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Quadrics of rank four in the ideal of a canonical curve. (English) Zbl 0542.14018
Let C denote a smooth curve of genus g, K its canonical bundle and $$\phi:C\to P(H^ 0(C,K)^*),$$ the canonical map. If J(C) is the Jacobian of line bundles of degree g-1 on C and $$\Theta$$ a (suitably chosen) Theta-divisor on J(C) then according to a theorem of Riemann we have for any line bundle $$L\in J(C): mult_ L\Theta =\dim H^ 0(C,L)=\dim H^ 0(C,K\otimes L^{-1}).$$ In particular if D is an effective divisor of degree g-1 on C such that $${\mathcal O}_ C(D)$$ is a double point of $$\Theta$$, the space of rational functions $$g_ 1$$ with $$(g_ 1)\geq -D$$ has a base $$\{$$ 1,$$f\}$$ and $$H^ 0(C,K(-D))$$ has a base $$\{w_ 1,w_ 2\}$$. It follows that $$\eta_ i=fw_ i\quad(i=1,2)$$ are holomorphic differentials and one can show that the 2 quadratic differentials $$\eta_ 1w_ 2$$ and $$\eta_ 2w_ 1$$ are equal. - This implies that $$q_ D=\eta_ 1w_ 2-\eta_ 2w_ 1,$$ considered as a quadratic form on $$H^ 0(C,K)^*$$, vanishes on the affine cone over $$\phi$$ (C). Thus to every double point of $$\Theta$$ one can associate in a natural way a quadric (of rank 4) in $$P(H^ 0(C,K)^*)$$ vanishing on $$\phi$$ (C). In the paper in question it is proven that the space of all quadrics in $$P(H^ 0(C,K)^*)$$ through $$\phi$$ (C) is spanned by the above quadrics arizing from double points of $$\Theta$$. - This was proven for hyperelliptic and trigonal curves by A. Andreotti and A. L. Mayer [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 21, 189- 238 (1967; Zbl 0222.14024)] for $$g\leq 6$$ by E. Arbarello and J. Harris [Compos. Math. 43, 145-179 (1981; Zbl 0494.14011)] and was a long standing question in general. A consequence is an alternate proof of Torelli’s theorem for curves which are neither hyperelliptic, nor trigonal nor plane quintics.
Reviewer: H.Lange

##### MSC:
 14H40 Jacobians, Prym varieties 14K25 Theta functions and abelian varieties 14C20 Divisors, linear systems, invertible sheaves
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##### References:
 [1] Andreotti, A.: On a theorem of Torelli. Am. J. Math.80, 801-828 (1958) · Zbl 0084.17304 [2] Andreotti, A., Mayer, A.: On period relations for abelian integrals and algebraic curves. Ann. Scuola Norm. Sup. Pisa21, 189-238 (1967) · Zbl 0222.14024 [3] Arbarello, E., Harris, J.: Canonical curves and quadrics of rank 4. Comp. Math.43, (Fasc. 2) 145-179 (1981) · Zbl 0494.14011 [4] Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Topics in the theory of algebraic curves. To appear · Zbl 0559.14017 [5] Kempf, G.: Deformations of symmetric products. Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook conference. Princeton (1981), pp. 319-341
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