Related papers: Plane quartics: the matrix of bitangents
Let S be the variety of irreducible sextics with six cusps as singularities. Let W be one of irreducible components of W. Denoting by M_4 the space of moduli of smooth curves of genus 4, the moduli map of W is the rational map from W to M_4…
A closed plane meander of order n is a closed self-avoiding loop intersecting an infinite line 2n times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on…
In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with four invariant lines, including the line at infinity…
Let $C/K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model…
We introduce a new technique that is used to show that the complex projective plane blown up at 6, 7, or 8 points has infinitely many distinct smooth structures. None of these smooth structures admit smoothly embedded spheres with…
This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the…
Over the complex numbers, there are 92 plane conics meeting 8 general lines in projective 3-space. Using the Euler class and local degree from motivic homotopy theory, we give an enriched version of this result over any perfect field. This…
We investigate the convex hulls of the eight dihedral symmetry classes of $n \times n$ alternating sign matrices, i.e., ASMs invariant under a subgroup of the symmetry group of the square. Extending the prefix-sum description of the ASM…
Plane quartics containing the ten vertices of a complete pentalateral and limits of them are called L\"uroth quartics. The locus of singular L\"uroth quartics has two irreducible components, both of codimension two in $\P^{14}$. We compute…
These notes are intended as an easy-to-read supplement to part of the background material presented in my talks on enumerative geometry. In particular, the numbers $n_3$ and $n_4$ of plane rational cubics through eight points and of plane…
In this paper we explicitly compute equations for the twists of all the smooth plane quartic curves defined over a number field k. Since the plane quartic curves are non-hyperelliptic curves of genus 3 we can apply the method developed by…
A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation…
We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations,…
At the time of writing, the general problem of finding the maximal Waring rank for homogeneous polynomials of fixed degree and number of variables (or, equivalently, the maximal symmetric rank for symmetric tensors of fixed order and in…
Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by…
Quaternionic tori are defined as quotients of the skew field $\mathbb{H}$ of quaternions by rank-4 lattices. Using slice regular functions, these tori are endowed with natural structures of quaternionic manifolds (in fact quaternionic…
By means of associated structural invariants, we efficiently construct four biplanes of order 9 - except the one with the smallest automorphism group, that is found by Janko and Trung. The notion of non-transversal vector is introduced…
Using the results of Dalla Piazza, Fiorentino and Salvati Manni, we determine an explicit modular form defining the locus of plane quartics with a hyperflex among all plane quartics. As a result, we provide a direct way to compute the…
We consider the problem of recovering a unitary eigendecomposition of a complex unitary matrix from that of its embedded real-valued formulation. Such formulations arise naturally in scientific computing workflows that employ…
Kippenhahn discovered that the numerical range of a complex square matrix is the convex hull of a plane real algebraic curve. Here, we present an example of a convex set, which has a similar algebraic description as the numerical range,…