Divisors, Invertible Sheaves, and Their Relationship (With Examples)

DIVISORS, INVERTIBLE SHEAVES, AND THEIR RELATIONSHIP (WITH EXAMPLES)

EOIN MACKALL

Notation and Conventions. For this talk a variety is going to be an integral scheme, separated of finite type over a field k. We make two assumptions throughout this talk to make some parts simpler: k is going to be a fixed algebraically closed base field and by variety I mean only the closed points.

0. introduction Let’s start with some recollections on terminology. Deﬁnition 0.1. A vector bundle V over a variety X is a variety together with a map π : V → X such that over a covering of X by open subvarieties {Ui}i there is a commuting diagram

n ∼ −1 A × Ui π (Ui)

π|Ui Ui with the top (horizontal) arrow an isomorphism. If n = 1, then V is called a line bundle.

Definition 0.2. A finite rank locally free sheaf L on X is a sheaf so that over an open cover {Ui}i of X there are isomorphisms L| =∼ O⊕n where O is the structure sheaf of U . If n = 1, then L Ui Ui Ui i is called an invertible sheaf. The next remark is purely for clarification – we won’t use it at all. We’re going to, in this talk, stay strictly with invertible sheaves. Remark 0.3. There is an equivalence between the category of finite rank locally free sheaves and vector bundles on a variety X. More precisely, if V is a vector bundle on X then there is a functor LX which assigns to V a finite rank locally free sheaf on X with sections over an open subvariety U of X: −1 LX (V )(U) := {σ : U → π (U): π ◦ σ = id|U }.

There is also a functor SpecX (Sym(−)) that assigns to a ﬁnite rank locally free sheaf L a vector bundle over X. On an open U of X, the variety SpecX (Sym(L)) is deﬁned to be Spec(Sym(L(U))), the spectrum of the symmetric algebra of the OU -module L(U). These varieties glue together over the open sets U to give a variety over X. The composition

L 7→ SpecX (Sym(L)) 7→ LX (SpecX (Sym(L))) is canonically isomorphic with the dual, taking L to Hom(L, OX ). Sometimes, to avoid this duality, one takes the dual of the bundle they’re working with before taking the associated vector bundle.

Date: October 21, 2018. 1 Now ﬁx a variety X, which we’re going to assume is projective (meaning it has a morphism into a projective space of some dimension that realizes X as a closed subvariety). A question one might ask is the following: in how many ways can I realize X as a subvariety of projective space? Another n question one might ask: what is the minimal dimension n so that I can embed X into P ? These questions can be answered, if one has a speciﬁc variety X in mind, using invertible sheaves. First, let’s give an example. 1 n Example 0.4. Let X = P . Then X admits morphisms into P for any n ≥ 1 by taking [x : y] to [x : y : 0 : ··· : 0] for example. But there are other morphisms one can choose as well. If n = 2 then X can be realized as a closed subvariety using the map [x : y] 7→ [x2 : xy : y2]. 2 2 In this case X is isomorphic with the subvariety of P given by the vanishing of x0x2 = x1. There’s 2 also the map [x : y] 7→ [x : y : 0] that realizes X as a linear subvariety of P . If n = 3 then X can be realized as a closed subvariety using the map [x : y] 7→ [x3 : x2y : xy2 : y3]. 3 In this case X is isomorphic with the subvariety of P given by the vanishing of the polynomials 2 2 x0x3 = x2x1, x1 = x0x2, and x3x1 = x2. This subvariety is called a twisted cubic.

1 There are clearly a lot of ways to embed even P into various projective spaces. However, for 2 3 embeddings into P and P , these are essentially the only cases (allowing for automorphisms of the target space which could change the coordinates for example). The main purpose of this talk is to elucidate the above concept. In the ﬁrst section of these notes, we make precise the relationship between invertible sheaves and morphisms into projective 2 space. In the second section, we begin to answer the questions posed above Example 0.4. More precisely, we introduce the Divisor class group, the Picard group, and their relationship (along with the relationships between these objects and other well-known objects like certain cohomology groups). A number of examples are included in each section although they are limited by the knowledge of the author (i.e. they might not be the simplest but they’re all I got!). Remark 0.5. A notable absence from these notes is the concept of ample invertible sheaves. These objects are incredibly useful – especially in examples and in areas like the theory of curves, for instance. However, this talk itself aims to cover a much more foundational area of algebraic geometry which is still interesting in its own right. I would suggest the interested reader take a look at this material, e.g. in [Har77, Chapter 2, Sections 5-6; Chapters 3-5] or [Laz04, Chapter 1].

1. maps to projective space Roughly, this is how we should continue: given an invertible sheaf L on a variety X and n + 1 n global sections of s0, ..., sn L, one can try to define a morphism to P by sending a point x to the point [s0(x): ··· : sn(x)], where we use s0(x), ..., sn(x) to denote the images of these sections in the residue field ∼ Lx = OX,x k(x). There are some immediate problems one might have with this method at first glance though. E.g. ∼ “Why is this well-defined? We had to use a choice of an isomorphism Lx = OX,x that relies on an open set about x that trivializes L!” and “Even if this was well-defined, is it ever the case that these sections are all 0 and so do not define a point of projective space?” Both of which we’ll try to answer. First, let’s set up the morphisms we want to talk about in precise terms. We still fix a variety X over k, an invertible sheaf L, and we assume s0, ..., sn are global sections of L, possibly with repetitions and possibly with some being the 0 section. But, to solve the second question above, we add some assumptions. We’ll need to assume that, for any point x, the values s0(x), ..., sn(x) are not all 0. So in the definition x 7→ [s0(x): ··· : sn(x)], the target is always a point of projective space (even if the map is not well-defined for other reasons). Notice also that it’s not necessarily true these assumptions can be met for our particular L but, we’ll assume they hold anyway (these conditions are very frequently satisfied). Of course the map we defined above turns out to be well-defined. By definition, L comes with a ∼ collection of open sets {Ui}i covering X and isomorphisms ϕi : L|Ui = OUi . In the morphism we’re ∼ defining, we first take the isomorphism ϕi and restrict it to an isomorphism on stalks Lx = OX,x. It does not have to be the case, if x were in two open sets Ui,Uj, that s0(x), ..., sn(x) have the same images in k(x) under the two different isomorphisms ϕi, ϕj. However, if we take two such isomorphisms, over open sets Ui,Uj, and consider the morphism

−1 ϕ |U ∩U ϕUi |Ui∩Uj Uj i j OUi∩Uj −−−−−−→L|Ui∩Uj −−−−−−→OUi∩Uj of OUi∩Uj -modules then this composition is completely determined by where the section corresponding to 1 in OUi∩Uj (Ui ∩ Uj) goes. Being an isomorphism, the 1-section must be sent to an invertible element t of OUi∩Uj (Ui ∩ Uj). The diﬀerence in the two maps over our two diﬀerent choices of open set Ui,Uj is then given by multiplication by t

x 7→ [s0(x): ··· : sn(x)] and x 7→ [ts0(x): ··· : tsn(x)]. Due to the definition of projective space, these are actually the same point showing this map is well-defined. Altogether we’ve sketched out a proof of the following statement. 3 Lemma 1.1. Let X be a variety over k and L an invertible sheaf on X. Assume L has global sections s0, ..., sn such that for every point x in X the values s0(x), ..., sn(x) are not simultaneously n 0, then there is a well-defined morphism X → P . Proof. All that’s left is show that there is a morphism # f : OPn → f∗OX n where f : X → P is the map defined (on topological spaces) in the previous paragraphs. Let {Ui}i be a collection of open subsets of X that trivialize L and cover X, i.e. we assume there ∼ are isomorphisms ϕi : L|Ui = OUi on these opens. Fix one such Ui. Since not all of the sj are simultaneously vanishing, we can assume (by possibly making Ui smaller) that sj is not 0 in k(x) n for any point x in Ui. Now the image of Ui under f lands in the open affine of P where xj is # invertible (by the definition of f). We define f by a gluing construction: on the open set Ui the map f # is the sheafified ring map ∼ k[x0/xj, ..., xn/xj] → L(Ui) = OUi (Ui) that sends x0/xj, ..., xn/xj to s0/sj, ..., s0/sj. Note this map a priori depends on the choice of # trivialization of L but, for the same reason f is well-defined, the map f is also well-defined. It turns out morphisms of these type completely characterize maps to projective space. The converse to Lemma 1.1 is the following: n Lemma 1.2. If f : X → P is any morphism to projective space, then f is determined by the mor- ∗ ∗ ∗ phism constructed in the proof of Lemma 1.1 using the sheaf f O(1), and sections f (x0), ..., f (xn). n Proof. The proof is surprisingly fundamental in nature. In fact, since we know P can be constructed n by the gluing of affine opens A = Spec(k[x0/xj, ..., xn/xj]) we also know that any morphism from n X to P is determined as a gluing of the maps to these open sets (we can consider maps from n the preimages of these opens to the opens themselves; since P glues together in a specific way, these maps must glue together in the same way in order to define a morphism). Now we can use a canonical bijection n HomVar/k(V, A ) = Homk−Alg(k[x0/xj, ..., xn/xj], OV (V )) to determine that any morphism to projective space is determined by the images of the xi/xj. To conclude, we notice that if g is the morphism constructed as in Lemma 1.1 from the in- ∗ ∗ ∗ ∗ ∗ ∗ vertible sheaf f O(1) and the sections f (x0), ..., f (xn) then g O(1) = f O(1) and g (x0) = ∗ ∗ ∗ f (x0), ..., g (xn) = f (xn). By the first paragraph of this proof, the two morphisms g, f must be the same. 1 Example 1.3. Let X = P . The global sections of O(1), i.e. the k-vector space Γ(X, O(1)), is spanned freely by the sections x0, x1. These two sections turn out to generate O(1): indeed for any point x = [t0 : t1] if x0 vanishes then t0 = 0 so that x1 6= 0 and visa versa exchanging x1 for x0. 1 The sections x0, x1 determine a unique map to P and it’s an uninteresting one (it’s just the 1 identity X → P ). If we were interested in maps to other projective spaces, we could just start including other sections and wondering what happens. For example, taking x0, x1, 0 defines a map 2 to P with [x : y] 7→ [x : y : 0] 1 2 which isn’t very interesting either since it’s just a linear embedding of P into P . 2 2 2 2 If instead we used O(2), with Γ(X, O(2)) = Spank{x0, x1x0, x1}, and used the sections x0, x1x0, x1 then we are looking at the map [x : y] 7→ [x2 : xy : y2] as in Example 0.4. 4 3 2 2 3 The last case in Example 0.4 is given by the sections x0, x0x1, x1x0, x1 of O(3). 1 1 1 Example 1.4. Let X = P × P and let π1, π2 : X → P be the first and second projection ∗ ∗ respectively. Then there is an invertible sheaf on X given by O(1, 1) := π1O(1) ⊗ π2O(1). If we let x0, x1 be coordinates on the first component and y0, y1 coordinates on the second, then we have

Γ(X, O(1, 1)) = Spank{x0y0, x0y1, x1y0, x1y1}. To see this, we use the existence of a Künnethisomorphism [Sta17, Tag 0BEC] i ∼ M j n i−j n H (X, O(1, 1)) = H (P , O(1)) ⊗ H (P , O(1)) j and set i = 0. 3 The morphism from X to P given by the invertible sheaf O(1, 1) and the sections x0y0, x0y1, x1y0, x1y1 define a morphism ([x : y], [w : z]) 7→ [xw : xz : yw : yz]. 3 This is a closed immersion and the subvariety in P that X is identified with is cut out by the polynomial x0x3 − x1x2 = 0.

Example 1.5. Let X = Gr(n, m) be the Grassmannian of m-planes in an n-dimensional vector space. Then there is a vector bundle on X whose fiber over any given point is the m-plane that point represents. Denote by S the locally free sheaf corresponding to this vector bundle. There is an exact sequence ⊕n 0 → S → OX → Q → 0 called the universal sub (or quotient) exact sequence. Taking the determinant of Q provides an invertible sheaf on X that has a canonical collection of sections s0, ..., sr generating det(Q). The morphism n−m ! ^ ⊕n ( n )−1 X = Gr(n, m) → P k = P m defined by the invertible sheaf det(Q) and the sections s0, ..., sr is called the Plücker embedding; it is a closed immersion. For example, let X = Gr(4, 2). Any point x in Gr(4, 2) represents a plane P , spanned by ⊕4 two linearly independent vectors v0, v1. The quotient k /P is also a plane, spanned by two ⊕4 complementary vectors w0, w1 of k . The Plücker embedding in this situation is the morphism V2 ⊕4 sending x to the point that represents the line spanned by the vector w0 ∧ w1 of k . In this case, the embedding 5 X → P 5 identifies X with the closed subvariety cut out by the equation x0x5 − x1x4 + x2x3 = 0. For more examples, we’ll need an easier method to construct invertible sheaves on a variety. This is accomplished in the next section through the introduction of divisors.

2. divisors, invertible sheaves, and their relation to cohomology Let X = Spec(R) be an aﬃne variety for the moment, and suppose f 6= 0 is an irreducible element of R with fR ( R (since we’re assuming varieties are integral, f is not a zero divisor). The subvariety V = Spec(R/fR) of X has codimension 1 by Krull’s principal ideal theorem [Sta17, Tag 00KV]. The natural embedding i of V into X moreover induces a short exact sequence of sheaves

0 → I → OX → i∗OV → 0 which is just the sheaf-version of the exact sequence 0 → fR → R → R/fR → 0. The sheaf I turns out to be an invertible sheaf on X. Indeed, the following map is commutative 0 fR R R/fR 0

·f ·f 0 R R R/fR 0 and, after going to sheaves, shows I is isomorphic with OX . It turns out the above example can be globalized. If X is now an arbitrary variety, not necessarily affine, and V is a subvariety which locally is cut out by a single equation (i.e. on a cover of open ∼ affines Ui = Spec(Ri) there exist elements fi so that Spec(Ri/fi) = V ∩ Ui), then the above reasoning shows that the ideal sheaf of V is locally trivial. In other words, for such a subvariety V its ideal sheaf is an invertible sheaf. In some sense this shows that subvarieties on X of codimension 1 should be dual to invertible sheaves. We’re going to make this correspondence precise. Definition 2.1. Let Div(X) be the free abelian group generated formally by the subvarieties of X of codimension 1. This literally means that M Div(X) = Z · D codim(D)=1 is a possibly very large sum of copies of Z with basis elements those subvarieties D having codimension 1 in X. P An element of Div(X) is called a divisor. A divisor D = niDi is said to be irreducible if ni = 0 P for all i 6= j, and nj = 1. A divisor D 6= 0 is said to be effective if D = niDi and ni ≥ 0 for all i. We’ll use the shorthand D ≥ 0 to say that D is effective, and D ≥ E to say (D − E) ≥ 0. It may not seem immediately obvious that we’ve done anything interesting by introducing the concept of a divisor. And in reality, the group Div(X) is often much too large to be of any direct use. To remedy this problem, we’re going to introduce a quotient of the group Div(X) and we’ll call elements of the quotient divisor classes. However, it will be important that we distinguish between a divisor and the class of a divisor in the quotient, if only to have a more explicit correspondence between divisors and line bundles. We’re going to make a definition (Definition 2.7) where the difference is important but first we need a way to construct divisors from functions: Definition 2.2. Let X be a variety, and k(X) its function field. Let D be a subvariety of X that has codimension 1 in X. There are a number of open subvarieties intersecting D nontrivially and we let U = Spec(R) be one of them. Since X is integral, R is an integral domain. The ring OX,D is the localization of R at a prime ideal p corresponding to D, hence OX,D admits a natural embedding 6 into k(X). Moreover, the fraction field of OX,D, being the localization at 0, is isomorphic to the function field k(X) since (Rp)(0) = R(0) = k(X). This means any f ∈ k(X) can be written as a fraction f = g/h for two elements g, h in OX,D. For any such D we define the order of a function f = g/h ∈ k(X) in the following way:

(1) if OX,D is a discrete valuation ring, then we pick a uniformizing parameter t in the maximal ideal of OX,D and deﬁne the order of f at D to be

ordD(f) := vt(g) − vt(h).

(2) if OX,D is not a discrete valuation ring, we construct the normalization N : Xe → X and deﬁne X ∗ ordD(f) := ordDi (N (f))

Di⊂D×X Xe −1 1 where the sum runs over all subvarieties Di contained in D ×X Xe = N (D). Lemma 2.3. The order of f ∈ K(X) at D is well-defined and there is a well-defined homomorphism div : k(X)× → Div(X) given by X f 7→ ordD(f) · D. codim(D)=1 Proof. To see that the order is well-defined, we’ll need to show the choice of uniformizing parameter isn’t important and we’ll need to show the choice of representation f = g/h isn’t important. The first of these is clear because the definition of the valuation doesn’t depend on the choice of uniformizing parameter. The second follows because if we had two expressions f = g/h and f = g0/h0 then we’d find, using the definition of the localization, that gh0 − g0h = 0. Hence, by applying valuations 0 0 vt(gh ) = vt(g h). Using that the valuation takes products to sums and rearranging, it follows 0 0 vt(g) − vt(h) = vt(g ) − vt(h ). For similar reasons it follows that the map defined as div will be a homomorphism if we show that × almost all ordD(f) are 0 for a fixed function f ∈ k(X) . That is to say, it’s clear that ordD(1) = 0 for all irreducible divisors D and ordD(fg) = ordD(f) + ord(g) by following the definition. To see that ordD(f) = 0 for all but a finite number of irreducible divisors D we proceed like so. Since the function field k(X) is also the limit of OX (U) as U ranges over open subsets of X, there is an open U where f appears as a section of OX (U); also, we can assume, by possibly making U smaller, that U is affine. The complement of U, being a closed subset of a variety X, has strictly smaller dimension than X and since this complement is also a noetherian space, it has a finite number of irreducible components. Hence it contains only a finite number of irreducible subvarieties of X. To complete the proof then, it suffices to show the same claim but now on the affine U. Let U = Spec(R). Then ordD(f) = vt(f) ≥ 0 for any irreducible subvariety D of R since f appears as a global section of U. The valuation is moreover positive if and only if the prime ideal defining D contains the ideal fR. But the subset of U consisting of prime ideals containing fR is a closed subset of strictly smaller dimension since f 6= 0. We can then again use the fact that this closed subset is noetherian and hence has finitely many irreducible components, corresponding to finitely many irreducible subvarieties, to complete the proof ordD(f) is 0 for almost all D. 1 the normalization can be constructed like so: cover X in affine opens Ui = Spec(Ri) with varying i; let R˜i be the normalization of each Ri in k(X), i.e. the subset of elements satisfying a monic polynomial with coefficients in Ri; the variety X˜ is constructed from gluing the Spec(R˜i) along the normalizations Spec(Ri^⊗R Rj ). 7 Remark 2.4. In all of the cases we work with, it suffices to only know the first part, (1), of definition 2.2. In fact, if X is a variety then a sufficient condition to only work with (1) is to assume all of the local rings in codimension 1 of X are discrete valuation rings. This happens if X is normal or, in an even more strict setting, by assuming X is smooth.

1 Example 2.5. Let X = P and D = [0 : 1] be the point at infinity. The function field of X is isomorphic with k(t) where t = x0/x1. For any rational function n t + ··· + a0 p(t)/q(t) = m t + ··· + b0 the order of p at D can be computed as follows. The local ring OX,D is isomorphic with the localization R = k[t](t). To calculate the order one considers a uniformizing parameter in R, for example t will do, and applies the recipe of definition 2.2: n m ordD(p(t)/q(t)) = vt(t + ··· + a0) − vt(t + ··· + b0) = n − m. n Example 2.6. Let X = P and let D = V+(f) be the vanishing set of an irreducible homogeneous equation f of degree m in the coordinates x0, ..., xn. Without loss of generality, we assume the open set D+(x0) where x0 doesn’t vanish intersects D nontrivially (otherwise we could swap D+(x0) with another such open set D+(xi) for i 6= 0).The function field of X is isomorphic with k(t1, ..., tn) where ti = xi/x0. For any rational function h(t) = p(t)/q(t) the order of h at D can be computed as follows. The local ring OX,D is isomorphic to the ring R = k[t1, ..., tn](f). Using f as the uniformizing parameter of R we find

ordD(h(t)) = ordD(p(t)) − ordD(q(t)) = vf (p(t)) − vf (q(t)). In particular, if we consider two irreducible homogeneous polynomials g, h that are both of degree n, then X div(g/h) = ordD(g/h) · D = V+(g) − V+(h). codim(D)=1 n Indeed, every subvariety D of P is determined by a homogeneous prime ideal in the ring R = k[x0, ..., xn]. If such a prime ideal p has height 1 (meaning the associated subvariety has codimension 1), then it is principal (indeed, if f1, ..., fn are nonzero elements generating p then (f1), for example, is contained in p and since R is a unique factorization domain, we can find an irreducible factor g1 that divides f1; then (g1) is prime and contained in p, hence equals p since p has height 1). It follows from the first paragraph that if D = V+(f) for an irreducible homogeneous polynomial f then ordD(g/h) = vf (g) − vf (h) with vf (g), vf (h) the highest power of f dividing g, h respectively. Our construction of the group of divisors can be functorialized in a natural way. We can construct a presheaf Div(X) by setting Div(X)(U) = Div(U) for any open U ⊂ X and by setting the restriction map Div(V ) → Div(U) for any open subsets U ⊂ V to be defined by D 7→ D ∩ U for any irreducible divisor D and extending linearly (an empty intersection maps to 0). For any divisor D, we’ll write D|U to denote the restriction to the open subset U. 8 Definition 2.7. To any divisor D of a variety X we define a presheaf by setting, for any open U of X, O(D)(U) := {f ∈ k(X): f = 0 or (div(f) + D)|U ≥ 0} and with obvious restriction maps. If X satisfies the assumptions of Lemma 2.10, and with justifi- cation to be made there, we’ll call O(D) the invertible sheaf defined by D. Remark 2.8. It might not be immediately clear that if f, g are two elements of O(D)(U) then f + g and −f are also elements of O(D)(U). However, this can be proven using the fact that the order was defined by valuations. Indeed, X div(f + g) + D = ordDi (f + g) · Di + D X ≥ min{ordDi (f), ordDi (g)}Di + D ≥ div(f) + D ≥ 0. and div(−f) = div(−1) + div(f) = div(f) ≥ 0. Lemma 2.9. The presheaf O(D) is a quasi-coherent sheaf. Proof. To prove this lemma it suffices to show, for any affine open U = Spec(R) and for any f ∈ R, that there is a bijection ∼ O(D)(U)f = O(D)(Uf ) between the localization at powers of f of the sections O(D)(U) and the sections of the localization Uf = Spec(Rf ). We can start by working with the restriction map given in the presheaf definition. That is to say, the map O(D)(U) → O(D)(Uf ) which is defined by taking g 7→ g induces a map on the localization

O(D)(U)f → O(D)(Uf ).

To see this, we just need to check f is invertible in O(D)(Uf ). This is true since on Uf the function

1/f is an invertible section, i.e. div(f)|Uf = 0, so that for any section s of O(D)(Uf ) we have s/f is also a section of O(D)(Uf ) as

(div(s/f) + D)|Uf = (div(s) − div(f) + D)|Uf = (div(s) + D)|Uf ≥ 0. Next we check that our proposed map is injective and surjective. It’s injective tautologically since if g/f n is a section mapping to 0, then f n · g/f n = g maps to 0 as well. Checking surjectivity n is similarly easy since if g is a section of O(D)(Uf ) ⊂ k(X) then there is an n ≥ 0 with h = f g contained in O(D)(U) because n (∗) (div(f g) + D)|U = (ndiv(f) + div(g) + D)|U . Since for any irreducible divisor E ⊂ U which is contained in the vanishing set of f we have ordE(f) > 0, and for any irreducible divisor E with E ∩ Uf 6= ∅ we have ordE(f) = 0, together these imply div(f) ≥ 0 on all of U or, equivalently, that the right hand side of (∗) is effective. n Finally, h/f maps to g under the given map which completes the proof. Lemma 2.10. Suppose X is a normal variety and assume that all of the local rings of X are unique factorization domains. If D is a divisor on X, then O(D) is an invertible sheaf. 9 Remark 2.11. I’ve opted to include all of the assumptions in this theorem. The first thing I thought when I saw this lemma was probably somewhere along the lines of: “Wow this seems like a lot random assumptions, are there any varieties that even satisfy these requirements?” The answer to this question is that there are actually a lot of varieties that satisfy these requirements. For example if a variety X is smooth then all of the assumptions hold. Proof. The proof proceeds in a number of steps. First, we show that any irreducible divisor on such a variety is locally defined by one equation. This is the analogy we used in the beginning of this section and it will let us use the same reasoning as we did there.

Step 1 ): We want to show if D is an irreducible divisor, then there is an open cover {Ui = Spec(Ri)}i ∼ of X so that D ∩ Ui = Spec(Ri/fi). To do this, we fix a point x of X and consider the ideal defined by D in the local ring OX,x. Since this local ring is a unique factorization domain by assumption, all of its codimension 1 subvarieties are given by principal (prime) ideals. In particular, the ideal corresponding to D is also given by a principal ideal. Picking a generator f of this ideal, and noting that the ring OX,x is a limit of opens, we can find some open subset U around x where D ∩ U is locally cut out by f. Doing this for all points x, and taking only a finite number that cover X completes this step. Corollary 2.12. Suppose X is a normal variety and assume all of the local rings of X are UFD’s. Then, given an irreducible divisor D ⊂ X, there is a covering of X by affine opens Ui = Spec(Ri) along with irreducible elements fi of Ri so that D ∩ Ui is isomorphic to Spec(Ri/fiRi). In the next step, we observe that under the given assumptions, we can assume that the divisor D is given locally in the form div(f) for some rational function f ∈ k(X).

Step 2 ): If D is an irreducible divisor and x is a point of X, then due to Step 1 we can find an open subset U, about about x, which we can assume to be affine equal Spec(R), such that ∼ D ∩ U = Spec(R/f) for some f in R. Considering f as a rational function, we find ordD(f) = 1 ∼ P since OX,D = R(f). For an arbitrary divisor D = niDi we can consider the intersection of such Q ni open sets corresponding to each irreducible Di and consider the rational function f = fi to complete this step.

Finally, the last step constructs an explicit isomorphism that locally trivializes O(D), i.e. we construct an isomorphism that shows O(D) is isomorphic with OX |U = OU over some small open set U.

Step 3 ): Given a point x of X, let U be an open aﬃne subset containing x so that D∩U = div(f)|U for some f of k(X), as constructed in Step 2. We want to produce an isomorphism

ϕ : OU → O(D)|U . Since O(D) is quasi-coherent by Lemma 2.9, it suﬃces to produce an isomorphism

ϕ(U): OU (U) → O(D)(U). We do this by defining ϕ(U)(g) := g · f −1 and defining φ−1(U)(g) := f · g. The composition in either direction is the identity, so we only need to show that both are well-defined to conclude. Clearly if g is a global section of OU then div(g)|U ≥ 0 so that

div(g/f)|U = (div(g) + div(1/f))|U ≥ div(1/f)|U = −D ∩ U.

Adding D|U to the left side of this equation shows that g/f is a valid element of O(D)(U) as desired. Conversely, if g is an element of O(D)(U) then (div(g) + D)|U ≥ 0 so that

(div(g) + D)|U = div(g)|U + div(f)|U = div(gf)|U ≥ 0. 10 But, since U is affine we have div(gf) ≥ 0 only if gf has valuation ≥ 0 over all prime ideals of height one in the ring defining U, call it R say. Since X is normal, U is normal, and this implies (see Lemma 2.13 below) \ gf ∈ Rp = R ht(p)=1 where the intersection occurs inside of k(X). Since this exactly means that gf is a global section of OU , we’ve shown our two maps are well-defined and thus isomorphisms which completes the proof. The following lemma is a fairly difficult exercise in ring theory. It’s used in Lemma 2.10, among other places, as a criterion that allows one to check if a function f of the function field k(X) of an affine variety X is actually an element of the global sections OX (X). The criterion is surprisingly useful, since it only requires checking the coefficient of div(f) along finitely many divisors. The proof isn’t revealing in any sense, however (so, if it’s your first time seeing it, it could be a good idea to skip it; don’t miss the forest for the trees). Lemma 2.13. Let R be a noetherian domain integrally closed in its field of fractions Q. Then there’s an equality \ R = Rp ht(p)=1 inside of the fraction field Q of R. T Proof. (cf. [Rei95, Section 8.10]). It’s clear that R ⊂ Rp, where the intersection is over all height 1 prime ideals. To get the converse, we prove the contrapositive statement: an element x ∈ Q \ R lies outside of Rp ⊂ Q for some height one prime ideal p. Since x is in Q, we can write x = a/b for some a, b of R. We’re going to consider the set of denominators D = {r ∈ R : rx ∈ R}. Note that the existence of two denominators b, d, i.e. an expression a/b = c/d, implies there is an expression ad = bc; thus the set of denominators consists exactly of those elements d with ad ∈ bR. This means that the set D is precisely the set of elements which annihilate the imagea ¯ of a in the R-module R/bR. Since D is the annihilator ofa ¯, it is an ideal and either D = R (i.e. b is a unit and x ∈ R all along) or the ideal D is a proper subset of R. Since we’re assuming the latter case, we can find a prime ideal p containing D and annihilating the image f¯ in R/bR of an element f 6= 0 of R. To do this, we consider all ideals of R which annihilate an element of R/bR and use the fact R is noetherian to find a maximal such ideal p containing D; that p is prime follows from the observation that if p annihilates f¯ then qsf¯ = 0 for two elements q, s ∈ R implies either sf¯ = 0 (i.e. s ∈ p) or if sf¯ 6= 0 then there is an ideal I ⊃ p annihilatings ¯f¯ and, as p is maximal with respect to this property, we thus get I = p (i.e. q ∈ I = p). We’re going to show p has height 1. This will complete the proof since for all representations of x as a fraction a/b, the inclusion D ⊂ p implies b ∈ p or, in other words, x ∈ Q \ Rp. To do this, we work instead with pRp ⊂ Rp since localization doesn’t affect the height. Then it suffices to show pRp is principal. Indeed, as Rp is a local integrally closed domain the conditions “Rp has dimension 1”, “Rp is a discrete valuation ring”, and “pRp is principal” are equivalent. Due to the construction of pRp, there is an element f ∈ Rp with f · pRp ∈ bRp. Hence pRp ⊂ b/f · Rp inside of Q. We want to show the reverse inclusion. Well, f/b · pRp ⊂ Rp is an Rp-submodule of Rp, hence an ideal. We have two cases to consider: either this ideal is proper or it is the whole ring. Assume the former, that it is proper. Then multiplication by f/b is an endomorphism of pRp. Since R was noetherian, pRp is finitely generated 11 as an Rp-module. Thus, for any m ∈ pRp, the endomorphism of multiplication by f/b satisfies an equation n n−1 (f/b) + an−1(f/b) + ··· + a0 m = 0 for some an−1, ..., a0 of Rp. Choosing a nonzero m (hence not a zero divisor) shows f/b is contained in Rp since this ring is integrally closed. But we chose f as an element so that p annihilates f¯ ∈ R/bR. In other words, we chose f∈ / bR. Since b ∈ p (by our choice of p), we also have f∈ / bRp (indeed, if f ∈ bRp then there would exist an element z of R \ p with fz ∈ bR which can’t happen since p is maximal with respect to this property). But by assuming f/b · pRp was a proper ideal we’ve shown f/b ∈ Rp which implies f ∈ bRp. So the case f/b · pRp is a proper ideal does not occur and we must have f/b · pRp = Rp. In other words, we’ve shown pRp = b/f · Rp as desired. This gives a concrete way to generate lots of invertible sheaves on the class of varieties satisfying the above conditions. It means that whenever we know some codimension 1 subvariety of such a variety X, then we also know a typically nontrivial invertible sheaf on X. If X is projective, then a number of these turn out to have enough sections to define a morphism to projective space which was our original motivation to study such things.

n Example 2.14. If X = P and H = V+(x0) is the hyperplane at infinity, we can show an isomorphism O(H) =∼ O(1). We define a morphism f : O(H) → O(1) on sections over an open U ⊂ X using that both O(H) and O(1) are quasi-coherent to restrict our attention only to those open subsets D+(xi) for i = 0, ..., n where the variables x0, ..., xn don’t vanish. Now O(1)(D+(xi)) = k[x0/xi, ..., xn/xi] with each variable in degree 0. The map f is defined on D+(xi) by sending a section g of O(H)(D+(xi)) to g(x0/xi) in O(1)(D+(xi)). This is well-defined since if div(g) + H ≥ 0 it follows that g = f/h with h having at most one factor of x0 (see Example 2.6). It is injective, on sections, by construction and it is surjective, on stalks, since any of the global sections x0, ..., xn generate the stalk of O(1) at a point where they are nonvanishing. We turn now towards the goal of classifying invertible sheaves on a variety X. For this we write Pic(X) for the set of isomorphism classes of invertible sheaves on X. Note if L is an invertible sheaf on a variety X, then the dual of L is a tensor inverse to L, i.e. there is an isomorphism ∨ ∼ L ⊗ L = OX . The structure sheaf OX is a tensor identity, i.e. there are canonical isomorphisms ∼ ∼ L ⊗ OX = L = OX ⊗ L. Hence Pic(X) is a group with respect to the tensor product. Definition 2.15. We call the group Pic(X), with respect to the tensor product, the Picard group of the variety X. Now for any variety X satisfying the conditions of Lemma 2.10, for example a smooth variety X, we can define a map Div(X) → Pic(X) by sending D to O(D). The next two lemmas show this map is in fact a homomorphism.

Lemma 2.16. Assume X satisﬁes the assumptions of Lemma 2.10. If D1 and D2 are two divisors ∼ on X, then O(D1) ⊗ O(D2) = O(D1 + D2).

Proof. It suffices to show the claim for two irreducible divisors D1,D2 which are possibly equal. In this case we can define an isomorphism between the two by specifying what happens over a small affine open U where we can assume D1,D2 are defined by equations g1, g2 (see Corollary 2.12). Then the map g ⊗ f 7→ gf defines an isomorphism on stalks hence globally. Lemma 2.17. Assume X satisfies the assumptions of Lemma 2.10. If D is a divisor on X, then O(D)∨ =∼ O(−D). 12 Proof. We need to show that for the trivial (or empty or zero) divisor 0, the associated invertible sheaf O(0) is trivial, i.e. isomorphic with OX . The claim then follows from the uniqueness (up to isomorphism) of tensor inverses. We can define a map OX → O(0) by sending a section f of OX (U), for an open subset U ⊂ X, to f of O(0)(U). This is well-defined since div(f) ≥ 0 by definition of the divisor map. The map is surjective on sections also, since if a function f of k(X) has div(f) ≥ 0 then, as in the proof of Lemma 2.10, we can apply Lemma 2.13 to claim f is in fact a section over U. Hence the map defined above is an isomorphism of sheaves (at a point x ∈ X the map is surjective on stalks which are both rank 1 free OX,x-modules and hence the map is injective on stalks as well). Altogether, we’ve constructed a sequence of maps × × (D1) 0 → OX (X) → k(X) → Div(X) → Pic(X) → 0. The suggestive nature of how this sequence is written isn’t without reason, as the sequence turns out to be exact. To prove this, we’ll turn to cohomology. × × There are two sheaves (of abelian groups) on any variety X denoted OX and KX defined as × × follows: if U is an open subset of X then OX (U) := OX (U) are the sections invertible with × respect to multiplication, and KX (U) := k(X) is the constant sheaf associating to any open set the invertible elements of the field of rational functions on X. The canonical inclusion that happens on sections over any open set U induces an inclusion of sheaves. Forming the quotient sheaf we get an exact sequence × × × × 0 → OX → KX → KX /OX → 0. × × Proposition 2.18. Let X satisfy the conditions of Lemma 2.10. Then the qutoient sheaf KX /OX is isomorphic with the presheaf Div(X). Proof. We first check Div(X) is a sheaf. This is immediate from the isomorphism, which can be L checked on sections, between Div(X) and codim(x)=1 ix∗Z, the direct sum of the pushforwards of the constant sheaf Z along the inclusion ix : x → X of every point x of codimension 1. × × Now we’re going to define a map π : KX /OX → Div(X) that will induce an isomorphism on × × stalks which is the claim. Note that, even though KX /OX is defined as the sheafification of the × × quotient presheaf, the two sheaves have the same stalks a point x of X:(KX )x/(OX )x. Thus we × can first define a map KX (U)/OX (U) → Div(X) and pass to stalks in order to conclude. Thus we let π(U)(f) = div(f)|U for any f ∈ k(X). That this passes to the quotient follows from observing

div(f)|U ≥ 0 and − div(f)|U = div(1/f)|U ≥ 0 for any invertible section f over U. Hence we must have div(f)|U = 0. It follows from the snake lemma applied to the diagram

× × × × 0 OX KX KX /OX 0

× × 0 OX KX Div(X) that π is injective. It’s also clear that π is surjective since if x is a point of X then, for any irreducible divisor D passing through x, the divisor associated to a local equation for D is exactly D at x. In other words, the map π is surjective on stalks. ∼ 1 × Proposition 2.19. There is an isomorphism Pic(X) = H (X, OX ). 13 Proof. In the case X is a variety, it’s sufficient to prove this theorem using Cechˇ cohomology since it is canonically isomorphic with derived functor cohomology here. The proof will be accomplished 1 × through showing there is a well-defined bijective mapα ˜ : Pic(X) → H (X, OX ) which is induced by a map α from the collection of all invertible sheaves on X. To define α, let L be an invertible sheaf on X. We fix then a cover of X by affine opens

{Ui = Spec(Ri)}i such that on these Ui there exists an isomorphism fi : L|Ui → OUi . The 1 × collection of these fi will define an element [α(L)] ∈ H (X, OX ). To see this, recall that an element 1 × of H (X, OX ) is an equivalence class [{Vi ∩Vj, gij}i,j] where the Vi are sufficiently fine opens covering X and the g are elements of O× (V ∩ V ) satisfying the relation ij Vi∩Vj i j

(cc) gij|Ui∩Uj ∩Uk · gjk|Ui∩Uj ∩Uk = gik|Ui∩Uj ∩Uk .

Two such elements [{Vi ∩ Vj, gij}i,j] and [{Wi ∩ Wj, hij}i,j] are equivalent if there are smaller opens Yi ∩ Yj and elements pi,j satisfying (cc) along with equalities −1 (ch) pij · (gij|Yi∩YJ ) · pij = hij|Yi∩Yj . To define α(L) we make a collection f from the isomorphism f | ◦ (f )|−1 and set ij j Ui∩Uj i Ui∩Uj α(L) = {U ∩ U , f }. More precisely, since each f | ◦ (f )|−1 is an automorphism of O i j ij j Ui∩Uj i Ui∩Uj Ui∩Uj there is a determined section in OUi∩Uj (Ui ∩Uj) which is moreover invertible giving rise to the given automorphism, i.e. this morphism corresponds to multiplication by an element of O× (U ∩ U ). Ui∩Uj i j We define f to be the element of O× (U ∩ U ) corresponding to f | ◦ (f )|−1 . That the ij Ui∩Uj i j j Ui∩Uj i Ui∩Uj fij satisfy the cocycle condition (cc) is immediately clear from their construction. Now we wantα ˜ to be the induced map on Pic(X). However, we still need to show this makes sense. To do this, we check if L =∼ L0 is an isomorphism of invertible sheaves then [α(L)] = [α(L0)]. In this case the isomorphisms ∼ ∼ 0 ∼ OUi∩Uj = L|Ui∩Uj = L |Ui∩Uj = OUi∩Uj 1 × for every pair i, j also determine an element e of H (X, OX ) and this element e provides the information needed for equivalence in (ch), i.e. we have [α(L)] = e + [α(L0)] − e = [α(L0)]. We still need to checkα ˜ is bijective. Injectivity follows from checking [α(OX )] = 0 (which is obvious), and from [α(L⊗L0)] = [α(L)]+[α(L0)] (which can be observed by noting an isomorphism 0 ∼ ∼ 0 ∼ (L ⊗ L )|U = OU can be found from the tensor product of isomorphisms L|U = OU , L |U = OU ). Surjectivity follows from noting that (cc) is exactly a gluing condition for sheaves. If a = {Vi ∩ 1 × Vj, gij}i,j is a representative of an element of H (X, OX ) then we can define an invertible sheaf La by setting La|Vi = OVi and putting the gluing data

OVi∩Vj = (La|Vi )Vi∩Vj → (La|Vj )|Vj ∩Vi = OVj ∩Vi to be the morphism sending 1 to gij. Then α(La) = [a]. Proposition 2.20. The sequence (D1) is exact. Proof. By applying the global sections functor to the exact sequence × × 0 → OX → KX → Div(X) → 0 we get a long exact sequence × × 1 × 1 × 0 → OX (X) → k(X) → Div(X) → H (X, OX ) → H (X, KX ). 1 × Now we can use the isomorphism of 2.19 to substitute Pic(X) for H (X, OX ). Finally, we have 1 × × H (X, KX ) = 0 since KX is constant and hence flasque. 14 Definition 2.21. The cokernel of the divisor map div : k(X)× → Div(X) is called the divisor class group. We denote this group CH1(X). Corollary 2.22. There is an isomorphism CH1(X) −→∼ Pic(X) obtained from the exact sequence (D1). n Example 2.23. The Picard group of X = P is isomorphic with Z. To see this, we can define an 1 ∼ isomorphism CH (X) = Z where the positive generator of Z is given by the class of V+(x0). Indeed, any irreducible divisor of X is defined by one equation, see Corollary 2.12. So if D is an irreducible divisor then say D = V+(f). I claim [D] = deg(f)[V+(x0)]. This is because we can find a rational function g/h with div(g/h) = V+(g) − nV+(x0), see Example 2.6. Then

0 = [div(g/h)] = [V+(g)] − n[V+(x0)]. Due to Example 2.14, another description of Pic(X) is as the group Z generated by O(1). Now we can classify all maps from projective space to another projective space. For any morphism n m P → P the map is completely determined by m+1 sections of a line bundle O(a) for some a > 0. Compare with Example 1.3. Example 2.24. Let X = Gr(n, k) be the Grassmannian of k-planes in an n-dimensional vector ∼ Vn−k space. Then Pic(X) = Z with generator Q where Q is the universal quotient bundle (see Example 1.5).

Example 2.25. Let X be a smooth projective curve over C. Then Pic(X) = Z ⊕ C where C is a 1 certain abelian variety of dimension the genus g = H (X, OX ) of X. To see this, one can use the exponential exact sequence (in the analytic topology)

2πif f7→e × 0 → Z → OX −−−−−→OX → 1 to get a long exact cohomology sequence 0 0 0 × 0 → H (X, Z) → H (X, OX ) →H (X, OX ) → 1 1 1 × 2 → H (X, Z) → H (X, OX ) → H (X, OX ) → H (X, Z) → 0. 1 × ∼ By Proposition 2.19,H (X, OX ) = Pic(X) and we’ll compute the remaining groups. The group 2 2 H (X, Z) is isomorphic with the singular cohomology Hsing(X, Z) of X, hence is Z. The group 1 1 2g H (X, Z) is similarly equal to the singular cohomology Hsing(X, Z) = Z . Note that the map 0 0 × × 1 C = H (X, OX ) → H (X, OX ) = C is surjective as it is the exponential. The group H (X, OX ) is canonically a finite dimensional C vector space which we are assuming has dimension g. Altogether this yields an isomorphism ∼ g 2g Pic(X) = Z ⊕ C /Z as claimed. More generally, given any field k and a smooth projective curve X over k there is an abelian ∼ variety A over k so that Pic(X) = Z⊕A(k) where A(k) denotes the k-rational points of A. Moreover, 1 the dimension of A as a k-variety is equal to the genus g of X, i.e. g = dim H (X, OX ). This is shown in [Kle05] using heavy formalism from descent theory and representable functors. By Corollary 2.22, there is a natural isomorphism CH1(X) → Pic(X). The inverse of this map is 1 called the first Chern class and denoted by c1 : Pic(X) → CH (X). The next two lemmas explore this inverse in more detail. 15 Lemma 2.26. If D is an irreducible divisor on a variety X satisfying the conditions of Lemma 2.10, then O(−D) is isomorphic with the ideal sheaf of D.

Proof. We can define an isomorphism directly. Let ID be the ideal sheaf of D. Define ID → O(−D) locally on an open affine cover {Ui = Spec(Ri)}i by g 7→ g. Note that if we assume Ui are small enough then the ideal sheaf is canonically the sheaf associated to fiRi over Ui. The map we defined makes sense: if g ∈ ID(Ui) ⊂ OX (Ui) ⊂ k(X) is then g = hfi for a local equation fi of D ∩ Ui and

(div(g) − D)|Ui = (div(hfi) − div(fi))|Ui = div(h)|Ui ≥ 0.

This map is surjective on stalks: if g ∈ k(X) is such that (div(g) + D)|U ≥ 0 for some open U containing a point x, then, possibly shrinking U to a subset of some Ui, we can write

div(g/fi)|U = (div(g) + div(1/fi))|U = (div(g) − D)|U ≥ 0 which, by Lemma 2.13, shows g/fi ∈ OX (Ui) or g ∈ fiOX (Ui) = ID(Ui). Since these sheaves are both locally free of the same rank, surjectivity on stalks implies isomorphism. Lemma 2.27. If L is an invertible sheaf on X and s ∈ L(X) is a nonzero global section, then consider the subvariety of X deﬁned by Z(s) := {x ∈ X : s(x) = 0}.

In this case we have c1(L) = [Z(s)]. Proof. Under the map Div(X) → Pic(X) we have D 7→ O(D). It suffices by Lemma 2.26 to show there is a short exact sequence ∨ 0 → L → OX → OZ(s) → 0. ∨ ∨ One can define a map OX → L sending 1 to s and then take the dual to get the map L → OX = OX defined above. The rest of the Lemma can be checked directly. n Example 2.28. In this example we go into detail for the first Chern class of X = P . We’ve already 1 seen Pic(X) = Z, generated by the class of O(1) or O(−1). We’ve also seen that CH (X) = Z, generated by the class of a hyperplane. Since the global sections of O(1) are exactly the cooridinate functions x0, ..., xn on X, a nonzero global section vanishes along some hyperplane. Thus we find

c1(O(1)) = [H]. For higher n > 0, the nonzero global sections of O(n) have vanishing at the n hypersurfaces of X. By Example 2.23 this means c1(O(n)) = n[H]. Of course, we already knew this because of Example 2.14.

References [Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, Graduate Texts in Mathematics, No. 52. MR 0463157 [Kle05] Steven L. Kleiman, The Picard scheme, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 235–321. MR 2223410 [Laz04] Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004, Classical setting: line bundles and linear series. MR 2095471 [Rei95] Miles Reid, Undergraduate commutative algebra, London Mathematical Society Student Texts, vol. 29, Cam- bridge University Press, Cambridge, 1995. MR 1458066 [Sta17] The Stacks Project Authors, stacks project, http://stacks.math.columbia.edu, 2017. 16 Mathematical & Statistical Sciences, University of Alberta, Edmonton, CANADA E-mail address: mackall at ualberta.ca URL: www.ualberta.ca/~mackall