Related papers: Real forms on rational surfaces
A family of algebraic surfaces with many nondegenerate real singularities is introduced with the help of a construction, which has been used in previous works for the generation of substitution tilings.
In this paper we describe an algorithm for implicitizing rational hypersurfaces in case there exists at most a finite number of base points. It is based on a technique exposed in math.AG/0210096, where implicit equations are obtained as…
We give an elementary proof of a recent result by Fishman, Kleinbock, Merrill and Simmons about rational points on quadratic surfaces.
An expository description of smooth cubic curves in the real or complex projective plane.
A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally…
Smooth real cubic surfaces are birationally trivial (over $\R$) if and only if their real locus is connected or, equivalently, if and only if they have two skew real lines or two skew complex conjugate lines. In such a case a…
Let $X,Y$ be two irreducible subvarieties of the projective space $\mathbb{P}^n$, and $d\geq 1$ an integer number. The main result of this paper is an algorithm to construct {\bf explicitly}, in terms of $d$ and the ideals defining $X$ and…
Explicit representations of complex structures on closed manifolds are valuable, but relatively rare in the literature. Using isoparametric theory, we construct complex structures on isoparametric hypersurfaces with $g=4, m=1$ in the unit…
Let $X$ be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on $X$, with height at most $B$, for the case in which $X$ is a curve or a surface. In the latter case one excludes…
In this note, we study linear determinantal representations of smooth plane cubics over finite fields. We give an explicit formula of linear determinantal representations corresponding to rational points. Using Schoof's formula, we count…
Let $(X,D)$ be a pair where $X$ is a projective variety. We study in detail how the behavior of rational curves on $X$ as well as the positivity of $-(K_X+D)$ and $D$ influence the behavior of rational curves on $D$. In particular we give…
Computing occluding contours is a key building block of non-photorealistic rendering, but producing contours with consistent visibility has been notoriously challenging. This paper describes the first general-purpose smooth surface…
We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.
We will show that there is a smooth complex projective surface, birational to some Enriques surface, such that the automorphism group is discrete but not finitely generated.
Nontrivial infinitesimal bendings for a class of two-dimensional surfaces are constructed. The surfaces considered here are orientable; compact; with boundary; have positive curvature everywhere except at finitely many planar points; and…
Using a quartic surface and its rational curves we can give an infinite number of integer hexahedra; these are 6 sided 3d solids, each face a trapezoid, with all sides and diagonals having intger lengths.
A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be either a torus, a K3 surface, an Enriques surface or a rational surface. Automorphisms of rational surfaces are quite…
We give an effective iterative characterization of the classes of (smooth, rational) (-1)-curves on the blowup of the projective plane at general points. Such classes are characterized as having self-intersection -1, arithmetic genus 0, and…
We compare real and complex dynamics for automorphisms of rational surfaces that are obtained by lifting \chg{some} quadratic birational maps of the plane. In particular, we show how to exploit the existence of an invariant cubic curve to…
We construct a normal projective $\mathbb{Q}$-Gorenstein surface over an algebraically closed field whose canonical ring is not finitely generated. Moreover, we provide a counterexample to the minimal model program for…