Fourier-Mukai transforms and semi-stable sheaves on nodal Weierstraß cubics

Abstract

Abstract. We completely describe all semi-stable torsion free sheaves of degree zero on nodal cubic curves using the technique of Fourier-Mukai transforms. The Fourier-Mukai images of such sheaves are torsion sheaves of finite length, which we compute explicitly. We show that the twist functors, which are associated to the structure sheaf O and the structure sheaf k(p 0 ) of a smooth point p 0 , generate an SL(2, Z)-action (up to shifts) on the bounded derived category of coherent sheaves on any Weierstaß cubic.

On a Relative Fourier–Mukai Transform on Genus One Fibrations

“…If X is Gorenstein, this functor is a dualising functor on X and we obtained a generalisation of Mukai's result [25], (3.8) as well as a more general form of our result [12], Theorem 6.11.…”

“…In the singular case, however, it is difficult to prove that a given integral transform is an equivalence of categories, because the methods of Bridgeland [8] and Bondal, Orlov [6] do not apply. We overcome such difficulties by using a completely different strategy which allows us to use results from our earlier paper [12], in which the special case S = Spec(k) was studied.…”

Relative integral functors for singular fibrations and singular partners

“…We expect that Proposition 2.16 could be a useful tool for the study of the moduli spaces of relatively semistable sheaves on X → S with respect to a suitable relative ample divisor following [3,2,9,15], and for the study of the derived category of X generalising [10].…”

Classification of semistable sheaves on a rational curve with one node

“…However, until now only some partial results in this direction were known. The problem was solved for the sheaves of degree 0 by Burban and Kreußler [2] by reducing it to the classification of torsion sheaves made by Gelfand and Ponomarev [5]. In the case of coprime degree and rank Burban [1] classified stable locally free sheaves by detecting those sheaves that have a one-dimensional endomorphism ring.…”

“…In order to do this we introduce and analyze certain combinatorial objects-chains and cycles (see Definition 2.1), which are used for the classification of indecomposable torsion-free sheaves. With any aperiodic cycle a one associates an indecomposable locally free sheaf B(a) (see [2,3] and Section 2) and with any chain b one associates an indecomposable non-locally free sheaf S(b) (see [2] and Section 2). The conditions of (semi)stability of B(a) and S(b) imply certain conditions on the cycle a and the chain b.…”