Asian Journal of Mathematics
Volume 8 (2004)
Anticanonical divisors of a moduli space of parabolic vector bundles of half weight on ℙ 1
Pages: 395 – 408
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, Lℳ) (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 ). In , 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 , two vector spaces H0(ℳ0, L4) and H0(ℳ1, L2) are considered. In , 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.