Asian Journal of Mathematics

Volume 8 (2004)

Number 3

Anticanonical divisors of a moduli space of parabolic vector bundles of half weight on ℙ ¹

Pages: 395 – 408

DOI: https://dx.doi.org/10.4310/AJM.2004.v8.n3.a2

Author

Takeshi Abe

Abstract

Let ℳ₀ [resp. ℳ₁] be a coarse moduli space of rank 2 semistable vector bundles of even [resp. odd] degree with fixed determinant on a smooth projective curve X. The Picard group is infinite cyclic . Let L be the ample generator. The dimension of a vector space H⁰(ℳ_i, L^ℳ) (i = 0, 1) is given by the Verlinde formula. For small m \gt 0, the meaning of this dimension can be explained in the framework of algebraic geometry. For example, we have

dimH⁰(ℳ₀, L) = 2^g,

where g is the genus of X. On the other hand, we have

dimH⁰(Jac(X),O(2ϴ)) = 2^g.

In fact we have a natural isomorphism between these two vector spaces (See [1]). In [2], the meaning of the two equations

dimH⁰(ℳ₀, L²) = 2^g-1(2^g + 1)

dimH⁰(ℳ₁, L) = 2^g-1(2^g - 1)

are clarified. The above dimensions are the number of even or odd theta characterictics on X. Beauville associated to an even [resp. odd] theta characterictic κ a divisor D_κ on ℳ₀ [resp. ℳ₁] that can be described from a moduli-theoretic viewpoint, and proved that they form a basis of H⁰(ℳ₀, L²) [resp. H⁰(ℳ₁, L)]. In [13], two vector spaces H⁰(ℳ₀, L⁴) and H⁰(ℳ₁, L²) are considered. In [15], Pauly deals with a parabolic case.

The purpose of this paper is to carry out a similar study for a moduli space ℳ ^Par(ℙ¹; I) of rank 2 semistable parabolic vector bundles with half weights of degree zero on ℙ¹.

Published 1 January 2004