Variations of moduli of parabolic bundles

1. Introduction In this paper, we study the moduli spaces of semistable parabolic bundles of arbitrary rank over a smooth curve X with marked points in a nite subset P of X. A parabolic bundle consists of a holomorphic bundle E over X together with weighted ags in the bers E p for each p 2 P. The moduli space M of semistable parabolic bundles was constructed by Mehta and Seshadri as the space of semistable holomorphic structures modulo s-equivalence. In 19], it is proved that M is a normal projective variety which is also smooth for a generic choice of weights. They also observe that suuciently close generic weights have isomorphic (in fact, identical) moduli. We consider the problem further by studying the eeect on the moduli of varying the weights. The space of admissible weights is a simplicial subset of R N which we denote W. For 2 W, we denote by M the corresponding moduli. The collection of weights with respect to which there are some strictly semistable bundles is a union of hyperplanes H W. Our main result is that if and are generic weights which lie on either side of a given hyperplane H, then M and M are related by a special birational transformation which is similar to a ip in Mori theory. To see what this means, assume 2 H does not lie on any other hyperplane and let M denote the set of s-equivalence classes of strictly semistable bundles. This is in general the singular locus of M with a few possible exceptions (cf. Remark 3.3). Then is smooth and the theorem states that there are two canonical projective maps , M M &. M which are isomorphisms on the complement of and are P e ; P e (locally trivial) brations over. Moreover, e + e + 1 = codim. where P(Y) = P i dim H i (Y; Z)t i denotes the Poincar e polynomial of Y. Additionally, one checks easily that one of the two maps and must be a small resolution (cf. 12]). In fact, we believe that for each singular moduli M , there is a smooth moduli M so that the canonical map M ! M is a small resolution (Conjecture 4.8). Another related but much simpler problem is the eeect of choosing on the boundary of the weight space. This corresponds to the degeneration of …