Bertram, Aaron An existence theorem for Prym special divisors. (English) Zbl 0646.14006 Invent. Math. 90, 669-671 (1987). 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 Cited in 3 ReviewsCited in 9 Documents MSC: 14C20 Divisors, linear systems, invertible sheaves 14H25 Arithmetic ground fields for curves Keywords:theta-divisor of a Prym variety; norm map; linear series on an algebraic curve Citations:Zbl 0628.14036; Zbl 0549.14004 PDFBibTeX XMLCite \textit{A. Bertram}, Invent. Math. 90, 669--671 (1987; Zbl 0646.14006) Full Text: DOI EuDML References: [1] Arbarello, E., Cornalba, M., Griffiths, P., Harris, J.: Geometry of Algebraic Curves. Berlin Heidelberg New York: Springer 1984 · Zbl 0559.14017 [2] Debarre, O.: Variétés de Prym et ensembles d’Andreotti et Mayer (Preprint) · Zbl 0716.14029 [3] Fulton, W., Harris, J., Lazarsfeld, R.: Excess linear series on an algebraic curve. Proc. Am. Math. Soc.92, 320-322 (1984) · Zbl 0549.14004 [4] Gieseker, D.: Stable curves and special divisors: Petri’s conjecture. Invent. Math.66, 251-275 (1982) · Zbl 0522.14015 [5] Griffiths, P., Harris, J.: The dimension of the variety of special linear systems on a general curve. Duke Math. J.47, 233-272 (1980) · Zbl 0446.14011 [6] Harris, J.: Theta characteristics on algebraic curves. Trans. Am. Math. Soc.271, 611-638 (1982) · Zbl 0513.14025 [7] Kempf, G.: Schubert methods with an application to algebraic curves. Amsterdam: Publ. Math. Centrum 1972 · Zbl 0223.14018 [8] Kleiman, S., Laksov, D.: Another proof of the existence of special divisors. Acta Math.132, 163-176 (1974) · Zbl 0286.14005 [9] Mumford, D.: Prym varieties I. In: Contributions to analysis. New York: Academic Press, 1974, pp. 325-350 [10] Mumford, D.: Theta characteristics of an algebraic curve. Ann. Sci. Ec. Norm. Super.4, 181-192 (1971) · Zbl 0216.05904 [11] Welters, G.E.: A theorem of Gieseker-Petri type for Prym varieties. Ann. Sci. Ec. Norm. Super.18, 671-683 (1985) · Zbl 0628.14036 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.