Related papers: A sextic surface cannot have 66 nodes

We continue the development of methods for enumerating nodal curves on smooth complex surfaces, stressing the range of validity. We illustrate the new methods in three important examples. First, for up to eight nodes, we confirm…

In this paper, we study the property of bigness of the tangent bundle of a smooth projective rational surface with nef anticanonical divisor. We first show that the tangent bundle $T_S$ of $S$ is not big if $S$ is a rational elliptic…

Joret et al. proved that posets with cover graphs of treewidth at most 2 have dimension at most 1276. Their proof is long and very complex. We give a short and much simpler proof that the dimension of such posets is at most 12.

We classify lattice $3$-polytopes of width larger than one and with exactly $6$ lattice points. We show that there are $74$ polytopes of width $2$, two polytopes of width $3$, and none of larger width. We give explicit coordinates for…

Given a graph $G$ and an integer $k$, the $H$-free Edge Editing problem is to find whether there exists at most $k$ pairs of vertices in $G$ such that changing the adjacency of the pairs in $G$ results in a graph without any induced copy of…

We classify K3 surfaces with a non-symplectic finite automorphism of high order. It is shown that such an automorphism cannot be of order 60, and for each of the orders 38, 44, 48, 50, 54 and 66, there exists a unique K3 surface with such…

This paper deals with the Nash problem, which consists in proving that the number of families of arcs on a singular germ of a surface $S$ coincides with the number of irreducible components of the exceptional divisor in the minimal…

We give several new constructions for moderate rank elliptic curves over $\mathbb{Q}(T)$. In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over $\mathbb{Q}$ using polynomials of…

We study ramified covers of the projective plane. Given a smooth projective surface S and a generic enough projection of S to the projective plane, we get a cover of the plane ramified over a plane curve. The branch curve is usually…

We give necessary and sufficient criteria for a smooth Enriques surface S in P^r to be scheme-theoretically an intersection of quadrics. Moreover we prove in many cases that, when S contains plane cubic curves, the intersection of the…

The aim of the paper is to provide a series of new examples of smooth surfaces in P^4, not of general type, in degrees varying from 12 up to 14, and to describe their geometry. By using mainly syzygies and liaison techniques, we construct…

We study nodal quintic surfaces with an even set of 16 nodes as analogues of singular Kummer surfaces. The interpretation of the natural double cover of an even 16-nodal quintic as a certain Fano variety of lines could be viewed as a…

We analyze the configurations of conics and lines on a special class of Kummer octic surfaces. In particular, we bound the number of conics by $176$ and show that there is a unique surface with $176$ conics, all irreducible: it admits a…

Newton-Okounkov bodies serve as a bridge between algebraic geometry and convex geometry, enabling the application of combinatorial and geometric methods to the study of linear systems on algebraic varieties. This paper contributes to…

In this note it is shown that, given a smooth minimal complex surface of general type S with p_g(S)=0, K^2_S=3, for which the bicanonical map is a morphism, then the degree of the bicanonical map of S is not equal to 3. This completes our…

In the previous paper, we established an elementary bound for numbers of points of surfaces in the projective $3$-space over ${\Bbb F}_q$. In this paper, we give the complete list of surfaces that attain the elementary bound. Precisely…

Let $k$ be a field of arbitrary characteristic. Let $S$ be a singular surface defined over $k$ with multiple rational curve singularities and suppose that the Chow group of zero cycles of its normalisation $\tilde{S}$ is finite dimensional.…

We construct explicit geometric models for and compute the fundamental groups of all plane sextics with simple singularities only and with at least one type $\bold E_8$ singular point. In particular, we discover four new sextics with…

A Howe curve is defined as the normalization of the fiber product over a projective line of two hyperelliptic curves. Howe curves are very useful to produce important classes of curves over fields of positive characteristic, e.g., maximal,…

We consider elliptic surfaces whose coefficients are degree $2$ polynomials in a variable $T$. It was recently shown that for infinitely many rational values of $T$ the resulting elliptic curves have rank at least $1$. In this article, we…