Torsion divisors on algebraic curves

George Kempf

Vol. 97 (1981), No. 2, 437–441

DOI: 10.2140/pjm.1981.97.437

Abstract

Let D be a divisor on a smooth complete algebraic curve C such that the multiple dD is rationally equivalent to zero for some positive integer d. We may write D = D₀ − D_∞ where D₀ and D_∞ are distinct effective divisors.

The purpose of this note is to prove the

Theorem. Assume that C is a curve with general moduli in characteristic zero. For such a divisor D, the cohomology H¹(C,𝒪_C(D₀ + D_∞)) must be zero (equivalently |K − D₀ − D_∞| is empty).

Mathematical Subject Classification 2000

Primary: 14H10

Secondary: 14C20

Milestones

Received: 30 January 1980

Published: 1 December 1981