Birational Classification of Moduli Spaces

Abstract

In this paper, I want to introduce a new way to study moduli spaces of objects in certain kinds of abelian categories. I have applied this method to the study of moduli spaces of representations of quivers, of vector bundles over smooth projective curves and of vector bundles over P ². In the first and third case, this allows the rationality problem to be reduced to the case of a suitable number of suitably sized matrices up to simultaneous conjugacy; in the second case, one can reduce the rationality problem to that of vector bundles of degree 0, which leads to a proof that the moduli space of vector bundles of rank r and determinant line bundle L is rational provided that the rank r and the degree of the line bundle are co-prime. I shall give a general idea of the method largely without proof and present in incomplete detail how the method is applied in the case of representations of quivers where the preparatory work is of interest in its own right since it gives an efficient algorithm to compute the canonical decomposition of a dimension vector. Here the new idea is the notion of a rigid sub-dimension vector of a dimension vector. If α is a dimension vector then β is said to be a rigid sub-dimension vector of α if and only if a general representation of dimension vectorαhas a unique subrepresentation of dimension vector β. A dimension vector α is said to be uniform if and only if it is a multiple of a Schur root in which case it is either a Schur root or else the Schur root β in question is real or isotropic and the canonical decomposition of αis α= nβ. A major resuit proved in this paper is