The Euclid-Fourier-Mukai algorithm for elliptic surfaces.(English)Zbl 1311.14044

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 $$2t=2-\langle v,v\rangle$$.
Let 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.
In 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.

MSC:

 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations 14J28 $$K3$$ surfaces and Enriques surfaces 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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