A CRITERION FOR GETTING A BIG COMPONENT OF THE MODULI OF VECTOR BUNDLES BY CHANGING A POLARIZATION

Abstract

Let MH(c1, c2) be a coarse moduli scheme parameterizing all rank-two H-μ-stable vector bundles with Chern classes (c1, c2) on a smooth projective surface X over an algebraically closed field. For fixed two ample line bundles H and H′, it is known that if c2 is greater than some constant p(X, H, H′) depending on H and H′ then MH(c1, c2) and MH′(c1, c2) are birationally equivalent. In this paper we show that this constant p(X, H, H′) generally does depends on the choice of H and H′. More precisely, we give some example of surface (and c1) on which, for any number K, there exists an integer c2 with c2≥K such that sup H: ample dim MH(c1, c2) = + ∞. This result is available also for normal surfaces.