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An existence theorem for Prym special divisors. (English) Zbl 0646.14006

Let C be a smooth irreducible projective curve of genus g (over an algebraically closed field of characteristic \(\neq 2)\) and let \(\pi: \tilde C\to C\) be an irreducible, étale double cover of C. Associated to \(\pi\) we have the norm map \(Nm: Pic^{2g-2}(\tilde C)\to Pic^{2g- 2}(C).\) If \(\omega_ C\) denotes the canonical bundle of C, one can consider for any integer r the subvarieties \(V^ r\) of \(Nm^{- 1}(\omega_ C)\) formed by all line bundles L such that \(h^ 0(\tilde C,L)\geq r+1\), \(h^ 0(\tilde C,L)\equiv r+1\quad (mod 2).\) A known fact about these varieties \(V^ r\) is that \(\dim (V^ r)\geq g-1-r(r+1)/2\) if \(V^ r\neq \emptyset\). A recent theorem by G. E. Welters [Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 671-683 (1985; Zbl 0628.14036)] states that \(V^ r\), if not empty, has exactly the expected dimension \(g-1-r(r+1)/2\) and is smooth away from \(V^{r+2}\) if C is a general curve of genus \(g\) and \(\pi: \tilde C\to C\) is any irreducible, étale double covering. - Moreover Welters also conjectures that an existence result of Kempf’s and Kleiman-Laksov’s type should hold: namely that \(V^ r\neq \emptyset\) as soon as \(g-1-r(r+1)/2\geq 0\) for any curve C of genus \(g\) and any covering \(\pi: \tilde C\to C\) (in fact he proves a weaker form of this conjecture, namely that \(V^ r\neq \emptyset\) when \(g\geq (r+1)^ 2+1).\)
The present note answers Welter’s conjecture in the affirmative. The (easy and elegant) proof is based on the results on excess linear series on an algebraic curve contained in a paper by W. Fulton, J. Harris and R. Lazarsfeld [Proc. Am. Math. Soc. 92, 320-322 (1984; Zbl 0549.14004)].
Reviewer: C.Ciliberto

MSC:

14C20 Divisors, linear systems, invertible sheaves
14H25 Arithmetic ground fields for curves
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References:

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