Green's Conjecture for curves on rational surfaces with an anticanonical pencil

TL;DR

This paper proves Green's conjecture for certain smooth curves on rational surfaces with an anticanonical pencil, and shows invariance of key geometric invariants within the linear system.

Contribution

It establishes Green's conjecture for curves on rational surfaces with an anticanonical pencil and demonstrates the constancy of Clifford dimension, Clifford index, and gonality.

Findings

Green's conjecture is proved for the specified curves.

Constancy of Clifford dimension, Clifford index, and gonality is shown.

Results apply under mild hypotheses on the line bundle L.

Abstract

Green's conjecture is proved for smooth curves C lying on a rational surface S with an anticanonical pencil, under some mild hypotheses on the line bundle L defined by C. Constancy of Clifford dimension, Clifford index and gonality of curves in the linear system |L| is also obtained.