Related papers: Del Pezzo surfaces and local inequalities

Let $(X,D)$ be an open log del Pezzo surface of rank one, that is, $X$ is a normal projective surface of Picard rank one, the boundary $D$ is a reduced nonzero divisor on $X$, and the anti-log canonical divisor $-(K_X+D)$ is ample. We show…

In this note we shall introduce a simple, effective numerical method for solving partial differential equations for scalar and vector-valued data defined on surfaces. Even though we shall follow the traditional way to approximate the…

We prove Manin's conjecture for split smooth quintic del Pezzo surfaces over arbitrary number fields with respect to fairly general anticanonical height functions. After passing to universal torsors, we first show that we may restrict the…

We complete the classification of regular generically free actions of finite groups on del Pezzo surfaces, up to birational equivalence. As a byproduct, we settle several open problems in equivariant birational geometry, e.g., we classify…

Let X be a smooth projective surface of irregularity 0. The Hilbert scheme of n points on X parameterizes zero-dimensional subschemes of X of length n. In this paper, we discuss general methods for studying the cone of ample divisors on the…

The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study…

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…

We introduce the notion of strongly asymptotically log del Pezzo flags, and classify such flags under the assumption that their zero-dimensional part lies in the boundary. We use this result to give a new and conceptual proof of the…

Manin's conjecture for the asymptotic behavior of the number of rational points of bounded height on del Pezzo surfaces can be approached through universal torsors. We prove several auxiliary results for the estimation of the number of…

We prove that the spaces of rational curves on del Pezzo surfaces are either irreducible or empty, with a unique exception.

We prove a Poincare type inequality for differential forms on compact manifolds by means of a constructive 'globalization' of a local Poincare inequality on convex sets.

The exceptional log Del Pezzo surfaces with delta=1 are classified.

In an earlier work we identified the types and numbers of static equilibrium points of solids arising from fine, equidistant $n$-discretrizations of smooth, convex surfaces. We showed that such discretizations carry equilibrium points on…

We compute the complexity of del Pezzo surfaces with du Val singularities.

For a linear system $|C|$ on a smooth projective surface $S$, whose general element is a smooth, irreducible curve, the Severi variety $V_{|C|, \delta}$ is the locally closed subscheme of $|C|$ which parametrizes irreducible curves with…

In this paper we study the problem of the optimal distribution of two materials on smooth submanifolds $M$ of dimension $d-1$ in $\mathbf R^d$ without boundary by means of the topological derivative. We consider a class of shape…

This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of…

In this article a new upper bounds for the multiple trigonometrical integrals are found. The method of the work based on a new method of estimation for the areas of algebraic surfaces.

In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of level set function, the…

In this paper, we establish the Zariski decompositions of arithmetic R-divisors of continuous type on arithmetic surfaces and investigate several properties. We also develop the general theory of arithmetic R-divisors on arithmetic…