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 ≥ 3, L ∈ Pic^k(X), h⁰(X,L) = 2, L spanned and such that L has certain prescribed scrollar invariants, i.e. essentially we prescribe the sequence of integers h⁰(X,L^⊗t), t ≥ 1.