Complete curves in moduli spaces of stable bundles on surfaces (Q1318119)

If \(X\) is a smooth algebraic surface with an ample divisor \(H\), \({\mathcal Q}\) a line bundle on \(X\) and \(M_ H({\mathcal Q},c_ 2)\) the moduli space of \(\mu\)-stable vector bundles of rank two, determinant \({\mathcal Q}\) and second Chern class \(c_ 2\), then the following result is proven: For \(c_ 2 \gg 0\) the quasi-projective varieties \(M_ H({\mathcal Q}, c_ 2)\) contain complete curves, in particular they are not affine. This gives a partial answer to a question posed by Hirschowitz and Hulek about the maximal dimension of complete subvarieties of \(M_ H({\mathcal Q},c_ 2)\). Le Potier and Strømme have shown that for \(X=\mathbb{P}_ 2\) the dual of the canonical line bundle \(K_{M_ H({\mathcal Q},c_ 2)}\) is ample. This is shown to be true for any Del Pezzo surface \(X\).