Linear Series on Semistable Curves

We study h 0 (X, L) for line bundles L on a semistable curve X of genus g, parametrized by the compactified Picard scheme. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: X has two components; X is any semistable curve and d = 0 or d = 2g − 2; X is stable, free from separating nodes, and d ≤ 4. These results are shown to be sharp. Applications to the Clifford index, the combinatorial description of hyperelliptic curves, and plane quintics are given. Contents 1. Introduction and preliminaries 1 1.1. Gluing global sections 2 1.2. Clifford index of a line bundle 5 2. Riemann's theorem for semistable curves 6 2.1. Balanced line bundles. 7 2.2. Positivity properties of balanced line bundles. 7 3. Clifford's Theorem for all degrees 9 3.