A Clifford inequality for semistable curves

Let $X$ be a semistable curve and $L$ a line bundle whose multidegree is uniform, i.e., in the range between those of the structure sheaf and the dualizing sheaf of $X$. We establish an upper bound for $h^0(X,L)$, which generalizes the classic Clifford inequality for smooth curves. The bound depends on the total degree of $L$ and connectivity properties of the dual graph of $X$. It is sharp, in the sense that on any semistable curve there exist line bundles with uniform multidegree that achieve the bound.