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European Mathematical Society Publishing House
2016-09-19 17:04:52
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
87
2012
4
Equivariant classes of matrix matroid varieties
László
Fehér
Eötvös Loránd University, BUDAPEST, HUNGARY
András
Némethi
Hungarian Academy of Sciences, BUDAPEST, HUNGARY
Richárd
Rimányi
University of North Carolina at Chapel Hill, CHAPEL HILL, UNITED STATES
Equivariant class, matroid, enumerative geometry
To each subset $I$ of $\{1,\dots,k\}$ associate an integer $r(I)$. Denote by $X$ the collection of those $n\times k$ matrices for which the rank of a union of columns corresponding to a subset $I$ is $r(I)$, for all $I$. We study the equivariant cohomology class represented by the Zariski closure $Y=\overline{X}$. This class is an invariant of the underlying matroid structure. Its calculation incorporates challenges similar to the calculation of the ideal of $Y$, namely, the determination of the geometric theorems for the matroid. This class also gives information on the degenerations and hierarchy of matroids. New developments in the theory of Thom polynomials of contact singularities (namely, a recently found stability property) help us to calculate these classes and present their basic properties. We also show that the coefficients of this class are solutions to problems in enumerative geometry, which are natural generalization of the linear Gromov–Witten invariants of projective spaces.
Algebraic topology
Algebraic geometry
Convex and discrete geometry
General
861
889
10.4171/CMH/271
http://www.ems-ph.org/doi/10.4171/CMH/271