Theta-characteristics on singular curves

Abstract

On a smooth curve a theta–characteristic is a line bundle L with square that is the canonical line bundle ω. The equivalent conditionHom(L, ω) ∼= L generalizes well to singular curves, as applications show. More precisely, a theta–characteristic is a torsion–free sheaf F of rank 1 with Hom(F , ω) ∼= F . If the curve has non ADE–singularities then there are infinitely many theta–characteristics. Therefore, theta–characteristics are distinguished by their local type. The main purpose of this article is to compute the number of even and odd theta–characteristics (i.e. F with h(C,F) ≡ 0 resp. h(C,F) ≡ 1 modulo 2) in terms of the geometric genus of the curve and certain discrete invariants of a fixed local type. Mathematics Subject Classification (2000). 14H60, 14H42, 14H40. A theta–characteristic on a smooth projective curve C over C is a line bundle L whose square is the canonical bundle K. Let Θ be the set of all theta–characteristics. If g is the genus of the curve, then there are 2 theta– characteristics on C. Naturally, one would like to know the dimension of the linear systems |L|, L ∈ Θ. However, these dimensions depend not only on the genus alone, but also on the complex structure of the curve. Nevertheless, using the his theory of theta–functions Riemann proved that the dimension modulo 2 depends only on the genus. If Θ = {L ∈ Θ | h(L) is even.} Θ− = {L ∈ Θ | h(L) is odd.} are the even resp. odd theta–characteristics, then there are 2g−1(2g + 1) even and 2g−1(2g − 1) odd theta–characteristics. Atiyah gave another analytic proof of this, and Mumford the first algebraic one [A, M]. Mumford’s ideas were refined and extended by Harris to include singular curves into the theory — at least Gorenstein curves, for example plane curves [H]. On a singular curve a line bundle is defined to be a locally free sheaf of rank 1. Harris showed that the number of even and odd theta–characteristics can be computed in terms of the genus of the curve and certain discrete invariants of the singularaties. He also remarked that it would be desirable to be able to treat torsion–free sheaves of rank 1 alongside with the line bundles. This is what we want to do in this article.