相关论文: Tropical Invariants from the Secondary Fan

We give a tropical description of the counting of real log curves in toric degenerations of toric varieties. We treat the case of genus zero curves and all non-superabundant higher-genus situations. The proof relies on log deformation…

We prove that the quadratically enriched count of rational curves in a smooth toric del Pezzo surface passing through $k$-rational points and pairs of conjugate points in quadratic field extensions $k\subset k(\sqrt{d_i})$ can be determined…

G. Mikhalkin introduced a refined count for real rational curves in a toric surface which pass through some points on the toric boundary of the surface. The refinement is provided by the value of a so-called quantum index. Moreover, he…

Tropical counting tools are useful for many enumerative questions. We count tropical multinodal surfaces using floor plans, looking at the case when two nodes are tropically close together, i.e., unseparated. We generalize tropical floor…

We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is arXiv:1509.07453, where a tropical-algebraic correspondence theorem…

We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on…

We generalize Givental's Theorem for complete intersections in smooth toric varieties in the Fano case. In particular, we find Gromov--Witten invariants of Fano varieties of dimension $\geq 3$, which are complete intersections in weighted…

We introduce the notion of tropical Lagrangian multi-sections over a fan and study its relation with toric vector bundles. We also introduce a "SYZ-type" construction for toric vector bundles which gives a reinterpretation of Kaneyama's…

We define scrollar invariants of tropical curves with a fixed divisor of rank 1. We examine the behavior of scrollar invariants under specialization, and provide an algorithm for computing these invariants for a much-studied family of…

Lattice gauge theories of permutation groups with a simple topological action (henceforth permutation-TFTs) have recently found several applications in the combinatorics of quantum field theories (QFTs). They have been used to solve…

We give an introduction to Tropical Geometry and prove some results in Tropical Intersection Theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of…

We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts. First, we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure…

In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface $\fn$. We compute the number of such curves through the appropriate number of fixed general points on $\fn$, and the number of…

We discuss some properties of the relative Gromov--Witten invariants counting rational curves with maximal contact order at one point. We compute the number of Cayley's sextactic conics to any smooth plane curve $Y$. In particular, we…

In this paper, we study tropical Weierstrass points. These are the analogues for tropical curves of ramification points of line bundles on algebraic curves. For a divisor on a tropical curve, we associate intrinsic weights to the connected…

The first part of this work constructs positive-genus real Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the present part focuses on their properties that are essential for actually working…

We define and study the cyclic Bergman fan of a matroid M, which is a simplicial polyhedral fan supported on the tropical linear space T(M) of M and is amenable to computational purposes. It slightly refines the nested set structure on…

The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it…

We investigate the problem of counting tropical genus g curves in g-dimensional tropical abelian varieties. For g = 2, 3, we prove that the tropical count matches the count provided by G\"ottsche, Bryan-Leung, and Lange-Sernesi in the…

We give several characterizations of stable intersections of tropical cycles and establish their fundamental properties. We prove that the stable intersection of two tropical varieties is the tropicalization of the intersection of the…