Characteristic classes of affine varieties and Plucker formulas for affine morphisms
Alexander Esterov

TL;DR
This paper introduces affine characteristic classes and affine Thom polynomials for affine varieties, extending intersection theory to affine cohomology, and applies these concepts to classical enumerative problems and tropical geometry.
Contribution
It develops the theory of affine characteristic classes and affine Thom polynomials, providing new tools for enumerative geometry and tropical correspondence theorems.
Findings
Derived affine versions of classical Thom polynomials (Plücker formulas)
Classified toric varieties with dual hypersurfaces
Computed tropical fans and Newton polytopes related to affine varieties
Abstract
An enumerative problem on a variety is usually solved by reduction to intersection theory in the cohomology of a compactification of . However, if the problem is invariant under a "nice" group action on (so that is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of . We call this limit the affine cohomology of and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus. This allows us to make the first steps in computing affine Thom polynomials. Classical Thom polynomials count how many fibers of a generic proper map of a smooth variety have a prescribed collection of singularities, and our affine version addresses the same question for generic polynomial maps of affine algebraic…
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