Related papers: Counting Zariski chambers on Del Pezzo surfaces
We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first example, to our knowledge, of a simply connected smooth variety whose sets of integral points are never…
Let a,b,c,d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in PP^3 defined by ax^4+by^4+cz^4+dw^4=0. We prove that if V contains a rational point that does not lie on any of the 48 lines on…
As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of…
We prove Manin's conjecture for a split singular quartic del Pezzo surface with singularity type $2\Aone$ and eight lines. This is achieved by equipping the surface with a conic bundle structure. To handle the sum over the family of conics,…
Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational…
In algebraic geometry specialisations and valuations play and important role. In this paper we start investigating analogous structures for Zariski structures. Specifically, we look into the existence and uniqueness properties of extensions…
The image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the…
From a scalar field defined at the corner of a cube, an isosurface can be extracted using the Marching Cube algorithm. The isosurface separates the cell into two or more partial cells. A similar situation arises when an material interface…
Illumination of scenes is usually generated in computer graphics using polygonal meshes. In this paper, we present a geometric method using projections. Starting from an implicit polynomial equation of a surface in 3-D or a curve in 2-D, we…
This paper is an enhancement of the previous note "Explicit computations of Zariski decompositions on P_Z^1". In this paper, we observe several properties of a certain kind of an arithmetic divisor D on the n-dimensional projective space…
Zernike polynomials are a basis of orthogonal polynomials on the unit disk that are a natural basis for representing smooth functions. They arise in a number of applications including optics and atmospheric sciences. In this paper, we…
Two classical results in algebraic geometry are that the branch curve of a del Pezzo surface of degree 1 can be embedded as a space sextic curve and that every space sextic curve has exactly 120 tritangents corresponding to its odd theta…
This is an expanded version of the two papers "Interpolation of Varieties of Minimal Degree" and "Interpolation Problems: Del Pezzo Surfaces." It is well known that one can find a rational normal curve in $\mathbb P^n$ through $n+3$ general…
A series of Zariski pairs and four Zariski triplets were found by using lattice theory of K3 surfaces. There is a Zariski triplet of which one member is a deformation of another.
I prove new local inequality for divisors on smooth surfaces, describe its applications, and compare it to a similar local inequality that is already known by experts.
We study a class of semialgebraic convex bodies called discotopes. These are instances of zonoids, objects of interest in real algebraic geometry and random geometry. We focus on the face structure and on the boundary hypersurface of…
A couple of complex projective plane curves are said to make a Zariski pair if they have the same degree and the same type of singularities, but their embeddings in the projective plane are topologically different. In this paper, we present…
We characterize the integral Zariski decomposition of a smooth projective surface with Picard number 2 to partially solve a problem of B. Harbourne, P. Pokora, and H. Tutaj-Gasinska [Electron. Res. Announc. Math. Sci. 22 (2015), 103--108].
In this article we prove the following theorems about weak approximation of smooth cubic hypersurfaces and del Pezzo surfaces of degree 4 defined over global fields. (1) For cubic hypersurfaces defined over global function fields, if there…
In this paper, we prove that a pair of the minimal resolution of a del Pezzo surface with rational double points whose general anti-canonical member is smooth and its exceptional divisor lifts to the Witt ring. We also classify a del Pezzo…