Related papers: Smooth tropical complete intersection curves of ge…

The first part of this paper is a survey on Teichmueller curves and Veech groups, with emphasis on the special case of origamis where much stronger tools for the investigation are available than in the general case. In the second part we…

Let $C$ be a $4$-cover of an elliptic curve $E$, written as a quadric intersection in $\mathbb{P}^3$. Let $E'$ be another elliptic curve with $4$-torsion isomorphic to that of $E$. We show how to write down the $4$-cover $C'$ of $E'$ with…

The family of complex projective surfaces in projective three space of degree $d$ having precisely $\delta$ nodes as their only singularities has codimension $\delta$ in the linear system of surfaces of degree $d$ for sufficiently large $d$…

We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group $\pi_3$. This generalizes a result of Gutierrez-Ratcliffe who relate the intersection of two subgroups with the…

Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface…

We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also…

A $3$-polytope is a $3$-connected, planar graph. It is called unigraphic if it does not share its vertex degree sequence with any other $3$-polytope, up to graph isomorphism. The classification of unigraphic $3$-polytopes appears to be a…

In this work we classify all smooth surfaces with geometric genus equal to three and an action of a group G isomorphic to (Z/2)^k such that the quotient is a plane. We find 11 families. We compute the canonical map of all of them, finding…

We study K3 surfaces with 9 cusps, i.e. 9 disjoint $A_2$ configurations of smooth rational curves, over algebraically closed fields of characteristic $p\neq 3$. Much like in the complex situation studied by Barth, we prove that each such…

This paper is a follow-up of a previous work in which we show that, for a $3$-edge connected tropical curve $\Gamma$, the existence of a divisor of degree $3$ and Baker-Norine rank at least $1$ in $\Gamma$ is equivalent to the existence of…

We construct examples of $C^\infty$ smooth submanifolds in ${\Bbb C}^n$ and ${\Bbb R}^n$ of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact…

We investigate the number of nodal intersections of random Gaussian Laplace eigenfunctions on the standard three-dimensional flat torus with a fixed smooth reference curve, which has nowhere vanishing curvature. The expected intersection…

A curve over a perfect field $K$ of characteristic $p > 0$ is said to be superspecial if its Jacobian is isomorphic to a product of supersingular elliptic curves over the algebraic closure $\overline{K}$. In recent years, isomorphism…

We show that every possible value for the Clifford index and gonality of a curve of a given genus on a $K3$ surface occurs.

We prove that there are intersection graphs of axis-aligned boxes in $\mathbb{R}^3$ and intersection graphs of straight lines in $\mathbb{R}^3$ that have arbitrarily large girth and chromatic number.

Let $(S,H)$ be a general primitively polarized $K3$ surface of genus $\p$ and let $p_a(nH)$ be the arithmetic genus of $nH.$ We prove the existence in $|\mathcal O_S(nH)|$ of curves with a triple point and $A_k$-singularities. In…

In this paper we study a construction of algebraic curves from combinatorial data. In the study of algebraic curves through degeneration, graphs usually appear as the dual intersection graph of the central fiber. Properties of such graphs…

We study abelian surfaces defined over finite fields which do not contain any possibly singular curve of genus less than or equal to $3$. Firstly, we complete and expand the characterisation of isogeny classes of abelian surfaces with no…

We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over $\mathbb C$. The family of genus 3 hyperelliptic curves which have a degree 2 cover to an elliptic curve $E$ and degree 4 covers to elliptic curves…

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…