Birational maps of moduli spaces of vector bundles on \(K3\) surfaces (Q410129)

Let \(X\) be a complex projective \(K3\) surface with \(\mathrm{Pic} (X) = \mathbb {Z}H\). For each Mukai vector \((r,c_1,c_2)\) let \(M_H(r,c_1,c_2)\) be the moduli space of stable sheaves on \(X\) with Mukai vector \((r,c_1,c_2)\). In this paper the authors construct birational maps between some of these moduli spaces using the reflection functor associated to the Fourier-Mukai transform of the diagonal of \(X\times X\). They prove that if \(a\leq 0\), then \(M_H(r,dH,a)\) and \(M_H(-a,dH,-r)\) are birational; if \(a\in \{0,1\}\), then \(M_H(r,dH,a)\) and \(M_H(a,dH,r)\) are birational, except in cases with \(H^2=2\) described completely. As a corollary they get that if \(d^2(H^2)-2ra > 0\), \(a\leq 1\) and \(\mathrm{gcd}(r,d(H^2),a) =1\), then \(M_H(r,dH,a)\) and \(M_H(a,dH,r)\) are birational. Taking \(a=0\) they get that if \(\mathrm{gcd}(r,d(H^2)) =1\) and \(d>0\), then \(M_H(r,dH,0)\) is birationally equivalent to a holomorphic symplectic manifold with a Lagrangian fibration.