81 research outputs found
Derived categories of irreducible projective curves of arithmetic genus one
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 auto-equivalences 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.Comment: 44 pages; minor modifications; to appear in Compositio Mat
On the Hall algebra of an elliptic curve, I
In this article we describe the Hall algebra H_X of an elliptic curve X
defined over a finite field and show that the group SL(2,Z) of exact
auto-equivalences of the derived category D^b(Coh(X)) acts on the Drinfeld
double DH_X of H_X by algebra automorphisms. Next, we study a certain natural
subalgebra U_X of DH_X for which we give a presentation by generators and
relations. This algebra turns out to be a flat two-parameter deformation of the
ring of diagonal invariants C[x_1^{\pm 1}, ..., y_1^{\pm 1},...]^{S_{\infty}},
i.e. the ring of symmetric Laurent polynomials in two sets of countably many
variables under the simultaneous symmetric group action.Comment: 47 pages, Latex; several changes in the presentatio
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