Algebraic Geometry 3-Homework 7 1) Divisors and Invertible Sheaves. Let X Be a Noetherian Scheme

Algebraic Geometry 3-Homework 7

1) Divisors and invertible sheaves. Let X be a noetherian scheme; for sim- plicity we assume that X is integral. Let K be the function field of X, KX the corresponding constant sheaf on X. For U = Spec A affine, we have the inclusion A → K; these inclusions induce the monomorphism of sheaves of rings on X, OX → KX . Restricting to the group of units gives the inclusion × × of sheaves of groups OX → KX and the short exact sequence × × × × 1 → OX → KX → KX /OX → 1 a. Recall that a sheaf F of abelian groups on a topological space T is flasque if for each inclusion of open subsets U ⊂ V , the restriction map F(V ) → F(U) is surjective. By a well-known theorem Hi(T, F) = 0 for i > 0 if F is a flasque sheaf on T . Show that there is an exact sequence of abelian groups (*) 0 × 0 × div 0 × × cl 1 × 1 → H (X, OX ) → H (X,KX ) −−→ H (X,KX /OX ) −→ H (X, OX ) → 1. 0 × × The group CartX := H (X,KX /OX ) is called the group of Cartier divisors 0 × × × on X. We have H (X,KX ) = K and the map div : K → CartX is the divisor map. 1 × ∼ ˇ 1 × ˇ 1 × b. Using (*), show that H (X, OX ) = H (X, OX ). Recall that H (X, OX ) is isomorphic to the group Pic(X) of isomorphism classes of invertible sheaves on X. n c. Let ξ be a Cartier divisor. Show there is a finite open cover X = ∪i=1Ui, × × elements fi ∈ K , uij ∈ OX (Ui ∩ Uj) such that i. for each i, j, fi = uijfj ii. {fi,Ui} defines an element of CartX , equal to ξ. d. Take ξ ∈ CartX , {fi,Ui} as in (c) representing ξ. Define a sheaf of OX -modules OX (ξ) on X by

OX (ξ)(U) = {g ∈ K | gfi ∈ OX (U ∩ Ui), i = 1, . . . , n} ∼ Show that OX (ξ)|Ui = OUi for each i, so OX (ξ) is an invertible sheaf. Show that cl(ξ) is the class of OX (ξ) in Pic(X). d. The group DivX is the free abelian group on the integral codimension one closed subvarieties of X, an element of DivX is called a Weil divisor, or simply a divisor on X. A point x ∈ X has codimension c if the closure {x}, as integral closed subscheme of X, has codimension c. We let X(c) denote the set of codimension c points of X, so DivX may be described as the free abelian group on the set X(1). P Deﬁne a partial order ≥ on DivX by saying D ≥ 0 if D = x nx · x and 0 0 nx ≥ 0 for all x; D ≥ D if D − D ≥ 0. Assume that each x ∈ X(1) is a regular point of X, that is, the local ring OX,x is a regular ring. Then OX,x is a regular local ring of dimension one, × hence a DVR with quotient ﬁeld K. Let ordx : K → Z be the associated 1 2

× order function, that is, if (t) ⊂ OX,x is the maximal ideal, and a ∈ K , then ordx(a) × a = ut with u ∈ OX,x. For ξ ∈ CartX , we may choose a representative ˜ × × × ˜ ξx of the image of ξx ∈ (KX /OX )x in K and let |ξ|x = ord(ξx). Deﬁne |ξ| ∈ DivX by X |ξ| := |ξ|x · x. x∈X(1) × Deﬁne Div : K → DivX by Div(f) = |div(f)|. Equivalently, represent ξ by {fi,Ui} as in (c), and then |ξ|x = ordx(fi) if x ∈ Ui. Since fi = uijfj with × uij ∈ OX (Ui ∩ Uj) , ordx(fi) is independent of the choice of Ui containing x. Assume now that X is a regular scheme. Recall that a noetherian regular local ring is a UFD (theorem of Auslander-Buchsbaum). Show that the map

| − | : CartX → DivX is an isomorphism of abelian groups. Hint: Use the fact that each local ring OX,y is a UFD to deﬁne an inverse to | − |. Conclude that there is an exact sequence

0 × 0 × Div cl 1 → H (X, OX ) → H (X,KX ) −−→ DivX −→ Pic(X) → 1. 2a. Suppose that X is a smooth scheme of finite type over a field k. We eff have defined in class the functor CartX/k: for T a k-scheme eff CartX/k(T ) := {D ⊂ X × T | D is a closed subscheme, flat over T with ideal sheaf ID locally principal and locally defined by a non-zero divisor}

eff Define a map β : CartX/k(Spec k) → CartX which gives a bijection of eff CartX/k(Spec k) with the subset of CartX consisting of elements ξ ∈ CartX such that the associated divisor |ξ| satisfies |ξ| ≥ 0. eff b. For a k-scheme T and D ∈ CartX/k(T ), we have the ideal sheaf ID ⊂ −1 OX×T . ID is an invertible sheaf on X × T and we wrote OX (D) for ID . Show that ∼ OX (β(D)) = OX (D) eff for all D ∈ CartX/k(Spec k). c. Again, suppose that X is a smooth scheme, separated and of finite type over a field k. Take D ∈ DivX . Let OX (D) be the sheaf of OX -modules on X with

OX (D)(U) = {f ∈ K | f = 0 or (Div(f) + D) ∩ U ≥ 0 in DivU } ∼ If D = |ξ| for ξ ∈ CartX , show that OX (D) = OX (ξ). Therefore, OX (D) is an invertible sheaf on X and cl(ξ) is the isomorphism class of OX (D). 3

3. Linear systems. Let X be as in (2). Let T be a k-scheme and L an invertible sheaf on X × T . We have defined the functor L LinX×T/T : SchT → Sets with L 0 eff ∼ ∗ LinX×T/T (f : T → T ) := {D ∈ CartX/k(Spec k) |OX×T 0 (D) = (IdX ×f) F} a. Take D ∈ DivX and let L be the invertible sheaf OX (D) on X. Show L that the map β defined in (2a) gives a bijection of LinX/Spec k(Spec k) with Lin(D) := {D0 ∈ Div(X) | D0 ≥ 0 and there is an f ∈ k(X)× with D0 = D + div(f)} b. Take X and D as in (3a), with now X a smooth projective geometrically 0 integral k-scheme. Let V = H (X, OX (D)) and let P(V ) be the projective space on V , that is ∗ ∨ × P(V ) = Projk(Sym (V )) = V \{0}/v ∼ λv; λ ∈ k . Show that sending [f] ∈ P(V )(k) to D +div(f) gives a well-defined bijection of Lin(D) with P(V )(k). c. Pullback maps. Let f : Y → X be a morphism of smooth irreducible finite type separated k-schemes and let D > 0 be in DivX = CartX . Suppose ∗ that f(Y ) 6⊂ supp (D). Show that f (ID) ⊂ OY is a locally principal ideal, and define f ∗(D) to be the divisor associated to the effective Cartier divisor ∗ f with ideal sheaf f (ID). Letting DivX ⊂ DivX be the subgroup generated by D > 0 with f(Y ) 6⊂ supp (D), show that sending D > 0 to f ∗(D) extends ∗ f to a homomorphism f : DivX → DivY . 0 d. The base locus of the linear system Lin(D) is ∩D0∈Lin(D)supp (D ). Sup- pose X is a smooth projective geometrically integral k-scheme. Show that the base locus of Lin(D) is empty if and only if OX (D) is generated by 0 global sections (H (X, OX (D)) ⊗k OX → OX (D) is surjective) and in this n 0 case, there is a morphism fD : X → Pk , n = dimkH (X, OX (D)) − 1, such n that f(X) is not contained in any hyperplane of Pk and 0 n 0 ∗ Lin(D) = {D ∈ DivX | there is a hyperplane H ⊂ Pk with D = f (H)}. e. Let X be a smooth projective geometrically integral k-scheme, and f : n X → Pk a morphism such that f(X) is not contained in any hyperplane. n ∗ Choose a hyperplane H0 ⊂ Pk and let D0 = f (H0) ∈ DivX . Following (b), we give Lin(D0) the structure of the set of k-points in a projective space 0 ∗ P := P(H (X, OX (D0))). Show that the set of effective divisors {f (H)}, n as H runs over all hyperplanes in Pk , is the set of k-points in a (projective) ∼ n linear subspace L of P, with L = Pk .