On a Conjecture on Linear Systems

Abstract

Let C be a smooth projective curve of genus g ≥ 2 and let L be a globally generated line bundle on C. The evaluation map gives rise to an exact sequence 0 → E ∗ L → Γ(C, L)C → L → 0 of vector bundles on C and E is a vector bundle of rank h 0 (C, L) − 1. Let Σi ⊂ Γ(C, ∧ iE) be the cone of locally decomposable sections in ∧ iE. We state: Conjecture The cone Σi spans Γ(C, ∧ iE) for all i and for all globally generated line bundles L on all curves C. We prove: Main Result (Simplified) Above conjecture is true for a hyperelliptic curve C with a globally generated line bundle L of degree d ≥ 2g + 3.