A note on moduli of vector bundles on rational surfaces (Q1890273)

Let \(X\) be a smooth projective rational surface and let \(H\) be an ample divisor on \(X\) such that \(H K_X< 0\). Let \(\overline{M}_H(r,c_1,\chi)\) be the moduli space of semistable sheaves \(E\) on \(X\) with \(\text{rank}(E)=r\), \(c_1(E)=c_1\) and \(\chi(E)=\chi\). When \(H c_1=0\) and \(\chi \leq 0\), \textit{M. Maruyama} [in: Algebraic geometry and commutative algebra. I, Kinokuniya, Tokyo, 233--260 (1988; Zbl 0743.14015)] constructed a contraction map \(\phi:\overline{M}_H(r,c_1,\chi) \to \overline{M}_H(r- \chi,c_1,0)\) associating to any \(E\) as above the semistable sheaf \(F\) defined by the universal extension \(0 \to E \to F \to H^1(X,E) \otimes \mathcal O_X \to 0\). The author generalizes this construction by replacing \(\mathcal O_X\) with a rigid and stable vector bundle \(E_0\). Provided that \(E\) has \(E_0\)-twisted degree \(\deg_{E_0}(E)= H c_1(E_0^{\vee} \otimes E)=0\) he obtains results in the same direction as Maruyama's, showing that the analogue of \(\phi\) is a morphism which is an immersion on the open subscheme consisting of \(\mu\)-stable vector bundles and its image is Cohen--Macaulay and normal. Motivated by results of \textit{G. Ellingsrud} and \textit{S. A. Strømme} [J. Reine Angew. Math. 441, 33--44 (1993; Zbl 0814.14003)] the author also investigates the structure of the map when \(\deg_{E_0}(E)=1\). In particular, he finds relations on the Hodge polynomials of some moduli spaces, which are particularly meaningful when \(X=\mathbb P^2\).