The Euclid-Fourier-Mukai algorithm for elliptic surfaces (Q407362)

Let \(X\) be a \(K3\) surface with an elliptic fibration \(X\to {\mathbb P}^1\) which has a section and only nodal singular fibers. Then its relative Jacobian \(\hat{X}\to {\mathbb P}^1\) is isomorphic to \(X\). Let us fix a Mukai vector \(v\) of rank \(r>0\) and some polarization. Let \(M_X(v)\) be the moduli space of semistable torsion free sheaves with Mukai vector \(v\). For some choices of the polarization the moduli space \(M_X(v)\) is an irreducible, smooth projective variety of dimension \(2t=2-\langle v,v\rangle\).NEWLINENEWLINELet us consider on \(\hat{X}\) the Mukai vector \(\hat{v}\) corresponding to \(v\). The authors study the birational rational map \(\Psi: M_X(v)\dashrightarrow M_{\hat{X}}(\hat{v})\). They prove that if \(t<r\) then this map is an isomorphism of moduli spaces, whereas if \(r\leq t<r+d\), where \(d\) is minus the fiber degree of \(v\), then \(\Psi\) is a Mukai flop unless \(t=r=2\), in which case \(\Psi\) is an isomorphism.NEWLINENEWLINEIn general, when \(\Psi\) is not an isomorphism, the authors describe some new compactifications of moduli spaces of vector bundles using objects of the derived category of coherent sheaves.