SMOOTH CURVES WITH A PENCIL WITH CERTAIN SCROLLAR INVARIANTS

Abstract

Here we use nodal curves on Hirzebruch surfaces to construct pairs $ (X,L) $ with $ X $ a smooth projective curve of genus $ g geqq 3 $, $ L in {mathrm{Pic}^k}(X) $, $ h^0(X,L)=2 $, $ L $ spanned and such that L has certain prescribed scrollar invariants, i.e. essentially we prescribe the sequence of integers $ h^0(X,L^{otimes t}) $, $ t geqq 1 $.