Quadrics of rank four in the ideal of a canonical curve.

*(English)*Zbl 0542.14018Let 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 |

##### Keywords:

ideal of canonical curve; Torelli theorem; Jacobian; Theta-divisor; quadrics; double points**OpenURL**

##### 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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