The Euclid-Fourier-Mukai algorithm for elliptic surfaces

Understand

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.