Characteristic classes of affine varieties and Plücker formulas for affine morphisms

Abstract

An enumerative problem on a variety V is usually solved by reduction to intersection theory in the cohomology of a compactification of V . However, if the problem is invariant under a "nice" group action on V (so that V is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of V . We call this limit the affine cohomology of V 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 varieites. This notion is also motivated by developing an intersection-theoretic approach to tropical correspondence theorems: they can be reduced to the computation of affine Thom polynomials, because the fundamental class of a variety in the affine cohomology is encoded by the tropical fan of this variety.The first concrete answer that we obtain is the affine version of what were, historically speaking, the first three Thom polynomials -the Plücker formulas for the degree and the number of cusps and nodes of a projectively dual curve. This, in particular, characterizes toric varieties, whose projective dual is a hypersurface, computes the tropical fan of the variety of double tangent hyperplanes to a toric variety, and describes the Newton polytope of the hypersurface of non-Morse polynomials of a given degree. We also make a conjecture on the general form of affine Thom polynomials -a key ingredient is the n-ary fan, generalizing the secondary polytope.

Galois theory for general systems of polynomial equations

“…The situation with reduced tuples is similar: the codimension 1 components D i of the naive A-discriminant come with natural multiplicities equal to the number of singular roots of the system f = 0 for a generic tuple f ∈ D i . By Theorem 3.8, an irreducible tuple A is reduced if and only if D A is reduced in the sense of the aforementioned multiplicity (see [Est13] for the computation of the multiplicities for non-reduced and reducible tuples).…”

Arithmetics and combinatorics of tropical Severi varieties of univariate polynomials

“…As an example, we use Theorem 2.6 to present a new explanation of the cones in the tropicalization of the set of bivariate polynomials with two nodes given in Example 3.9 of [8]. …”

“…as the union of the two cones giving the subdivisions and hidden tie presented in Example 3.9 of [8]. These cones are explicitly C ⋄ , with ⋄ equal to ≤ and to ≥:…”

“…However, we restricted our attention to the particular case δ = 2, which highlights the new phenomena that occur in type III cones and that already requires many technical results. Our approach is used in Example 2.7 to clarify the two dimensional case worked out in [8].…”

“…A recent interesting paper of Esterov [8] defines affine characteristic classes of affine varieties, which govern equivariant enumerative problems. The computation of the fundamental class of a hypersurface of the torus in the corresponding affine cohomology ring, amounts to the computation of the Newton polytope of a defining equation.…”