Related papers: Conic-line arrangements in the complex projective …

We introduce arrangements of rational sections over curves. They generalize line arrangements on P^2. Each arrangement of d sections defines a single curve in P^{d-2} through the Kapranov's construction of \bar{M}_{0,d+1}. We show a…

Using linear projections one gets new inequalities for the successive minima of the lattice of sections of an hermitian line bundle on an arithmetic surface.

A recurring task in particle physics and statistics is to compute the complex critical points of a product of powers of affine-linear functions. The logarithmic discriminant characterizes exponents for which such a function has a degenerate…

We study second order focal loci of nondegenerate plane congruences in P4(C) with degenerate focal conic. We show the projective generation of such congruences when the second order focal locus fills a component of the focal conic,…

We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of K\"ahler-Einstein metrics with cone singularities.

We classify nets of conics in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent…

We classify projective symmetries of irreducible plane sextics with simple singularities which are stable under equivariant deformations. We also outline a connection between order~2 stable symmetries and maximal trigonal curves.

We first show that the union of a projective curve with one of its extremal secant lines satisfies the linear general position principle for hyperplane sections. We use this to give an improved approximation of the Betti numbers of curves…

We present a procedure which allows one to integrate explicitly the class of checkerboard IC-nets which has recently been introduced as a generalisation of incircular (IC) nets. The latter class of privileged congruences of lines in the…

We study intersections of projective convex sets in the sense of Steinitz. In a projective space, an intersection of a nonempty family of convex sets splits into multiple connected components each of which is a convex set. Hence, such an…

We describe a new infinite family of line arrangements in the projective plane with only triple points singularities and recover previously known examples.

This paper is the first in a series. The main goal of the series is to present a geometric construction of certain remarkable tensor categories arising from quantum groups coresponding to the value of deformation parameter $q$ equal to a…

We study linear systems of surfaces in $\mathbb{P}^3$ singular along general lines. Our purpose is to identify and classify special systems of such surfaces, i.e., those nonempty systems where the conditions imposed by the multiple lines…

Zauner's conjecture concerns the existence of $d^2$ equiangular lines in $\mathbb{C}^d$; such a system of lines is known as a SIC. In this paper, we construct infinitely many new SICs over finite fields. While all previously known SICs…

We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski…

Let $f\colon C \rightarrow \mathbb{P}^1$ be a degree $k$ genus $g$ cover. The stratification of line bundles $L \in \mathrm{Pic}^d(C)$ by the splitting type of $f_*L$ is a refinement of the stratification by Brill-Noether loci $W^r_d(C)$.…

We study the orchard problem on cubic surfaces. We classify possibly reducible cubic surfaces $X\subseteq \mathbb{P}^3(\C)$ with smooth components on which there exist families of finite sets (of unbounded size) with quadratically many…

This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative…

We study Hardy type inequalities involving mixed cylindrical and spherical weights, for functions supported in cones. These inequalities are related to some singular or degenerate differential operators.

Consider a smooth complex surface $X$ which is a double cover of the projective plane $\mathbb{P}^2$ branched along a smooth curve of degree $2s$. In this article, we study the geometric conditions which are equivalent to the existence of…