A Clifford inequality for semistable curves

TL;DR

This paper extends the classical Clifford inequality to semistable curves, providing an upper bound for the number of sections of certain line bundles based on the curve's structure.

Contribution

It introduces a generalized Clifford inequality for semistable curves with uniform multidegree line bundles, incorporating dual graph connectivity.

Findings

Established a sharp upper bound for $h^0(X,L)$ on semistable curves.

The bound depends on total degree and dual graph connectivity.

Existence of line bundles achieving the bound on any semistable curve.

Abstract

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.