Related papers: Conics on smooth quartic surfaces

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.

We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.

We prove that a K3 quartic surface defined over a field of characteristic 2 can contain at most 68 lines. If it contains 68 lines, then it is projectively equivalent to a member of a 1-dimensional family found by Rams and Sch\"utt.

We prove that a del Pezzo surface with Picard number one has at most four singular points.

We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in…

A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears that many (but not all) of them can be relaxed to smooth minimal…

We use floor decompositions of tropical curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in $\RP^n$ is maximal. That is, there exist generic configurations of real linear spaces such…

We construct a smooth complex projective rational surface with infinitely many mutually non-isomorphic real forms. This gives the first definite answer to a long standing open question if a smooth complex projective rational surface has…

We give upper bounds for the number of rational points of bounded anti-canonical height on del Pezzo surfaces of degree at most five over any global field whose characteristic is not equal to two or three. For number fields these results…

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…

We study the effective cones of cycles on universal hypersurfaces on a projective variety $X$, particularly focusing on the case of universal hypersurfaces in $\mathbb{P}^n$. We determine the effective cones of cycles on the universal conic…

We describe convex quadric surfaces in n dimensions and characterize them as convex surfaces with quadric sections by a continuous family of hyperplanes.

This paper proves that the maximum number of rational points on a smooth, absolutely irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible using techniques from algebraic…

We investigate the number of straight lines contained in a K3 quartic surface \(X\) defined over an algebraically closed field of characteristic 3. We prove that if \(X\) contains 112 lines, then \(X\) is projectively equivalent to the…

In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an irreducible lattice…

We study the tropical lines contained in smooth tropical surfaces in R^3. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however,…

A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible…

To a generic configuration of eight points in convex position in the plane, we associate a list consisting of the following information: for all of the 56 conics determined by five of the points, we specify the position of each remaining…

Let $f$ be a positive definite (non-classic) integral quaternary quadratic form. We say $f$ is strongly $s$-regular if it satisfies a regularity property on the number of representations of squares of integers. In this article, we prove…

We give a defining equation of a complex smooth quartic surface containing 56 lines, and investigate its reductions to positive characteristics. This surface is isomorphic to the complex Fermat quartic surface, which contains only 48 lines.…