Related papers: Disc counting on toric varieties via tropical curv…

In this paper, we investigate tropical secant varieties of ordinary linear spaces. These correspond to the log-limit sets of ordinary toric varieties; we show that their interesting parts are combinatorially isomorphic to a certain natural…

The Welschinger numbers, a kind of a real analog of the Gromov-Witten numbers which count the complex rational curves through a given generic collection of points, bound from below the number of real rational curves for any real generic…

We give new upper bounds for the number of nonconstant holomorphic maps depending only on the genus. Our estimates improve previously known bounds. The proof is based on the study of pullbacks of holomorphic differentials, together with…

We identify certain Gromov-Witten invariants counting rational curves with given incidence and tangency conditions with the Betti numbers of moduli spaces of point configurations in projective spaces. On the Gromov-Witten side, S. Fomin and…

Hurwitz numbers are a weighted count of degree d ramified covers of curves with specified ramification profiles at marked points on the codomain curve. Isomorphism classes of these covers can be included as a dense open set in a moduli…

In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even when it is singular) in terms of combinatorial data coming from the fan of the toric variety.

We give a definition of tropical orbit spaces and their morphisms. We show that, under certain conditions, the weighted number of preimages of a point in the target of such a morphism does not depend on the choice of this point. We equip…

In recent work, the authors derived a tropical interpretation of monotone and strictly monotone double Hurwitz numbers. In this paper, we apply the technique of tropical flows to this interpretation in order to provide a new proof of the…

In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves…

We describe the tropical curves in toric varieties and define the tropical Gromov-Witten invariants. We introduce amplitudes for the higher topological quantum mechanics (HTQM) on special trees and show that the amplitudes are equal to the…

We generalize the author's formula for Gromov-Witten invariants of symplectic toric manifolds (see math.AG/0006156) to those needed to compute the quantum product of more than two classes directly, i.e. involving the pull-back of the…

In this paper we introduce broccoli curves, certain plane tropical curves of genus zero related to real algebraic curves. The numbers of these broccoli curves through given points are independent of the chosen points - for arbitrary choices…

Double Hurwitz numbers have at least four equivalent definitions. Most naturally, they count covers of the Riemann sphere by genus g curves with certain specified ramification data. This is classically equivalent to counting certain…

This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated…

Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky. The tropical A-discriminant, which is the tropicalization of the dual variety of the…

I first recall the various problems of real enumerative geometry out of which I could extract some integer valued invariants, providing some real counterpart to Gromov-Witten invariants. I then discuss sharpness of the lower bounds given by…

For real toric surfaces and conjugation invariant point conditions with all conjugate pairs on the boundary divisors, we prove that the signed count of real curves of arbitrary genus in the linear system through the given points is…

There are two types of $J$-holomorphic spheres in a symplectic manifold invariant under an anti-symplectic involution: those that have a fixed point locus and those that do not. The former are described by moduli spaces of $J$-holomorphic…

We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation…

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows…