Scrollar invariants of smooth projective curves

Abstract

Let X be a smooth projective curve of genus g⩾3 and R∈Pic k ( X) with h 0( X, R)=2 and R spanned. There are k−1 integers e i , 1⩽ i⩽ k−1, with e 1⩾⋯⩾ e k−1 ⩾0 and e 1+⋯+ e k−1 = g− k+1 associated to R (the so-called scrollar invariants of R). Here we prove that for all integers k⩾3 and e⩾0 there is an integer g( k, e) such that for all integers g⩾ g( k, e) and all integers e i , 1⩽ i⩽ k−1, with e 1⩾⋯⩾ e k−1 ⩾0, e 1⩽ e k−1 + e and e 1+…+ e k−1 = g− k+1 there exist a smooth curve X of genus g and R∈Pic k ( X) such that e 1, …, e k−1 are the scrollar invariants of R.