Related papers: Counting Zariski chambers on Del Pezzo surfaces
We present an improved algorithm for the computation of Zariski chambers on algebraic surfaces. The new algorithm significantly outperforms the so far available method and allows therefore to treat surfaces of high Picard number, where huge…
Zariski chambers are natural pieces into which the big cone of an algebraic surface decomposes. They have so far been studied both from a geometric and from a combinatorial perspective. In the present paper we complement the picture with a…
The big cone of every smooth projective surface $X$ admits the natural decomposition into Zariski chambers. The purpose of this note is to give a simple criterion for the interiors of all Zariski chambers on $X$ to be numerically determined…
Let $X$ be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of $X$. We study these two decompositions and determine when a Weyl chamber is contained in the interior…
The big cone of every K3 surface admits two natural chamber decompositions: the decomposition into Zariski chambers, and the decomposition into simple Weyl chambers. In the present paper we compare these two decompositions and we study…
In this note we give a quick and simple proof of the existence (and uniqueness) of Zariski decompositions on surfaces. While Zariski's original proof employs a rather sophisticated procedure to construct the negative part of the…
In this note we consider the problem of integrality of Zariski decompositions for pseudoeffective integral divisors on algebraic surfaces. We show that while sometimes integrality of Zariski decompositions forces all negative curves to be…
For a Del Pezzo surface of degree 8 given over the rationals we decide whether there is a rational parametrization of the surface and construct one in the affirmative case. We define and use the Lie algebra of the surface to reach the aim.…
We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The obtained formulae express cohomologies in terms…
Zariski decompositions play an important role in the theory of algebraic surfaces. For making geometric use of the decomposition of a given divisor, one needs to pass to a multiple of the divisor in order to clear denominators. It is…
We construct algebraic geometric codes from weak del Pezzo surfaces. The codes are associated to the anti-canonical class of the anti-canonical model and to the set of rational points of these models. Since we consider weak Del Pezzo…
We give a characterization of all del Pezzo surfaces of degree 6 over an arbitrary field $F$. A surface is determined by a pair of separable algebras. These algebras are used to compute the Quillen $K$-theory of the surface. As a…
We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic…
We discuss the problem of existence of rational curves on a certain del Pezzo surface from a computational point of view and suggest a computer algorithm implementing search. In particular, our computations reveal that the surface contains…
In a 1962 paper, Zariski introduced the decomposition theory that now bears his name. Although it arose in the context of algebraic geometry and deals with the configuration of curves on an algebraic surface, we have recently observed that…
We investigate the characteristic numbers of Del Pezzo surfaces using degenerations.
Given a point S (the light position) in P^3 and an algebraic surface Z (the mirror) of P^3, the caustic by reflection of Z from S is the Zariski closure of the envelope of the reflected lines got by reflection of the incident lines (Sm) on…
Let $f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$ and let us consider a del Pezzo surface of degree one given by the equation $\cal{E}_{f}: x^2-y^3-f(z)=0$. In this note we prove that if the set of rational points on the curve $E_{a,…
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 construct a Zariski decomposition for cycle classes of arbitrary codimension. This decomposition is an analogue of well-known constructions for divisors. Examples illustrate how Zariski decompositions of cycle classes reflect the…