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

Bernardara, Marcello; Hein, Georg

Asian J. Math. 18, No. 2, 345-364 (2014).

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.

Reviewer: Adrian Langer (Warszawa)

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)

Keywords:

moduli spaces; elliptic \(K3\) surfaces; semistable sheaves; Fourier-Mukai transform