Green's Conjecture for curves on rational surfaces with an anticanonical   pencil
  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. 
Margherita Lelli-Chiesa
math.AG