MIT OpenCourseWare http://ocw.mit.edu 18.727 Topics in Algebraic Geometry: Algebraic Surfaces Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Recall from last time that we defined the group scheme PicX over k as well as the group scheme Pic0 X , which is the connected component of 0 (i.e. X ) in PicX (and is a proper scheme over k). Now, let L Obe a line bundle in the class corresponding to the universal element. L is a line bundle on X × Pic 0 X . Choose a closed point x of X and let M = L| 1{ }×0 x Pic. Then replace L by L ⊗ (p∗M)−X2so that we get L | x Pic0 ∼=O Pic0and, for every closed point a {∈ Pic 0 the }×X XX ,line bundle La = LX ais algebraically equivalent to 0. Such an L is called a ×{ }Poincar´e line bundle on X × Pic 0 X . Given a choice of basepoint a, it is unique up to isomorphism. Now, note further that the Zariski tangent space at 0 of Pic 0 X is canonically isomorphic to H1(X, OX ) and Pic0X is a commutative group scheme. If it is reduced, then it is an abelian variety. If char(k) = 0, it is automatically reduced (by a theorem of Grothendieck-Cartier). Theorem 1. Let X be a surface, q = h1(X, X ) its irregularity, s the dimension of the Picard Ovariety of X. Let b1 be the first Betti number = h1et´(X, Q�). Thenb1 = 2s and Δ = 2q − b1 = 2(q − s) lies between 0 and 2pg, while Δ = 0 if char(k) = 0. Proof. Note that, for � relatively prime to p = char(k), � >> 0 )b1 = H1 (Z/�Z et´(X, Z/�Z) = {a ∈ Pic X|� (1)· a = 0}= {a ∈ Pic 0X|� · a = 0} = (Z/�Z)2s where the second equality follows from Kummer theory on 0 → µ� → F m →�Fm → 0, the second from the fact that Pic /Pic is finitely generated, so thetorsion group is finite and � can be chosen larger than the size of the torsion group, and the third because Pic 0(X) is the underlying abelian group of the 0 0 Picard variety of X. The closed points of PicX = (PicX )red, so b1 = 2s.Now, (2) Δ = 2q − b1= 2(q − s) = 2dim TPic0,0dim T(Pic0X−)Xred,0 ≥ 0 and (3) q − s = dim H1(X, OX ) − dim (∩i∞=1Ker βi)where the βi are the Bockstein homomorphisms defined inductively by (4) β : H1(X, O ) → H11 X (X, OX ), βi : Ker βi−1 1 → coker βi−1 0 ALGEBRAIC SURFACES, LECTURE 11Thus, q − s ≤ dim (∪∞i=1Im βi) ≤ h2(X, OX ) = pg. In characteristic 0, propergroup schemes of finite type are reduced, so Pic0X is already an abelian variety. �0.1. The Albanese Variety. Let X be a smooth projective variety, x0 ∈ X a fixed closed point. A pair (A, α) consisting of an abelian variety and a morphism α : X→A s.t. αx0 = 0 is called the Albanese variety of X. For every morphismf : X → B s.t. B is an abelian variety and f(x0) = 0, ∃ a unique homomorphismof abelian varieties g : A → B s.t. the diagram below commutes.f X B �(5) �α ��� �g ��A Note that a rigidity theorem for abelian varieties implies that any morphism (as varieties) g� : A → B is of the form g�(a) = g(a) + b where g : A → B is ahomomorphism of abelian varieties and b = g�(0) ∈ B. Thus, we can formulate the definition without the closed point x0, where we say that there exists a unique homomorphism g : A → B s.t. g ◦ α = f. It is clearly unique if it exists.For existence, let X be a smooth projective variety, and let P (X) be the re-duced Picard variety of X, and P (X)∨ its dual abelian variety. Then Pic0P (X) = P (X)∨ (for an abelian variety, Pic 0 is automatically reduced). We have a univer-sal Poincar´e line bundle L on X × Pic0X and therefore on the reduced subschemeX × P (X). Let µ : X × P (X) → X × P (X) be the switch (y, x) �→ (x, y). µ∗L is a line bundle on P (X) X and therefore comes from the Poincar´e bundle on P (X) × ×P (X)∨ by a map X → P (X)∨ (by the universal property of Pic 0 P (X)). ����� 2 ��One can check that this gives P (X)∨as the Albanese variety of X using general nonsense, so Alb (X) exists and is unique up to unique isomorphism. Further-more, it is dual to the Picard variety of X. Note: If X is a smooth projective curve, then Pic 0(X) is reduced and carries a principal polarization, so P (X)∨ ∼= = Pic 0 is the Jacobian of X. ForP (X) ∼X a surface, we showed that the dimension of the Albanese variety is ≤ q, with equality holding Δ = 0 (e.g. if char(k) = 0 or if pg = 0).⇔If k = C, there is an explicit way to see the Albanese variety. We have a map i : H1(X, Z) H0(X, Ω1 )∗ defined by �i(γ), ω� = ω. The image of i is a→ Xγ lattice in H0(X, Ω1 )∗, and the quotient is an abelian variety (a priori a complexX torus, but a Riemann form exists). It is Alb (X), and is functorial in X, i.e. X ��Y (6) αX �� ��αY Alb (X) ∃! ��Alb (Y )� � � � � ������������ 3 ALGEBRAIC SURFACES, LECTURE 11 It follows that the image of X in Alb (X) generates the abelian variety (else the subvariety that X generates inside Alb (X) would satisfy the universal property). In particular, if Alb (X) = 0, α(X) is not a point, and if X Y is a surjection,� → so is Alb (X) Alb (Y ). Over C, our construction gives us an isomorphism α∗ : H1(X, Z)→→ H1(Alb (X), Z), so the inverse image under α of any ´etale covering of Alb (X) is connected. All Abelian coverings are obtained in this way. For now, assume that char(k) = 0. Proposition 1. Let X be a surface, α : X Alb (X) the Albanese map. Sup-→ pose α(X) is a curve C. Then C is a smooth curve of genus q, and the fibers of α are connected. We first prove the following lemma: Lemma 1. Suppose α factors as X f T j Alb (X) with f surjective. Then→ →˜j : Alb (T ) Alb (X) is an isomorphism.→ Proof. The functoriality of Alb gives a surjective morphism Alb (X) Alb (T )→(since X T is surjective), along with a commutative diagram→ f X ��T αT(7) αx = Alb (X) ∼��Alb (T ) ��Alb (X)f ˜j ˜j f is the identity by the universal property, so ˜j must be an isomorphism.