Related papers: Tropical trigonal curves

We study the Berkovich analytification of the space of genus $0$ logarithmic stable maps to a toric variety $X$ and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two…

Let $ \mathcal{D} = \{D_{1}, ..., D_{\ell}\} $ be a multi-degree arrangement with normal crossings on the complex projective space $ \mathbf{P}^{n} $, with degrees $ d_{1}, ..., d_{\ell} $; let $ \Omega_{\mathbf{P}^{n}}^{1}(\log…

We study the moduli of trigonal curves. We establish the exact upper bound of ${36(g+1)}/(5g+1)$ for the slope of trigonal fibrations. Here, the slope of any fibration $X\to B$ of stable curves with smooth general member is the ratio…

We classify trivalent graphs with 16 vertices and 16 edges that arise from intersecting two quadratic surfaces in tropical 3-space. There are 4,009 such graphs, representing maximally degenerate stable models of elliptic curves realized as…

Let $X$ be an algebraic variety and let $S$ be a tropical variety associated to $X$. We study the tropicalization map from the moduli space of stable maps into $X$ to the moduli space of tropical curves in $S$. We prove that it is a…

We introduce a notion of tropical vector bundle on a tropical toric variety which is a tropical analogue of a torus equivariant vector bundle on a toric variety. Alternatively it can be called a toric matroid bundle. We define equivariant…

We establish faithful tropicalisation for point configurations on algebraic tori. Building on ideas from enumerative geometry, we introduce tropical scaffolds and use them to construct a system of modular fan structures on the tropical…

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…

Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every $q$-holonomic sequence of rational functions. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the…

We study the topology of the moduli space of unramified $\mathbb{Z}/p$-covers of tropical curves of genus $g \geq 2$, where $p$ is a prime number. We use recent techniques by Chan--Galatius--Payne to identify contractible subcomplexes of…

In this paper, we define two numbers. One comes from counting tropical curves with a stop and the other is the number of holomorphic discs in toric varieties with Lagrangian boundary condition. Both of these curves should satisfy some…

We show, for each algebraically closed field, the rationality of the following two moduli spaces: M(3,3) parametrizing pairs (C, \eta) where C has genus 3 and \eta is a 3-torsion divisor class, respectively of M(3,<3>) parametrizing pairs…

The moduli space of degree $d$ morphisms on $\mathbb{P}^1$ has received much study. McMullen showed that, except for certain families of Latt\`es maps, there is a finite-to-one correspondence (over $\mathbb{C}$) between classes of morphisms…

We explain the correspondences between twisted monopoles with Dirac type singularity and polystable twisted mini-holomorphic bundles with Dirac type singularity on a 3-dimensional torus. We also explain that they are equivalent to…

We prove that, for any g greater or equal than 3, a matrix g x 5 with tropical rank 3 has Kapranov rank 3.

In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of…

We use tropical and nonarchimedean geometry to study the moduli space of genus $0$ stable maps to $\mathbb{P}^1$ relative to two points. This space is exhibited as a tropical compactification in a toric variety. Moreover, the fan of this…

The moduli space of Gieseker vector bundles is a compactification of moduli of vector bundles on a nodal curve. This moduli space has only normal crossing singularity and it provides a flat degeneration. We prove a Torelli type theorem for…

We consider the moduli space of genus 4 curves endowed with a $g^1_3$ (which maps with degree 2 onto the moduli space of genus 4 curves). We prove that it defines a degree $\frac{1}{2}(3^{10}-1)$ cover of the 9-dimensional Deligne-Mostow…

The moduli stack of Deligne-Mumford stable curves of genus g admits a stratification, so that the number of nodes of the curves belonging to one stratum is constant. The irreducible components of the stratum corresponding to curves with…