Compactified Jacobians and q,t-Catalan numbers, II

Details Compactified Jacobians and q,t-Catalan numbers, II Compactified Jacobians and q,t-Catalan numbers, II

2012-04-24

arxiv.org/abs/1204.5448v1

Mathematics

Algebraic Geometry

35 pages

Scientific paper

We continue the combinatorial study of the homology of compactified Jacobians of plane curve singularities with one Puiseux pair (m,n) and its relation to the generalized q,t-Catalan numbers. We describe two generalizations of the bijective constructions of J. Haglund and N. Loehr and use them to prove a simple formula for the Poincare polynomials for the homology for m=kn\pm 1. Using a construction of B. Fantechi, L. Goettsche and D. van Straten, we give a bijective proof of the (q,t)-symmetry for n<4. We also give a geometric interpretation of a relation to the theory of (m,n)-cores discovered by J. Anderson.

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