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: http://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(ℳi, L) (i = 0, 1) is given by the Verlinde formula. For small m > 0, the meaning of this dimension can be explained in the framework of algebraic geometry. For example, we have

dimH0(ℳ0, L) = 2g,

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

dimH0(Jac(X),O(2ϴ)) = 2g.

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) = 2g-1(2g + 1)

dimH0(ℳ1, L) = 2g-1(2g - 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 Dκ 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.

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