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Sur les variétés de Prym des courbes tétragonales. (On the Prym varieties of tetragonal curves). (French) Zbl 0674.14034

The author proves the conjecture of R. Donagi [Bull. Am. Math. Soc., New Ser. 4, 181-185 (1981; Zbl 0491.14016)] in the “tetragonal case”. More precisely, let \(\pi: \tilde C\to C\) be a double covering of a tetragonal curve C (where C is non hyperelliptic, non trigonal and non bielliptic) and let the associate Prym variety \(P(\tilde C,\pi,C)\) be isomorphic to some Prym variety \(P(\tilde D,\rho,D)\) (as a principal polarized abelian variety). The author proves that for \(g(C)\geq 13\) the pair \((\tilde D,D)\) can be either \((\tilde C,C)\) or one of the two following pairs \((\tilde C,C')\), \((\tilde C'',C'')\) which correspond to \((\tilde C,C)\) by the tetragonal construction (corollary 4.4). To prove that he applies the methods of G. E. Welters [Am. J. Math. 109, 165-182 (1987; Zbl 0639.14026)], especially to the “tetragonal case”. The first idea in the proof is that the dimension of the locus \(Sing_{ex}\Xi\) of the exceptional singularities of the theta divisor \(\Theta\) of a Prym variety \(P=P(\tilde C,C)\) is equal to dim(P)-6 only if the curve C is tetragonal (theorem 3.1). (Theorem 4.2) gives that Sing(\(\Theta)\) is a union of three components \(S,S'\) and \(S''\) of dimension \(\dim(P)-6.\) The triple \((S,S',S'')\) determines a triple of surfaces \((\Sigma,\Sigma',\Sigma'')\) in the Prym variety P (theorem 3.2). The surfaces \(\Sigma,\Sigma',\Sigma ''\) are singular at \(0\in P\) and the associate projective tangent cones at 0 are actually the semicanonical curves \(C,C',C''\) which determine uniquely some two-sheeted coverings of \(C,C',C''\), namely those from the tetragonal triple (cf. (lemma 4.5).
Reviewer: A.I.Iliev

MSC:

14K30 Picard schemes, higher Jacobians
14H45 Special algebraic curves and curves of low genus
14K25 Theta functions and abelian varieties
14H40 Jacobians, Prym varieties
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References:

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