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# 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

#### Abstract

Let ℳ_{0} [resp. ℳ_{1}] 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 H^{0}(ℳ_{i}, *L*^{ℳ}) (*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

dimH^{0}(ℳ_{0}, *L*) = 2^{g},

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

dimH^{0}(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

dimH^{0}(ℳ_{0}, *L*^{2}) = 2^{g-1}(2^{g} + 1)

dimH^{0}(ℳ_{1}, *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 ℳ_{0} [resp. ℳ_{1}] that can be described from a moduli-theoretic viewpoint, and proved that they form a basis of H^{0}(ℳ_{0}, *L*^{2}) [resp. H^{0}(ℳ_{1}, *L*)]. In [13], two vector spaces H^{0}(ℳ_{0}, *L*^{4}) and H^{0}(ℳ_{1}, *L*^{2}) 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}(ℙ^{1}; *I*) of rank 2 semistable parabolic vector bundles with half weights of degree zero on ℙ^{1}.