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256 10 Variations of Hodge Structure this means that for every curve X and every universal family of deformations X -> В of X, the set of points b e В such that Xb is hyperelliptic is a proper analytic subset of В if g > 2, and the set of points b e В such that Xb is trigonal or isomorphic to a smooth plane curve of degree 5 is a proper analytic subset of В if g > 5. Theorem 10.24 (See Arbarello et al. 1985) (a) (Noether) Let X be a non-hyper elliptic curve. Then the map given by the product x : Я°(Х, Kxf1 -> H°(X, Kf) is surjective. (b) (Petri) Let X be a curve which is non-hype relliptic, non-trigonal and not isomorphic to a planar quintic. Then X is determined by Ker x : Я°(Х, Kxf2 -» Я°(Х, Kf) in the following way: the canonical map фкх :X->F8~X is an embedding since X is not hyperelliptic, and the symmetric elements of Ker x are exactly the homogeneous polynomials of degree 2 over?8~x which vanish on X; when X is neither trigonal nor isomorphic to a planar quintic, X С ?8~х is isomorphic to the algebraic subscheme or complex submanifold defined by these equations. These algebraic statements now give us the following results on the period map for curves. Corollary 10.25 (Infinitesimal Torelli theorem for curves) Let X be a non- hyperelliptic curve. Then the local period map V : В is an embedding at the point 0 € В corresponding to X. Proof By theorem 10.24 and lemma 10.22, dVl is injective at 0 when X is not hyperelliptic. D Corollary 10.26 Ifg>5,a generic curve X is determined by its infinitesimal variation of Hodge structure and by the isomorphism Я1H(Х) (H X, C) #1i0(X))*. A0.10) 10.3 Applications 25 7 In particular, assume that we have two curves X and X , an isomorphism compatible with the intersection forms i : Hl(X, C) HX(X , C), and a germ of isomorphisms j : (B,0) (Bf,0f) between the bases of the local universal deformations ofX and X respectively, giving an identification of the variations of Hodge structure in the neighbourhood of X and ofX : Vх : В Vх : B Then X and X are isomorphic. V Grass(g, Hx(Xf, Q). Proof The first statement follows from the fact that the differential dVx at the point 0 6 В corresponding to X gives a symmetric map (relative to the Serre duality Hh0(X) (HX(X, C) HX °(X))*): W Нот(Я1H(Х), Hl(X, The isomorphism Я1 0^) (Hl(X, С) Я1>0(Х))* induced by the intersection form on HX(X, C) makes it possible to dualise this map to a symmetric map x : HX °(X) °(X) -» W*. Lemma 10.22 and theorem 10.24 then sho>v that if X is not hyperelliptic, trigonal or a planar quintic, X can be identified with the subscheme of Р(Я10(Х)*) defined by the symmetric elements of Ker x. The second result follows immediately from this, since by differentiation, the commutative diagram above gives an identification of the infinitesimal vari- variations of Hodge structures at the points b and j(b), Vb e B, compatible with the duality isomorphisms A0.10) since i preserves the intersection form, i.e. a commutative diagram where the vertical arrows are isomorphisms: Тв,ь Uom(Hx0(Xb), Hx(Xb, C) Hx<°(Xb)) ), Hx{X Kbh But then for generic b, we must have Xb Xjqj), since Xb and Xj^) are determined by Ker [ib and Ker x^^) respectively. It then follows that Xb X^) for every b e B. (The moduli space of the curves is separated (Deligne & Mumford 1967).) D The result shown above is the generic Torelli theorem for curves of genus > 5. It says essentially that the global period map, which to a curve X associates the polarised Hodge structure on Я1 (X, Z), is of degree 1 on its image. In the second volume, we will prove a similar statement due to Donagi for most of the families