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.� � � ALGEBRAIC SURFACES, LECTURE 15 LECTURES: ABHINAV KUMAR 1. Elliptic Surfaces (contd.) Assume we are over C. Given f : X C, we can associate the functional in-P1→variant j : C , c �→ j(fc) and a homological invariant: let c1, . . . , cn be points →over which the fibers are singular, C∗ = C � {c1, . . . , cn} and X∗ = f−1(C∗); then we have the sheaf R1f Z, which is a homological invariant; (R1f Z) ⊗C C = ∗ ∗H1(FC, Z), so it is a locally constant sheaf. We get a representation G : π1(C∗) →SL2(Z) called the global monodromy. The equivalence class of this representation is called the homological invariant. The local monodromy at a point ci is the image of a loop around that point in SL2(Z). We can write down the conjugacy 1 n class of the local monodromy for the bad fibers: for a In fiber, it is .0 1 J and G determine the elliptic fibration up to isomorphism. Given any elliptic fibration, we get a corresponding Jacobian fibration which does have a section O : C X.→Let X C be an elliptic fibration with section ), and let the class of the → � fiber be F . For each singular fiber, let Fv = Θv,0 + Mv,iΘv,i, where Θv,0 is the identity component. Algebraic and numerical equivalence are the same for elliptic surfaces, and KX = f∗(KC − N), where N is isomorphic to the normal bundle of the zero section O in X. Note that OC (N) = R1f∗OX . Thus, KS ≡ (2g−2+χ)F , where χ = the “arithmetic genus” of X = h0(OX ) − h1(OX ) + h2(OX ), with (KS )2 = 0 and deg N = 02 = −χ. For any section P we have (1) 2g − 2 = P (P + K) = P 2 + P K = P 2 + 2g − 2 + χ· so P 2 = −χ. Let T be the subgroup of NS(X) generated by the zero section O and all irreducible components of fibers (called the trivial lattice) X(C) ∼= NS(X)/T(2) = E(K) ∼where K = k(C). Here T is torsion free, and the map is given by P ∈ E(K) �→ P mod T . We can obtain the Shioda-Tate formula: rk NS(X) = 2 + (mv − 1) + rk , E(k), where rk E(k) is the Mordell-Weil rank. 1� 2 LECTURES: ABHINAV KUMAR Remark. An open arithmetic question: is the rank of E(K) unbounded for K = K(C), the function field of a curve (e.g. C = P1 , E(Q(x)))? For Fp(k), the answer is yes (Shafarevich-Tate). For E(Q), we can write down a finite list of the possible torsion (Mazur’s theorem). 2. Kodaira dimension 0 If X is a surface with κ(X) = 0, then (K2) = 0, pg ≤ 1 (since K2 < 0 would imply K ruled): we will show this later. Noether’s formula says that 10 − 8q + 12pg = b2 + 2Δ, where Δ = 2(q − s). Then 0 ≤ Δ ≤ 2pg and Δ even implies that it is 0 or 2. Now, the four possibilities based on combinations of (b1, b2) are: (0, 10) are Enriques surfaces, “classical” if (pg, q, Δ) = (0, 0, 0) and “non-classical” if (pg, q, Δ) = (1, 1, 2). (2, 2) are bielliptic or hyperelliptic surfaces if (pg, q, Δ) = (0, 1, 0), and quasi-hyperelliptic surfaces if (pg, q, Δ) = (1, 2, 2). (0, 22) are K3 surfaces, with χ(OX ) = 0, q = 0, pg = 1, Δ = 0, and (4, 6) are abelian surfaces, with χ(OX ) = 0, q = 2, pg = 1, Δ = 0. Note that, by the above formulae, there is another possible combination: (b1, b2) = (2, 14), q1, pg = 1, Δ = 0. Proposition 1. No surface exists with this combination of invariants. Proof. Since b1 = 2 = 2s, the Picard variety of X has dimension 1. So ∃ a line bundle L0 on X difference from OX but algebraically equivalent to 0. Since K ≡ 0 and pg = 1, we get KX Applying Riemann-Roch to L1 ∼ OX . gives χ(L) = 2 L · (L ⊗ K−1) + χ(OX ) = χ(OX ) = 1. Thus, H0(L > 0 or H0(L−1) = H2(L) > 0. But L ≡ 0 gives L ∼ OX , a contradiction. � 2.1. Ab elian surfaces. These are smooth, complete group varieties of dimen-sion 2 over k. For the general theory, we refer to Mumford’s Abelian Varieties and Birkenhaake-Lange’s Complex Abelian Varieties. Over C, these are com-plex tori: C2/Λ, equipped with a Riemann form ψ : Λ × Λ → Z. This form is alternating, bilinear, and satisfies the following two conditions: • ψR : C2 × C2 → R satisfies ψR(iv, iw) = ψR(v, w) for v, w ∈ C × C. • The associated Hermitian form H(v, w) = ψR(iv, w)+iψR(v, w) is positive definite. This gives rise to a line bundle L on C2/Λ which is the class of an ample line bundle. L is called a polarization of the abelian variety, and (3) h0(L) = disc (φ) = |Pf (ψ)| In general, over any characteristic, A∨ = Pic 0A is the dual abelian variety, and it is automatically reduced. A polarization is an isogeny A A∨: for example, →3 ALGEBRAIC SURFACES, LECTURE 15 whenever we have an ample line bundle L on A, the subgroup K(L) = {x ∈A T ∗ = L} is finite. The map A A∨, x �→ Tx ∗L ⊗ L−1 gives an isogeny with | x L∼ →finite kernel K(L), i.e. a polarization. For a line bundle L, (4) χ(L) = Lg = L2 , χ(L)2 = deg φLdim A 2 The polarization is principal if it has degree 1. There are two common examples of abelian surfaces with principal polarization: (1) J(C), C a genus 2 curve. Here, we can explicitly write down equations for J(C) (e.g. Cassels-Flynn, Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2). The image of C is J(C) under x �→ [x] − [x0], where x0 ∈ C is a fixed point, is the theta divisor and it generates NS(J(C)) for a generic C. We have C2 = C (C + K) = 2g − 2 = 2 since K ≡ 0· for an abelian variety. The theta divisor gives the principal polarization χ(OX (C)) = C2 = 1, so deg φC = χ(OX (C))2 = 1. 2 (2) Fix E2 with E1, E2 elliptic curves. Then E1 + E2 gives a principally polarized abelian variety, with E1 ·E2 = 1, Ei 2 = 0 = (E1 +E2)2/2 = 1. ⇒ The moduli space of principally polarized abelian varieties is made up of these two types of points, with the second type forming the boundary. There are lots of arithmetic questions here. By the Mordell-Weil theorem, if …