On the rank of general linear series on stable curves

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.