On some moduli spaces of bundles on \(K3\) surfaces. II (Q2845857)

Let \(S\subset\mathbb P^g\) be a smooth \(K3\) surface of genus \(g\geq 3\). Let \(H\) be a polarization of degree \(H^2=2g-2\). The author studies conditions for which the moduli space \(\mathcal M(v)\) of \(H\)-stable sheaves with Mukai vector \(v=(r,H,s)\) is birational to an Hilbert scheme \(M\) of points in \(S\), with length \(v^2/2+1\). The author addresses the case where \(v^2>0\). When the Picard number of \(S\) is \(2\) and \(H\) is primitive, under some numerical conditions on \(r,s\) (e.g. \(r\) even or \(r\) multiple of \(g-1\)), the author finds infinitely many cases of surfaces for which \(\mathcal M(v)\) is birational to \(M\). These surfaces are characterized by the values \(-d\) of the determinant of the Neron-Severi lattice. The author indeed shows that for infinitely many values of \(d\), the moduli space \(\mathcal M(r,H,s)\) is isomorphic to a moduli space \(\mathcal M(r,H+rD,\pm 1)\) for a suitable divisor \(D\) which depends on \(d\), and moreover \(\mathcal M(r,H+rD,\pm 1)\) is birational to the Hilbert scheme \(M\).