## 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

### Citations:

Zbl 0628.14036; Zbl 0549.14004
Full Text:

### References:

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