Rational surfaces and moduli spaces of vector bundles on rational surfaces

摘要

Let X be a smooth algebraic surface, $ L in textrm{Pic}(X) $ and H an ample divisor on X. Set M X,H (2; L, c 2) the moduli space of rank 2, H-stable vector bundles F on X with det(F) = L and c 2(F) = c 2. In this paper, we show that the geometry of X and of M X,H (2; L, c 2) are closely related. More precisely, we prove that for any ample divisor H on X and any $ L in textrm{Pic}(X) $ , there exists $ n_0 in mathbb{Z} $ such that for all $ n_0 leqq c_2 in mathbb{Z} $ , M X,H (2; L, c 2) is rational if and only if X is rational.