Rational surfaces and moduli spaces of vector bundles on rational surfaces

Let X be a smooth algebraic surface, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ L \in \textrm{Pic}(X) $\end{document} and H an ample divisor on X. Set MX,H(2; L, c2) the moduli space of rank 2, H-stable vector bundles F on X with det(F) = L and c2(F) = c2. In this paper, we show that the geometry of X and of MX,H(2; L, c2) are closely related. More precisely, we prove that for any ample divisor H on X and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ L \in \textrm{Pic}(X) $\end{document}, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ n_0 \in \mathbb{Z} $\end{document} such that for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ n_0 \leqq c_2 \in \mathbb{Z} $\end{document}, MX,H(2; L, c2) is rational if and only if X is rational.