19 research outputs found
Equivariant Hirzebruch classes and Molien series of quotient singularities
We study properties of the Hirzebruch class of quotient singularities â„‚ n /G, where G is a finite matrix group. The main result states that the Hirzebruch class coincides with the Molien series of G under suitable substitution of variables. The Hirzebruch class of a crepant resolution can be described specializing the orbifold elliptic genus constructed by Borisov and Libgober. It is equal to the combination of Molien series of centralizers of elements of G. This is an incarnation of the McKay correspondence. The results are illustrated with several examples, in particular of 4-dimensional symplectic quotient singularities
Towards classifying toric degenerations of cubic surfaces
We investigate the class of degenerations of smooth cubic surfaces which are
obtained from degenerating their Cox rings to toric algebras. More precisely,
we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii
bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and
who leave the question of classifying these degenerations in the degree 3 case
as an open problem. In order to carry out this classification we describe an
approach which is closely related to tropical geometry and present partial
results in this direction.Comment: v2: 21 pages, section 1 rewritten, added sections 6 and
Complexity of the usual torus action on Kazhdan-Lusztig varieties
We investigate the class of Kazhdan-Lusztig varieties, and its subclass of
matrix Schubert varieties, endowed with a naturally defined torus action.
Writing a matrix Schubert variety as (where is maximal possible), we show that can be of
complexity- exactly when . Also, we give a combinatorial
description of the extremal rays of the weight cone of a Kazhdan-Lusztig
variety, which in particular turns out to be the edge cone of an acyclic
directed graph. As a consequence we show that given permutations and ,
the complexity of Kazhdan-Lusztig variety indexed by is the same as the
complexity of the Richardson variety indexed by . Finally, we use this
description to compute the complexity of certain Kazhdan-Lusztig varieties.Comment: 28 page