On the rank of general linear series on stable curves

Abstract

We study the dimension of loci of special line bundles on stable curves and for a fixed semistable multidegree. In case of total degree $d=g-1$, we characterize when the effective locus gives a Theta divisor. In case of degree $g-2$ and $g$, we show that the locus is either empty or has the expected dimension. This leads to a new characterization of semistability in these degrees. In the remaining cases, we show that the special locus has codimension at least $2$. If the multidegree in addition is non-negative on each irreducible component of the curve, we show that the special locus contains an irrreducible component of expected dimension.

23 pages, 4 Figures. v2: The paper has been split in two, and this is a rewritten version of the first part. v3: some clarifications added and minor changes in presentation. Final version, to appear in Math. Ann. 20 pages, 2 Figures