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ON THE BASE LOCUS OF THE LINEAR SYSTEM OF GENERALIZED THETA FUNCTIONS

4 pages
ON THE BASE LOCUS OF THE LINEAR SYSTEM OF GENERALIZED THETA FUNCTIONS CHRISTIAN PAULY Abstract. Let Mr denote the moduli space of semi-stable rank-r vector bundles with trivial determinant over a smooth projective curve C of genus g. In this paper we study the base locus Br ? Mr of the linear system of the determinant line bundle L over Mr, i.e., the set of semi-stable rank-r vector bundles without theta divisor. We construct base points in Bg+2 over any curve C, and base points in B4 over any hyperelliptic curve. 1. Introduction Let C be a complex smooth projective curve of genus g and let Mr denote the coarse moduli space parametrizing semi-stable rank-r vector bundles with trivial determinant over the curve C. Let L be the determinant line bundle over the moduli space Mr and let ? ? Pic g?1(C) be the Riemann theta divisor in the degree g ? 1 component of the Picard variety of C. By [BNR] there is a canonical isomorphism |L|? ? ?? |r?|, under which the natural rational map ?L : Mr 99K |L|? is identified with the so-called theta map ? : Mr 99K |r?|, E 7? ?(E) ? Pic g?1(C).

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ON THE BASE LOCUS OF THE LINEAR SYSTEM OF GENERALIZED THETA FUNCTIONS
CHRISTIAN PAULY
Abstract.LetMrdenote the moduli space of semi-stable rank-rvector bundles with trivial determinant over a smooth projective curveCof genusg. In this paper we study the base locusBr⊂ Mrof the linear system of the determinant line bundleLoverMr, i.e., the set of semi-stable rank-rvector bundles without theta divisor. We construct base points inBg+2over any curveC, and base points inB4over any hyperelliptic curve.
1.Introduction LetCbe a complex smooth projective curve of genusgand letMrdenote the coarse moduli space parametrizing semi-stable rank-rvector bundles with trivial determinant over the curve g1 C. LetLbe the determinant line bundle over the moduli spaceMrand let ΘPic (C) be the Riemann theta divisor in the degreeg1 component of the Picard variety ofC. By [BNR] there is a canonical isomorphism|L| −→|rΘ|, under which the natural rational map ϕL:Mr99K|L|is identified with the so-called theta map g1 θ:Mr99K|rΘ|, E7→θ(E)Pic (C). g1 0 The underlying set ofθ(E) consists of line bundlesLPic (C) withh(C, EL)>0. For g1 a general semi-stable vector bundleE,θ(E) is a divisor. Ifθ(E) = Pic(C), we say thatE has no theta divisor. We note that the indeterminacy locus of the theta mapθ, i.e., the set of bundlesEwithout theta divisor, coincides with the base locusBr⊂ Mrof the linear system |L|. Over the past years many authors [A], [B2], [He], [Hi], [P], [R], [S] have studied the base k locusBrof|L|and their analogues for the powers|L |. For a recent survey of this subject we refer to [B1]. It is natural to introduce for a curveCthe integerr(C) defined as the minimal rank for which there exists a semi-stable rank-r(C) vector bundle with trivial determinant overCwithout theta divisor (see also [B1] section 6). It is known [R] thatr(C)3 for any curveCand that r(C)4 for a generic curveC. Our main result shows the existence of vector bundles of low ranks without theta divisor. Theorem 1.1.We assume thatg2. Then we have the following bounds. (1)r(C)g+ 2. (2)r(C)4, ifCis hyperelliptic. (g+1)(g+2) The first part of the theorem improves the upper boundr(C)given in [A]. The 2 statements of the theorem are equivalent to the existence of a semi-stable rank-(g(resp.+ 2) 2000Mathematics Subject Classification.Primary 14H60, 14D20. 1
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