In this paper we introduce the notion of a geometric associative rmatrix attached to a genus one fibration with a section and irreducible fibres. It allows us to study degenerations of solutions of the classical Yang-Baxter equation using the approach of Polishchuk. We also calculate certain solutions of the classical, quantum and associative Yang-Baxter equations obtained from moduli spaces of (semi-)stable vector bundles on Weierstraß cubic curves.Since the complex manifold M (n,d) E is a homogeneous space over the algebraic group J = Pic 0 (E), it turns out that r(v 1 , v 2 ; y 1 , y 2 ) ∼ r(v 1 − v 2 ; y 1 , y 2 ) = r(v; y 1 , y 2 ),

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.

In [K2] Moishezon twistor spaces over the connected sum nCP 2 (n ≥ 4), which do not contain effective divisors of degree one, were constructed as deformations of the twistor spaces introduced in [LeB]. We study their structure for n ≥ 4 by constructing a modification which is a conic bundle over P 2 . We show that they are rational. In case n = 4 we give explicit equations for such conic bundles and use them to construct explicit birational maps between these conic bundles and P 3 .

We investigate the bounded derived category of coherent sheaves on irreducible singular projective curves of arithmetic genus one. A description of the group of exact autoequivalences and the set of all t-structures of this category is given. We describe the moduli space of stability conditions, obtain a complete classification of all spherical objects in this category and show that the group of exact auto-equivalences acts transitively on them. Harder-Narasimhan filtrations in the sense of Bridgeland are used as our main technical tool.

Abstract. We study relative Fourier-Mukai transforms on genus one fibrations with section, allowing explicitly the total space of the fibration to be singular and non-projective. Grothendieck duality is used to prove a skew-commutativity relation between this equivalence of categories and certain duality functors. We use our results to explicitly construct examples of semi-stable sheaves on degenerating families of elliptic curves.