Related papers: Conics on Barth--Bauer octics

We combine classical Vinberg's algorithms with the lattice-theoretic/arithmetic approach from arXiv:1706.05734 [math.AG] to give a method of classifying large line configurations on complex quasi-polarized K3-surfaces. We apply our method…

An efficient way to get implicit equations of conics on five points and quadrics on nine, using pencils of conics and quadrics, is revealed. Parallel axis right cones intersect on a conic. An example, to show how to place five coplanar…

We consider singular Q-acyclic surfaces with smooth locus of non-general type. We prove that if the singularities are topologically rational then the smooth locus is C^1- or C*-ruled or the surface is up to isomorphism one of two…

In this article, we continue to study the geometry of bisections of certain rational elliptic surfaces. As an application, we give examples of Zariski N + 1-plets of degree 2N + 4 whose irreducible components are an irreducible quartic…

For each $1\leq n\leq6$ we present formulas for the number of $n-$nodal curves in an $n-$dimensional linear system on a smooth, projective surface. This yields in particular the numbers of rational curves in the system of hyperplane…

A 2-uniform tiling is an edge-to-edge tiling by regular polygons having $2$ distinct transitivity classes of vertices. There are 20 distinct 2-uniform tilings (these are of $14$ different types) on the plane, and since the plane is the…

We describe all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X and the symmetric…

In this paper, we study the restrictions on the number $m$ of conic-line curves in special pencils. The most general result we obtain is the relation between upper bounds on $m$ and the number $p$ of concurrent lines in these pencils. We…

We formulate an equivariant conservation of number, which proves that a generalized Euler number of a complex equivariant vector bundle can be computed as a sum of local indices of an arbitrary section. This involves an expansion of the…

Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that, the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the…

We consider K3 surfaces which are double cover of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are…

Using equivariant geometry, we find a universal formula that computes the number of times a general cubic surface arises in a family. As applications, we show that the PGL(4) orbit closure of a generic cubic surface has degree 96120, and…

We explore the connection between the rank of a polynomial and the singularities of its vanishing locus. We first describe the singularity of generic polynomials of fixed rank. We then focus on cubic surfaces. Cubic surfaces with isolated…

Miyaoka proved that a smooth surface of degree $d$ in ${\mathbf{P}}^3({\mathbb{C}})$ contains at most $2d(d-2)$ pairwise disjoint lines. In this note, we verify that the Maschke octic contains $96$ pairwise disjoint lines, thereby proving…

An effective divisor D on a smooth (compact complex) surface X is called even, if its class $[D] \in H^2(X,\Z)$ is divisible by 2. D may be assumed reduced w.l.o.g. Then D being even is equivalent to the existence of a double cover $Y \to…

We show, in this first part, that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic $2$ is at most $16$. We produce examples with…

Let $X$ be a smooth irreducible projective curve of genus $g$ and gonality 4. We show that the canonical model of $X$ is contained in a uniquely defined surface, ruled by conics, whose geometry is deeply related to that of $X$. This surface…

In 1884 the German mathematician Karl Rohn published a substantial paper on \cite{ROH} on the properties of quartic surfaces with triple points, proving (among many other things) that the maximum number of lines contained in a quartic…

We consider elliptic curves whose coefficients are degree 2 polynomials in a variable t. We prove that for infinitely many values of t the resulting elliptic curve has rank at least 1. All such curves together form an algebraic surface…

Using the techniques of Bayer--Macr\`i, we determine the walls in the movable cone of the Mukai system of rank two for a general K3 surface $S$ of genus two. We study the (essentially unique) birational map to $S^{[5]}$ and decompose it…