then, 1. The natural maps are injections: Pic X/S Pic X/S(zar) Pic X/S(ét) Pic X/S(fppf). (6) 2. Furthermore, if f has a section, then the maps are all isomorphisms. Proof To prove (1) we first show that Pic X/S Pic X/S(fppf). Fix λ Pic X/S (T ) and suppose it sheafifies to 0. We can represent λ by making a choice of L Pic X T in its coset class. Then L Pic X (T ) sheafifies to 0. Thus there is an fppf cover p : T T so that p X L = O XT. Since O S f O X holds universally, this implies that f T p X L = O T. But p is flat, so due to [Har77, (III.9.3)] we know that f T p XL = p f T L so that in particular, f T L is locally free of rank 1. Thus, L = f T f T L f T Pic T, and so the class of L in Pic X/S (T ) is trivial, and λ was 0 to begin with. The rest follows immediately, because sheafification is exact. For a detailed proof of (2), see [Kle05, (2.5)]. Thus in particular, injectivity holds in the following situation. Lemma 2.5 Suppose f : X S is proper and flat, and its geometric fibers are reduced and connected. Then f O X holds universally. O S Proof [Kle05, (3.11)]. Lemma 2.6 Suppose T = Spec A where A is a local ring. Then the natural maps are isomorphisms: Pic X (A) Pic X/S (A) Pic X/S(zar) (A). If A is Artin local with algebraically closed residue field, Pic X (A) and if A = k = k is an algebraically closed field, Pic X (k) Pic X/S(ét) (A), Pic X/S(fppf) (k). 6

Proof If A is local, it has trivial Picard group, so the first map is an isomorphism. Suppose L Pic X (A) maps to 0. Then there is a Zariski cover T T such that L XT = O XT. But since A is local, the only open set containing the closed point is T. Thus by restricting further, we may assume T = T, and so L is trivial. Thus the map is injective. Suppose λ Pic X/S(zar) (T ), and represent it locally by a Zariski cover T T and an invertible sheaf L on X T. Again, we may assume T = T, so that λ is actually the image of a sheaf on X T, and the map is surjective. Thus it is an isomorphism. Suppose A is Artin local with an algebraically closed residue field. Then any étale cover of T is a disjoint union of open subschemes, each isomorphic to T by [GD66, (4.17.6.2 and 4.17.6.3)], so by identical reasoning to the previous paragraph, the third isomorphism holds. Suppose A = k = k is an algebraically closed field. Then every fppf cover T T has a section. Thus the restriction of any sheaf to a cover can be further restricted back to T, so by the same reasoning the fourth isomorphism holds. In particular, we see that all 5 Picard functors have the same geometric points. Combine this with the fact that every section is locally an invertible sheaf as in Remark 2.3, and we see that that even though our first natural choice is rarely in practice representable, sheafification does not kill off all the desired information, and we still have a hope to get an interesting and useful parameter space as in Example 1.1. 3 The Picard Scheme Definition 3.1. If any of the functors defined in the previous chapter is representable by a scheme, we call the representing scheme the Picard scheme, and denote it by Pic X/S. If this is the case, we just say Pic X/S exists. Notice that even though we have multiple Picard functors, if more than one is representable, then they are representable by just one scheme. For example, suppose Pic X/S represents Pic X/S(zar). Then Pic X/S(zar) must already be sheaf in the étale and fppf topologies, so Pic X/S also represents the associated sheaves since they are the same. On the other hand, it doesn t necessarily represent Pic X/S, but if it did then Pic X/S would already have to be an fppf sheaf and thus must be equal to Pic X/S(zar), and therefore would also be represented by Pic X/S. In this section we will prove Grothendieck s main theorem on the Picard scheme, which states that Pic X/S exists if X S is flat, projective, and has integral geometric fibers. We will do this using the techniques we described in Section 1.2, by patching together the Picard functor using representable open subfunctors. Doing so will also give us insight into the geometric structure of the Picard scheme. In order to do the construction we will need a few definitions. 3.1 Relative Effective Divisors Definition 3.2. A closed subscheme D X is called an effective divisor if its ideal sheaf I D is invertible. Furthermore, if it is S-flat we call it a relative effective divisor on X/S. We say that D X is a relative effective divisor at x X if it is a relative effective divisor in a neighborhood of x. Lemma 3.3 Let D X be a closed subscheme, x D a point, and s S its image. Then D X is a relative effective divisor at x if and only if X s and D s are s-flat at x, and the fiber D s X s is a relative effective divisor of X s / Spec k(s) at x. Proof [Kle05, (3.4)]. 7

Definition 3.4. Define a functor Div X/S : C op Ab by the formula Div X/S (T ) := {Relative effective divisors on X T /T }. Notice that this is functor. Indeed, given p : T T, and a relative effective divisor D X T, we show that D T X T is a relative effective divisor by showing it is on all the fibers using Lemma 3.3. Fix t T and let t = p(t ). Consider the fibered tower D t ι X t p D p X D t ι X t Spec k(t ) p t Spec k(t). By Lemma 3.3, D t X t is a relative effective divisor and so, D t and X t are both t-flat. Thus D t and X t are both t -flat. Consider the exact sequence of sheaves on X t 0 I Dt O Xt ι O Dt 0. Since p t is a field extension, it is faithfully flat. Then p X is flat and the sequence, is exact. Also since p X is flat, by [Har77, (III.9.3)], 0 p XI Dt O X p Xι O Dt 0 (7) p Xι O Dt = ι p DO Dt = ι O D t. Thus the surjection in Sequence 7 is the comorphism of D t X t, and p X I D t is the ideal sheaf of D t, so that I D is invertible and D t t is a relative effective divisor of X t / Spec k(t ). Since t was arbitrary, Lemma 3.3 implies that D T X T is a relative effective divisor of X T /T. Theorem 3.5 Suppose X/S is flat and projective. Then Div X/S is representable by an open subscheme Div X/S of the Hilbert scheme. Proof [Kle05, (3.7)] Although we do not include the full proof here, it is interesting to see exactly which open subset of the Hilbert scheme represents Div X/S. Denote the Hilbert scheme by H. Recall from Example 1.1 that the Hilbert scheme parametrizes closed subschemes, and therefore has a universal object on X H in the way described in the example. Let W X H be this universal closed subscheme. Let q : W H be the restriction of the projection map. Let V be the set of points w W at which W is a relative effective divisor. This is open. Then let U = H \ q(w \ V ) H. It turns out that U represents Div X/S. Notice that a relative effective divisor is always a closed subscheme, thus it is some pullback of W. What we did was take the points of the Hilbert scheme at which W was not a relative effective divisor, and delete them. This is worth mentioning because it is quite interesting that a technique like this works so often in practice, and can be often used to create submoduli spaces, although we will not mention it again in this paper. Definition 3.6. There is a natural transformation of functors A X/S : Div X/S Pic X/S called the Abel map, which on T -points sends a relative effective divisor D to the class of the sheaf I 1 D. Note this also induces a map to all the sheafifications of the relative Picard functor. If both functors are representable, this induces a map A X/S : Div X/S Pic X/S, which we will also call the Abel map. 8

Definition 3.7. Fix an invertible sheaf L on X. We define a functor subfunctor LinSys L /X/S of Div X/S by mapping T to the set of relative effective divisors D X T on X T /T such that I 1 D = L XT ft N for some invertible sheaf N on T. It turns out that the representability of this functor is what will be our most basic building block with which we will construct the Picard scheme. 3.2 The Module Q One of the curious products of cohomology and base change is the following coherent sheaf described by Grothendieck in [GD63, (III.2.7.7.6)]. Suppose f : X S is proper, and let F be a coherent sheaf on X. Then there exists a coherent sheaf Q on S and a functorial isomorphism q such that for each quasicoherent O S -module N, q : H om(q, N ) f (F f N ). (8) In particular, letting N = O S, we see that the dual Q is isomorphic to f F. So we can think of Q as sort of an undual of the pushforward of F. For instance, if f F is invertible, the Q = F 1. Lemma 3.8 If H 1 (X s, F s ) = 0 for all s S, then Q is locally free. Proof [Kle05, (3.10)]. Theorem 3.9 Suppose f : X S is proper, flat, and has integral geometric fibers. Fix L Pic X, and let Q be the coherent sheaf on S associated to L. Then the projective bundle P(Q)/S represents LinSys L /X/S. Proof [Kle05, (3.13)] 3.3 Construction of the Picard Scheme This is the main theorem of this paper. We will construct the Picard scheme using the techniques described in Section 1.2. Theorem 3.10 Suppose f : X S is projective Zariski locally over a locally Noetherian scheme S, and is flat with integral geometric fibers. 1. Then Pic X/S exists, is separated and locally of finite type over S, and represents Pic X/S(ét). 2. If also S is Noetherian, and X/S is projective, then Pic X/S is a disjoint union of open subschemes, each of which is an increasing union of quasi-projective S-schemes. To lighten notation, I will denote by P the functor Pic X/S(ét). Notice that by Proposition 1.4, representability is Zariski local on S, so we may assume S is notherian and f is projective. If we show locally on S that Pic X/S is a disjoint union of open subschemes, each of which is an increasing union of a quasi-projective S-scheme, then plainly Pic X/S is separated and finite type over S. Thus it suffices to prove the second claim, as the first follows. We will do so in a number of steps. First, since f : X S is projective, fix a twisting sheaf O(1) on X. For an O X -module F, we will let P(F ) denote the Hilbert polynomial of F. Claim 3.11 For φ Q[n], let P φ to be the étale sheaf associated to the presheaf T {L Pic X T : P(L 1 t ) = φ for all t T }. Then P φ is a well defined sheaf, an open subfunctor of P, and {P φ P } forms an open cover of P. 9

Proof We begin by showing the presheaf is well defined, and it follows that the sheaf is. Fix a map p : T T, and t T. Let p(t ) = t. Then p t is a faithfully flat base change, so that by [Har77, (III.9.3)], H i (X t, Lt 1 ) k(t) k(t ) = H i (X t, L 1 t ). Thus in particular, dim k(t ) H i (X t, L 1 t ) = dim k(t) H i (X t, Lt 1 ), and so P(L 1 1 t ) = P(L t ). Since t T was arbitrary, if L (P φ ) pre (T ), then p X L (P φ ) pre (T ), and so the presheaf is a well defined functor. Thus P φ is a well defined sheaf. Furthermore, since sheafification is exact, it is a subsheaf of P. To show it is an open subfunctor, fix T P. We must show that the projection T P P φ T is an open immersion of schemes. By Remark 1.7, the map T P is equivalent to a section of P (T ), and can be represented by an étale cover p : T T and an invertible sheaf L on X T. Define an open subscheme T φ of T by T φ := { t T P(Lt 1 ) = φ }. It is open because L 1 T is T -flat, and for flat sheaves the Hilbert polynomial is locally constant [GD63, (III.2.7.9.11)]. Let T φ = p(t φ ). Étale maps are open so T φ T is an open immersion. We will show that this represents T P P φ by showing they have the same R points. First fix an R point of T φ. Then composing makes it an R-point of P, call it σ. Letting R = R T T, we have a commutative diagram R T φ T L R T φ T P. σ Thus σ R : R P is represented by the sheaf L R. Fix r R, and let t be its image in T φ. Then, P(L 1 1 r ) = P(L t ) = φ. Thus, since r R was arbitrary, R P factors through P φ by definition. But R R is an étale cover, and P φ an étale sheaf, so if σ R P φ (R ), then σ P φ (R). This implies that R T P factors through P φ, so that σ is a T P P φ point. Conversely, suppose we have an R-point of σ of T P P φ. This implies that R T P factors through P φ. Let R R be an étale cover so that the restriction σ R is an invertible sheaf L on X R. As before, letting R = R T T, we see that σ R : R P is represented by L R. Thus there is a cover R R R R on which the pullbacks of L R and L agree. Since this is a cover, this implies that for all r R, there is some r R such that P(L 1 r ) = P(L 1 ) = φ since R P factors through P φ, i.e., L P φ (R ). But this means R T factors through T φ, so that R T factors through T φ. Thus σ is an R point of T φ. So we can conclude that P φ is an open subfunctor of P. Finally, as φ ranges through Q[n], it is clear that the T φ cover T, so that the T φ cover T. Thus P φ forms an open cover of P. r This actually gives us a bit of insight into the geometry of the Picard scheme. In particular, as φ varies, we see that the T φ were disjoint, so that the T φ were disjoint. Thus by Theorem 1.3, if we can show that the P φ is representable, then P would be representable as the disjoint union of P φ. Thus what remains to show is that P φ is representable by an increasing union of quasi-projective S-schemes. 10

Claim 3.12 For φ Q[n], and m Z, define a subsheaf of P φ m of P φ to be the étale sheaf associated to the presheaf T { L (P φ ) pre : H i (X t, L 1 t (n)) = 0 for all t T, n m, i 1 }. Then P φ m is well defined, is an open subfunctor of P φ, and the collection {P φ m P φ } forms an open cover. Proof The proof of this claim is very similar to the previous one. The well definedness is again a consequence of [Har77, III.9.3] in the same way. Fix T P φ m and represent it locally by p : T T and L Pic XT. Then we can define T m := { t T : H i (X t, L 1 t (n))} = 0 for all i 1 and n m }. Then we let T m = p(t m). The fact that T m represents T P φ P φ m reduces again to a question about the cohomology groups of twists of L on the fibers, which is preserved by flat base change. So the proof is identical to the previous one. Thus the only difference is showing T m T is an open immersion. Since openness can be checked on an affine cover, finding a neighborhood of t T m that is contained within T m reduces precisely to the following application of Serre s Theorem [GD63, III.1.2.2.2]: given a coherent sheaf over a Noetherian ring A, there are finitely many i 1 and n m such that H i (F (n)) 0, and all of the nonzero modules are finitely generated. Suppose H i (F (n)) = 0 for all i 1, n m, at a point of Spec A. Considering H i (F (n)) as a coherent module, then we can find a neigborhood of the point at which it vanishes for all i and n. But for all by finitely many i and n, H i (F (n)) vanishes on all of Spec A, so intersect the neigbhorhoods corresponding to the nonvanishing cases. This gives a neighborhood of our original point on which H i (F (n)) = 0 for all i 1 and n m. Notice that if {T m T } is a covering by an increasing union of open subschemes, then so is {T m T }. Thus, if we can prove that each Pm φ is representable by a quasi-projective S-scheme, we will be done. In fact, it suffices to show that functors of the form P φ 0 are representable by a quasi projective S-scheme. To see this, let ε : Pic X Pic X be the natural isomorphism of functors which on T -points maps an invertible sheaf L to L (m). This is clearly an isomorphism. Furthermore, if we let ψ(n) := φ(n + m), the ε maps (Pm) φ pre isomorphically onto (P ψ 0 )pre. Thus sheafifying this functor shows ε + : Pm φ P ψ 0. Therefore, the construction will be finished by the following claim. Claim 3.13 For all φ Q[n], the étale sheaf P φ 0 is representable by a quasi-projective S-scheme. Proof Notice that s P(X s, O Xs ) is locally constant, and representability is local on S, so we may assume it is constant, say with image ψ. Let A : Div X/S P be the Abel map, and let Z := Div X/S P P φ 0. Then Z is an open subscheme of Div X/S, which in turn is an open subscheme of the Hilbert scheme. Furthermore, letting γ = ψ φ, we note that Z lies in Hilb γ X/S, which is projective. Thus Z is quasiprojective. Let α : Z P φ 0 be the projection map. We will show that α is surjective. Fix λ P φ 0 (T ), and represent it by a cover T T and an invertible sheaf L on X T. We have the following fibered tower F T Z P φ 0 Div X/S P. 11

Then we can show that F = LinSys L /XT /T. Consider an R point of F. This is the same data as the diagram R T σ A X/S Div X/S P. We see that R T P is represented by L R. Now σ can be viewed as a relative effective divisor D X R, and R Div X/S P is A X/S (σ) = I 1 D. Thus L R and I 1 D on X R generate the same map to P. But since f has integral geometric fibers, Pic X/S P by Corollary 2.5. Then I 1 D = L R fr N for some N Pic R. This is precisely the fact that D LinSys L /X/S(R). The converse follows essentially by definition. If D X R were in LinSys L /X/S (R), then I 1 D and L R differ by a pullback of an invertible sheaf on R, so that the maps to P in either direction must agree, and R P factors through F = Div X/S P T. Let Q be the coherent sheaf on T corresponding to L on X T, as described in Section 3.2. By Theorem 3.9, we see that the fibered product Z P φ T = P(Q). Furthermore, since all the higher 0 cohomology of L vanishes at each fiber, Q is locally free. Therefore there is a cover T T, such that Q T = (O T ) n. Thus in particular there is a surjection Q T O T. By [Har77, (II.7.12)], then T T factors through P(Q). Thus, to summarize, we have the following commutative diagram. T P(Q) T α T Z P φ 0 λ A X/S Div X/S P. Thus λ P φ 0 (T ) is represented by T P, which factors through Z. So λ T α : Z(T ) P φ 0 (T ). This implies that α is a surjection of étale sheaves. is in the image of Notice also that α : Z P factors through the Abel map, and therefore is represented by the image of a divisor D X Z, and therefore by a sheaf I 1 D. Therefore, if Q Z Pic Z is the coherent sheaf associated to I 1 D, we just showed that Z P Z φ = P(Q Z ) is a scheme. This, plus the fact that 0 α is a surjection, shows that Z P φ 0 α Z Z P φ 0 is a coequilizer diagram. But Z is quasi-projective, and the projection from the projective bundle of a locally free sheaf is smooth, flat and proper. Therefore the quotient P φ 0 is a quasi-projective scheme by Proposition 1.6. Remark 3.14 This construction gives us some intuition about what the Picard scheme looks like. First of all, we see that locally, it is a quotient of an open subscheme of the Hilbert scheme. In particular, it is locally 12

a quotient of an open set of relative effective divisors, where we associate those that are in the same linear system, i.e., we associate closed subschemes that have isomorphic ideal sheaves. 4 Basic facts about the Picard Scheme Now that we have constructed the Picard scheme, we can talk about its properties as a scheme and as a moduli space. Instead of making assumptions about f : X S that guarantee the existence of the Picard scheme, for this section we will just make the blanket assumption that Pic X/S exists and represents Pic X/S(fppf). More assumptions may be picked up as necessary. A reason for this is that our assumptions for Theorem 3.10 were sufficient for the existence of Pic X/S, but were by no means necessary. See [Kle05] for a discussion on the more general existence theorems. Proposition 4.1 The formation of the Picard scheme is compatible with base change. Indeed, if Pic X/S exists, and S S is any S-scheme, then Pic XS /S exists and is isomorphic to Pic X/S S S. Proof First, fix a T point of S. Then composing gives a T point of S, and Pic X/S(fppf) (T ) = Pic XS /S (fppf)(t ). Indeed, X S T = X S S T. Thus Pic XS /S (T ) = Pic X/S (T ), and so the relative Picard functors are isomorphic. But T T is an fppf covering over S if and only if it is one over S. Thus the associated sheaves are isomorphic. Note that Pic X/S S S is a scheme over S. Thus if T is an S scheme, a map T Pic X/S S S in the category of S schemes is the same as a map T Pic X/S in the category of S schemes (since the map to S is fixed). Thus, Hom S (T, Pic X/S S S ) = Hom S (T, Pic X/S ) = Pic X/S(fppf) (T ) = Pic XS /S (fppf)(t ). Recall from Example 1.1 the construction of the universal thing we like in a moduli space. This does not always exist for the Picard scheme, but we can see precisely when it does. Definition 4.2. An invertible sheaf P on X Pic X/S is called a universal sheaf or a Poincaré sheaf if it satisfies the following universal property. For all S-schemes T, and invertible sheaves L Pic X T, there is a unique map h : T Pic X/S such that L = (id X, h) P ft N for some N Pic T. Proposition 4.3 A Poincaré sheaf exists if and only if Pic X/S represents the relative Picard functor. If it exists, it is unique up to tensor product with f Pic X/S N for some invertible sheaf N on Pic X/S. Proof This is a simple consequence of Yoneda s Lemma. To lighten notation let P denote Pic X/S. Suppose P exists. Then each element of Pic X/S (T ) induces a unique map T P. If two elements induce the same map, the they must differ by a pullback of an invertible sheaf on T so they were in the same class to begin with. Thus Pic X/S (T ) = Hom(T, P ) and P represents Pic X/S (T ). Conversely, suppose P represents Pic X/S. Let P be a member of the class associated to id P P (P ). Fix T and L Pic X T. Then the class of [L ] is equivalent to a map h : T P, and by Yoneda s Lemma [(id, h) P] = [L ]. If h h then the associated classes in Pic X/S (T ) are not equal, so that (id, h) P and (id, h ) P cannot differ by a pullback of an invertible sheaf on T, so h is unique. Suppose P was also universal. Since P was equivalent to the identity map on P, we see that P = (id X, id P ) P f P N = P f P N. Corollary 4.4 Suppose a Poincaré sheaf exists, and O S f O X holds universally. Then the Poincaré sheaf is unique up to tensor product with a the pullback of a unique sheaf on Pic X/S. 13

Proof [Kle05, (4.3)]. Corollary 4.5 Suppose Pic X/S exists, O S f O X holds universally, and f has a section. Then a Poincaré sheaf exists. Proof This follows from Proposition 4.3 and Theorem 2.4. Lemma 4.6 Suppose X/S is projective and flat, and its geometric fibers are integral, and S is noetherian. If Z Pic X/S is a closed subscheme of finite type, then it is projective. Proof By the construction in Theorem 3.10, each connected component of Z lies in an increasing union of quasi-projective schemes. But Z is finite type over a noetherian scheme, so it is quasi-compact. Thus each component of Z is a closed subscheme of a single quasi-projective scheme, and as such each component of Z is quasi-projective. Again, since Z is quasi-compact, it has only finitely many components. So Z is quasi-projective. Definition 4.7. A Group Scheme G C is an S-scheme whose functor of points factors through the category of groups. Equivalently, it is an S-scheme equipped with three maps, m :G G G i :G G e :S G (multiplication) (inversion) (identity), such that the group law diagrams commute. It is a simple application of Yoneda s Lemma to see that these definitions are equivalent. In fact, the diagrams exactly say that the T -points of form a group. Plainly Pic X/S is a group scheme since it represents a sheaf of abelian groups. Suppose now that S = Spec k where k is a field. The group scheme structure of Pic X/k allows us to get a handle on its tangent space structure. Remark 4.8 Suppose G/k is a group scheme. Notice that e : Spec k G is a map over Spec k, and induces a map of local rings k O e k whose composition is the identity. Thus O e k so that k(e) = k. Then if k ε = k[ε]/ε 2 is the ring of dual numbers, we know by [Har77, (II.2.8)] that T e G = Hom e (Spec k ε, G), which denotes maps Spec k ε G whose image is e. Equivalently, this is maps Spec k Spec k ε G whose composition is G, which is precisely the kernel of G(k ε ) G(k). Proposition 4.9 If Pic X/k exists and represents Pic X/k(ét), then T 0 Pic X/k = H 1 (X, O X ). Proof Let X ε := X k k ε, and associate O Xε with (p 1 ) O Xε. Then there is an exact sequence of sheaves on X, 0 O X OX ε OX 0, where on an open set, the first map is b 1 + bε, and the second map is a + bε a. This sequence is surjective on global sections. Therefore the long exat sequence on cohomology induces the following diagram 0 H 1 (X, O X ) H 1 (X, O X ε ) H 1 (X, O X ) v + + (9) 0 T 0 Pic X/k Pic X/k (k ε ) Pic X/k (k). 14

Here + denotes sheafification as a map of abelian groups, and T 0 Pic X/k is the kernel of the second row by Remark 4.8. Thus v exists by the universal property of the kernel. Let K be an algebraically closed field extension. Then we can do the same construction after base changing to K, and so by [Har77, (III.9.3)], the following square H 1 (X, O X ) k K H 1 (X K, O XK ) v K T 0 Pic X/k k K v K T 0 Pic XK /K. commutes. Thus v K is an isomorphism if and only if v K is. But since K/k is faithfully flat, that holds if and only if v is an isomorphism. Thus we can assume without loss of generality that k is algebraically closed, and return to Diagram 9. Then both k and k ε are Artin local with algebraically closed residue field, so that by Lemma 2.6, each + is an isomorphism, and so v is also. Corollary 4.10 Suppose Pic X/k exists and represents Pic X/k(ét). Then, dim Pic X/k dim H 1 (X, O X ), with equality holding if and only if Pic X/k is smooth at 0. If so, then Pic X/k is smooth of dimension dim H 1 (X, O X ) everywhere. Proof As argued in the previous proof, we may assume that k is algebraically closed. λ Pic X/k (k), we define the translation automorphism T λ as the composition For a point T λ (id,λ) Pic X/k Pic X/k k k Pic X/k k Pic X/k Pic X/k. m This is an isomorphism which is multiplication by λ. Thus Pic X/k is smooth at 0 if and only if it is smooth at λ. But k points are dense, so that this holds if and only if Pic X/k is smooth everywhere. Now, dim 0 Pic X/k dim T 0 Pic X/k, with equality if and only if Pic X/k is regular at zero. But since k is algebraically closed this holds if and only if Pic X/k is smooth at 0. 5 The Dual of an Abelian Variety In the 19th century, Abel and Jacobi were investigating integrals of rational (algebraic) differentials ω on a compact Riemann Surface C. They showed that integrals of this form satisfied an addition law in the following sense. Let a 0 be a base point, and a 1,..., a n, b 1,..., b n C. Then there exist c 1,... c n C depending rationally on the a i and b i such that n i=1 ai a 0 ω + n i=1 bi a 0 ω = This looks like a group law, and can be rephrased in the following way. For any C there is a Lie group J such that the map a a a 0 ω can be factored as n i=1 ci a 0 ω. C \ {poles of ω} J exp Lie J l C, 15

where Lie J is the Lie algebra of J, and l is a linear map. This J is a sort of groupification of C, and is called the Jacobian. It can be constructed as we did in Section 1.3. Interestingly enough, if we view C as a complex algebraic curve, and define Pic 0 C/k to be the connected component of the identity Pic C/k, we have a purely algebraic construction of the Jacobian. Of note is the fact that the Jacobian of an elliptic curve is the elliptic curve itself. But with curves of higher genus g, dim J = g so that the analytic construction of the Jacobian cannot be repeated. Thus it is unclear what deeper structure an elliptic curve may have allowing for this strange fact. A thought could be that if the Jacobian is indeed a groupification of a variety, perhaps it is the group structure of the elliptic curve that allows for this. With this algebraic construction in hand, we hope to generalize and understand this result. As it turns out, Pic 0 X/k for an abelian variety X can be understand as a sort of dual of X, so that an elliptic curve is a special class of Abelian variety that is self dual. This is one of the first great triumphs of the algebric construction, as it detects this nice structural property that the analytic theory could never hope to detect. In this section we develop this theory further. 5.1 The Connected Component of the Identity of the Picard Scheme Let k be an algebraically closed field of characteristic 0. In fact, the results of this section do not require this, but the proofs we construct use it in a very important way. See, for example, Mumford s book [Mum70] for a more complete treatment. Definition 5.1. Suppose Pic X/k exists. Since it is a group scheme it has an identity element 0 : Spec k Pic X/k. Denote the connected component of the identity by Pic 0 X/k Definition 5.2. An abstract variety (or just variety) is an integral separated scheme X of finite type over k. If X is also a group scheme, it is called an algebraic group. If X is also projective, it is called an abelian variety. Note that in particular, X is irreducible, and so connected, and proper, so complete. Also, f : X Spec k is flat and projective, and X is integral, so by Theorem 3.10, Pic X/k exists, represents Pic X/k(ét), and is separated and finite type over k. Futhermore, since X is integral, Lemma 2.5 implies that O k f O X holds universally, and the identity element is a section of f, so that by Theorem 2.4, Pic X/k = PicX/k(ét), so in fact Pic X/k represents the relative Picard functor. By Corollary 4.5, a Poincaré sheaf exists on X Pic X/k. If we hope for Pic 0 X/k to be the dual of X, then at the very least it must be an abelian variety. Thus we must show it is a projective algebraic group. Lemma 5.3 Let G/k be a group scheme that is finite type. Then G 0, the connected component of the identity, is an open and closed sub-group scheme which is an algebraic group. Proof Since G is finite type over a field, it is noetherian, so it is compact, and has finitely many connected components. Thus each connected component is open. The closure of a connected component is plainly the component itself, thus each component is closed. Therefore G 0 is open and closed. Also, G 0 G 0 is connected, so that m(g 0 G 0 ) must beas well. But m(e, e) = e G 0, so that m(g 0 G 0 ) G 0. Thus G 0 is a subgroup. Since G 0 is a group scheme over an algebraically closed field of characteristic 0, it is smooth, and therefore it is reduced. Define α : G 0 G 0 G 0 on T -points by α(g, h) = gh 1. Then the diagonal is α 1 (e) which is a closed subscheme. Thus G 0 is separated. Finally, a smooth and connected scheme that is finite type over a field is irreducible. By Lemma 4.6, we know that Pic 0 X/k is quasi-projective, thus if we show it is proper, it is projective. 16

Lemma 5.4 Pic 0 X/k is proper. Proof We use the valuative criterion for properness [Har77, (II.4.7)]. Let A be a valuation ring, and K its field of fractions. Then for any map u : Spec K Pic 0 X/k as in the diagram below, we hope to extend it to a map v out of Spec A. Spec K v u Pic 0 X/k Spec A Spec k. If such a v exists, it is unique by the valuative criterion for separatedness [Har77, (II.4.3)]. Notice that u is a map to the Picard scheme, and therefore can be represented by an invertible sheaf L on X K. Thus if we can extend it to an invertible sheaf on X A we will have our map v. Without loss of generality, we can twist L until it has a global section. Thus assume L has a global section. A choice of section gives a map O XK L since X K is integral, so that L 1 O XK. Thus L 1 is an ideal sheaf for an effective divisor D X K. Let D denote the closure of D in X A. But since X A is regular, it follows that D X A is an effective divisor, with invertible ideal sheaf I D. Thus I 1 D extends L. Corollary 5.5 Pic 0 X/k is an abelian variety. 5.2 Basic Theory of Abelian Varieties While working in with a varieties over an algebraically closed field, we will use the term point more loosely. For instance, I will say x X to mean x : Spec k X is a closed point of X. Since closed points are dense, this abuse of notation is not catastrophic, and will lighten notation throughout. Also to lighten notation, denote the abelian variety Pic 0 X/k by ˆX, although we are not yet justified in calling it a dual. Recall how we defined the translation automorphism for Pic X/k in the proof of Corollary 4.10. We can do the same for any closed point x X, and denote it T x : X X. Definition 5.6. A translation invariant line bundle L of X is one where T x L = L for all x X. If T is locally of finite type over k, a line bundle L on X T is called translation invariant if L X {t} is for all closed points t T. It turns out that ˆX will parametrize translation invariant line bundles of X, and that the restriction of a Poincaré sheaf to ˆX serves as a universal translation invariant line bundle. But first, we must develop some theory. Definition 5.7. A homomorphism of abelian varieties is a morphism f : X Y, that is a homomorphism on T -points. Theorem 5.8 Let g : X Y be a morphism of abelian varieties. Then g(x) = h(x) + a for some a Y and h(x) a homomorphism. In particular, if g(0) = 0, then f is a homomorphism. Proof [Mum70, (p. 43)]. Definition 5.9. A homomorphism g : X Y of abelian varieties is called an isogeny if ker g is finite and g is surjective. Theorem 5.10 Let g : X Y be a homomorphism of abelian varieties. Then the following are equivalent: 17

(a) g is an isogeny. (b) dim X = dim Y and g surjects. (c) dim X = dim Y and g has finite kernel. (d) g is finite, flat, and surjective. Proof [Mil, (7.1)]. Theorem 5.11 (Seesaw Theorem) Let X be a complete variety, and T any variety. Let L be a line bundle on X T. Then the set T 1 := {t T L X {t} is trivial} is a closed subscheme of T. Furthermore, L XT1 = f T1 N for some line bundle N on T 1. Proof [Mum70, (p.54)]. Corollary 5.12 Let X be a complete variety, and T any variety. Suppose L Pic X T, and let x 0 X be such that 1. L {x0} T is trivial, and 2. L X {t} is trivial for all t T. Then L is trivial. Proof The set T 1 from the seesaw theorem is all of T, so that L = ft N for some N Pic T. The following composition is the identity. Therefore, Thus L = f T O T is trivial. T {x 0 } T X T f T T N = id T N = f T N {x0} T = L {x0} T = O T. Theorem 5.13 (Theorem of the cube) Let X, Y be complete varieties, and Z any variety, and let L be a line bundle on X Y Z. If there are points x 0 X, y 0 Y and z 0 Z such that 1. L {x0} Y Z is trivial, 2. L X {y0} Z is trivial, 3. L X Y {z0} is trivial, then L is trivial. Proof [Mum70, (pp. 55-58)]. 18

5.3 Translation Invariant Line Bundles In this section we will show that ˆX parametrizes translation invariant line bundles. Lemma 5.14 Let L Pic X. Then L is translation invariant if and only if m L = p 1L p 2L where p 1 and p 2 denote the projections from X X X. Proof For all x X, denote by x : X X the constant map X k x X. We refer to the following diagram. T x id X p 1 (id X,x) X X k X X X X p 2 x X m. (10) Suppose m L = p 1L p 2L. Then chasing 10 shows that T x L = (id, x) m L = (id, x) (p 1L p 2L ) = id XL x L = L. Conversely, let N := m L p 1L 1 p 2L 1. Then following the diagram around we see: N {0} X = O X, and for all x X, N X {x} = T x L L 1 = OX. Therefore N is trivial by Corollary 5.12. Proposition 5.15 Let L Pic X, and let λ Pic X/k be the corresponding closed point. If λ ˆX, then L is translation invariant. Proof The point 0 ˆX is the closed point corresponding to the structure sheaf of X. Then 0, λ ˆX. Let P be the restriction of a Poincaré sheaf to ˆX. Then P X {0} = OX, and P X {λ} = L. ( Notice that ) we can assume without loss of generality that P {0} ˆX is trivial. If not, tensor it with f ˆX P 1 {0} ˆX, which will have the same restriction to X {x} for all x ˆX. Define a line bundle N on X X ˆX by N := m P p 13P 1 p 23 P 1, where we abuse notation to let m denote (m, id). Then we can chase the following diagram around to see that N {0} X ˆX is trivial. 19