On the Existence of Special Divisors

Abstract

Let X be a complete nonsingular curve of genus g defined over an algebraically closed field Ic of any characteristic, let J be the jacobian variety. Denote by G;r the set of points of J which are images of linear series of degree n and projective dimension at least r, and put d== (r + 1) (n -r) -rg. Assume that G;r is formed by special series, that is (n r) < g. It is repeatedly asserted in the classical literature that G. r depends upon at least d parameters; in particular if d_: 0, then G,r is nonempty. For r= 1, the matter is treated in section 4 of Riemann's Theorie der Abel'schen Functionen [11] 1 and in lecture 31 of llensel-Landsberg [2] 1; the general case is treated in Brill-Noether [1] and in lecture 57 and appendix G of Severi [13].1 More recently, Martens [7] proved that G;r is an algebraic set whose components each have dimension at least d and at most (n -2r) and that if 1 r? (n-r) < (g-2), then the maximum occurs if and only if X is hyperelliptic, provided G.r is nonempty. Meis [9] gave an analytic proof that Gn1 is nonempt.y if d ? 0 or equivalently that X can be displayed as a branched covering of the sphere with at most (g + 3)/2 sheets. We offer a proof of the existence assertion in full generality. In fact, we prove that Gn meets any (closed) subvariety v of J with dimension at least (g d). It follows formally that given automorphisms f, of J (such as translations), an intersection (f (G,nri) n . nf1 (G,z,rp) n v) is a nonempty variety of dimension e = (dim (V) (r + 1) (g nj + ri)) whenever e ? 0. The proof involves constructing a vector bundle E on J, which algebraically deforms to the trivial bundle, and a section a over J of the Grassmann bundle B of rank-g-quotients of E such that a translate of G,r is the preimage of a certain special Schubert cycle o%. The cohomology ring of B is the tensor product of the rings of J and of a Grassmann variety, so by classical Schubert calculus, the class s of a is given by a certain polynomial in the g basic Schubert cycle classes, which themselves induce the classes w,,