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We construct algebraic surfaces with a large number of type A singularities. Bivariate polynomials presented in previous works for the construction of nodal surfaces and certain families of Belyi polynomials are used. In some cases explicit…
Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations…
In this paper, we issue an error analysis for integration over discrete surfaces using the surface parametrization presented in [PS22] as well as prove why even-degree polynomials exhibit a higher convergence rate than odd-degree…
In this note we construct an unlimited family of irregular algebraic surfaces of general type with canonical map of degree $ 8 $, irregularity $ 1 $ and arbitrarily large geometric genus such that the image of the canonical map is not a…
First we solve the problem of finding minimal degree families on toric surfaces by reducing it to lattice geometry. Then we describe how to find minimal degree families on, more generally, rational complex projective 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.
The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a…
This is a first graduate course in algebraic geometry. It aims to give the student a lift up into the subject at the research level, with lots of interesting topics taken from the classification of surfaces, and a human-oriented discussion…
This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic…
We present a method for computing projective isomorphisms between rational surfaces that are given in terms of their parametrizations. The main idea is to reduce the computation of such projective isomorphisms to five base cases by…
We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that…
Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from…
In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by…
We classify semi-algebraic surfaces in $\mathbb{R}^n$ with isolated singularities up to bi-Lipschitz homeomorphisms with respect to the inner distance. In particular, we obtain complete classifications for the Nash surfaces and the complex…
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 answer some enumerative questions about irreducible rational curves on Hirzebruch surfaces, by combining an idea of Kontsevich with the study of the geometry of certain natural parameter spaces. Our formulas generalize Kontsevich's…
We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such…
An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for…
The rich analytic structure of hadronic form factors makes a theoretically consistent yet easily applicable parametrisation cumbersome. Consequently, most parametrisations are limited to reproducing the simplest analytic features sufficient…
We give an algorithm that, for a given value of the geometric genus $p_g,$ computes all regular product-quotient surfaces with abelian group that have at most canonical singularities and have canonical system with at most isolated base…