Related papers: Refined tropical invariants and characteristic num…
This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus $g$ curves of fixed degree…
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…
We introduce a tropical enumerative invariant depending on a variable y which generalizes the tropical refined Severi degree. We show that this refined broccoli invariant is indeed independent of the point configuration, and that it…
We investigate combinatorial aspects of exceptional sequences in the derived category of coherent sheaves on certain smooth and complete algebraic surfaces. We show that to any such sequence there is canonically associated a complete toric…
We are interested in shapes of real algebraic curves in the plane and regions surrounded by them: they are named refined algebraic domains by the author. As characteristic finite sets, we consider points contained in two curves and the sets…
We prove a correspondence theorem for singular tropical surfaces in real three space, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we…
For any (possibly singular) hyperelliptic curve, we give the definition of a hyperelliptic refined spectral curve and the hyperelliptic refined topological recursion, generalising the formulation for a special class of genus-zero curves by…
We show that the commutator relations in the refined tropical vertex group can be expressed via the enumeration of suitable real rational curves in toric surfaces.
Since the first famous correspondence theorem by Mikhalkin appeared in 2005, tropical geometry has allowed a parallel treatment of real and complex counting problems. A prime example are the genus 0 Gromov-Witten invariants of the plane…
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…
We define a quadratically enriched count of rational curves in a given divisor class passing through a collection of points on a del Pezzo surface $S$ of degree $\geq 3$ over a perfect field $k$ of characteristic $\neq 2,3.$ When $S$ is…
We analyse the diagonal quotient for products of certain Artin--Schreier curves. The smooth models are almost always surfaces of general type, with Chern slopes tending asymptotically to 1. The calculation of numerical invariants relies on…
The purpose of this paper is to list the refined Humbert invariants for a given automorphism group of a curve $C/K$ of genus 2 over an algebraically closed field $K$ with characteristic $0$. This invariant is an algebraic generalization of…
In this paper, we consider weighted counts of tropical plane curves of particular combinatorial type through a certain number of generic points. We give a criterion, derived from tropical intersection theory on the secondary fan, for a…
We consider the issue of the slice invariance of refined topological string amplitudes, which means that they are independent of the choice of the preferred direction of the refined topological vertex. We work out two examples. The first…
The soft Barycentric refinement preserves manifolds with or without boundary. In every dimension larger than one, there is a universal spectral central limiting measure that has affinities with the Barycentric limiting measure one dimension…
The refined Humbert invariant is a positive definite quadratic form intrinsically attached to a curve $C$ of genus 2. This invariant is an algebraic generalization of the (usual) Humbert invariant. This invariant is useful because many…
In this paper, second installment in a series of three, we give a correspondence theorem to relate the count of genus $g$ curves in a fixed linear system in an abelian surface to a tropical count. To do this, we relate the linear system…
We obtain a sharp refinement of the strong multiplicity one theorem for the case of unitary non-dihedral cuspidal automorphic representations for GL(2). Given two unitary cuspidal automorphic representations for GL(2) that are not…
We show that the isolated invariant branches globalize to algebraic curves, when we consider weak toric type complex hyperbolic foliations on projective toric ambient surfaces. To do it, we pass through a characterization of weak toric type…