Bernd KREUSSLER
(Mary Immaculate College, Limerick)
Semi-stable sheaves on nodal cubics
Motivated by Kontsevich's homological mirror symmetry conjecture, in
recent years, derived categories of coherent sheaves on smooth projective
varieties and their groups of auto-equivalences attracted a lot of
interest. In this talk, which presents joint work with Igor Burban, we
study the first non-trivial singular example: nodal cubic curves. We use
techniques of P. Seidel and R. Thomas to construct a Fourier-Mukai
transform on such a curve. We apply this functor, which is an equivalence,
to give an explicit description of all indecomposable semi-stable torsion
free sheaves of degree zero on irreducible nodal cubic curves. For such
sheaves, we are able to calculate the Fourier-Mukai transform explicitly.
We apply results on the classification of modules of finite length over a
nodal ring, which were obtained by Gelfand and Ponomarev in the 1960s.
References:
-
I.Burban, B.Kreussler, Fourier-Mukai transforms and
semi-stable sheaves on nodal Weierstrass cubics,
arXiv:math.AG/0401437.
-
Seidel, P.; Thomas, R. Braid group actions on derived categories of
coherent sheaves, Duke Math. J. 108 (2001) 37--108,
arXiv:math.AG/0001043.
- Gelfand, I.M.; Ponomarev, V.A. Indecomposable
representations of the Lorentz group, Russ. Math. Surv. 23 (1968)
1--58; translation from Usp. Mat. Nauk 23 (1968) no. 2 (140), 3--60.