Asian Journal of Mathematics

The Euclid-Fourier-Mukai algorithm for elliptic surfaces

Marcello Bernardara and Georg Hein

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We describe the birational correspondences, induced by the Fourier-Mukai functor, between moduli spaces of semistable sheaves on elliptic surfaces with sections, using the notion of $P$-stability in the derived category. We give explicit conditions to determine whether these correspondences are isomorphisms. This is indeed not true in general and we describe the cases where the birational maps are Mukai flops. Moreover, this construction provides examples of new compactifications of the moduli spaces of vector bundles via sheaves with torsion and via complexes. We finally get for any fixed dimension an isomorphism between the Picard groups of the moduli spaces.

Article information

Asian J. Math., Volume 18, Number 2 (2014), 345-364.

First available in Project Euclid: 27 August 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 14J27: Elliptic surfaces 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]

Fourier-Mukai transform moduli spaces, Mukai flops elliptic surfaces stable sheaves Postnikov-stability


Bernardara, Marcello; Hein, Georg. The Euclid-Fourier-Mukai algorithm for elliptic surfaces. Asian J. Math. 18 (2014), no. 2, 345--364.

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