The Euclid-Fourier-Mukai algorithm for elliptic surfaces

Abstract

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.

Vertical sheaves and Fourier–Mukai transform on elliptic Calabi–Yau threefolds

“…A comprehensive review of the subject and a more complete list of results can be be found in [6]. More recently, t-structures and moduli problems of Bridgeland stable objects in the derived category have been studied in [41,36,42,38,59,9,37,15,35] using a similar approach.…”

“…A comprehensive review of the subject and a more complete list of results can be be found in [6]. More recently, t-structures and moduli problems of Bridgeland stable objects in the derived category have been studied in [41,36,42,38,59,9,37,15,35] using a similar approach.Of particular importance for the present paper is the relative Fourier-Mukai transform for elliptic fibrations. This was constructed by Bartocci et al [8] and Bridgeland [10,11] for elliptic surfaces and Friedman, Morgan and Witten [18,19] for stable bundles on elliptic threefolds.…”

On Verlinde sheaves and strange duality over elliptic Noether-Lefschetz divisors

“…We turn now to the proof that Ψ v is an isomorphism in codimension 1, which was given for a surface π : X → P 1 with irreducible fibers in [2], [14]. We thus take up the case when the fibration has at least one reducible fiber.…”

Wall crossing for moduli of stable sheaves on an elliptic surface

“…This is a fundamental tool for the study of coherent sheaves on elliptic surfaces, and many properties of moduli spaces are proved (cf. [1], [3], [8], [9], [13]). In [14] and [16], we studied the Hodge numbers and the Picard groups of the moduli spaces under the assumption that all fibers are irreducible.…”