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We compactify and regularize the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system turns out to have certain unusual…

The surface corresponding to the moduli space of quadratic endomorphisms of $\mathbb{P}^1$ with a marked periodic point of order $n$ is studied. It is shown that the surface is rational over $\mathbb{Q}$ when $n\le 5$ and is of general type…

We prove that a nodal quartic threefold $X$ containing no planes is $Q$-factorial provided that it has not more than 12 singular points, with the exception of a quartic with exactly 12 singularities containing a quadric surface. We give…

For each field $k$ of characteristic zero, we classify which groups act by automorphisms on a quartic del Pezzo surface over $k$. We also determine which groups act on $k$-rational, stably $k$-rational, or $k$-unirational quartic del Pezzo…

Given a nondegenerate ternary form $f=f(x_1,x_2,x_3)$ of degree 4 over an algebraically closed field of characteristic zero, we use the geometry of K3 surfaces and van den Bergh's correspondence between representations of the generalized…

Let S be a smooth cubic surface defined over a field K. As observed by Segre and Manin, there is a secant and tangent process on S that generates new K-rational points from old. It is natural to ask for the size of a minimal generating set…

The Eckardt hypersurface in $\mathbb{P}^{19}$ parameterizes smooth cubic surfaces with an Eckardt point, which is a point common to three of the $27$ lines on a smooth cubic surface. We describe the cubic surfaces lying on the singular…

We give quantitative and qualitative results on the family of surfaces in $\mathbb{CP}^3$ containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines $E$. We prove that its general element…

This paper further studies the matroid realization space of a specific deformation of the regular $n$-gon with its lines of symmetry. Recently, we obtained that these particular realization spaces are birational to the elliptic modular…

In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number 17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces and we use them…

The main focus of this paper is on models of quartic surfaces, especially so-called complex surfaces. These are special fourth-degree surfaces that Julius Pl\"ucker introduced in the 1860s for visualizing the local structure of a quadratic…

Let $Y$ be the complement of a plane quartic curve $D$ defined over a number field. Our main theorem confirms the Lang-Vojta conjecture for $Y$ when $D$ is a generic smooth quartic curve, by showing that its integral points are confined in…

In this paper we give an elementary proof of results on the structure of 4-class groups of real quadratic number fields originally due to A. Scholz. In a second (and independent) section we strengthen C. Maire's result that the 2-class…

A polygonal surface in the pseudo-hyperbolic space H^(2,n) is a complete maximal surface bounded by a lightlike polygon in the Einstein universe Ein^(1,n) with finitely many vertices. In this article, we give several characterizations of…

For each integer $D\ge3$, we give a sharp bound on the number of lines contained in a smooth complex $2D$-polarized $K3$-surface in $\mathbb{P}^{D+1}$. In the two most interesting cases of sextics in $\mathbb{P}^4$ and octics in…

The space of n (ordered) points on the projective line, modulo automorphisms of the line, is one of the most important and classical examples of an invariant theory quotient, and is one of the first examples given in any course. Generators…

We consider the ring of invariants of n points on the projective line. The space (P^1)^n // PGL_2 is perhaps the first nontrivial example of a Geometry Invariant Theory quotient. The construction depends on the weighting of the n points.…

A general net of quadric surfaces, together with a choice of a base point, defines a net of plane cubics via the Gale transformation of the remaining seven base points. To both nets, one can also naturally associate the same smooth plane…

Let $P$ be a set of $n$ points in the plane, not all on a line, each colored \emph{red} or \emph{blue}. The classical Motzkin--Rabin theorem guarantees the existence of a \emph{monochromatic} line. Motivated by the seminal work of Green and…

We consider the number \bar N(q) of points in the projective complement of graph hypersurfaces over \F_q and show that the smallest graphs with non-polynomial \bar N(q) have 14 edges. We give six examples which fall into two classes. One…