Linear Series on Semistable Curves

For a semistable curve X of genus g, the number h(0)(X, L) is studied for line bundles L of degree d parameterized 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, to the combinatorial description of hyperelliptic curves, and to plane quintics are given.