Related papers: Sums of residues on algebraic surfaces and applica…
The article surveys published and not yet published results about moduli spaces of algebraic surfaces.
We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.
This is a review paper about ADE bundles over surfaces. Based on the deep connections between the geometry of surfaces and ADE Lie theory, we construct the corresponding ADE bundles over surfaces and study some related problems.
Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive $\ell$-tuples of quadratic residues, studying corresponding algebraic curves and their…
We study the ring of algebraic functions on the space of persistence barcodes, with applications to pattern recognition.
This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.
We introduce the notions of mixed resolutions and simplicial sections, and prove a theorem relating them. This result is used (in another paper) to study deformation quantization in algebraic geometry.
The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We…
The simplest version of the Spin-polynomial invariants of the underlying differentiable structures of algebraic surfaces were considered and the simplest arguments were used in order to distinguish the underlying smooth structures of…
We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each…
Duadic group algebra codes are a generalization of quadratic residue codes. This paper settles an open problem raised by Zhu concerning the existence of duadic group algebra codes. These codes can be used to construct degenerate quantum…
In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…
We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli…
This paper is a survey of computational issues in algebraic geometry, with particular attention to the theory of Grobner bases and the regularity of an algebraic variety. 1. A geometric introduction to Grobner bases. 2. An algebraic…
The 2-matrix model has been introduced to study Ising model on random surfaces. Since then, the link between matrix models and combinatorics of discrete surfaces has strongly tightened. This manuscript aims to investigate these deep links…
In this paper we develop an abstract theory for the Codazzi equation on surfaces, and use it as an analytic tool to derive new global results for surfaces in the space forms ${\bb R}^3$, ${\bb S}^3$ and ${\bb H}^3$. We give essentially…
Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…
We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and…
We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on…
We extend finding geometrically-significant preserved quantities by solving specific PDEs to the affine transformations and subgroups. This can be viewed not only as a purely geometrical problem but also as a subcase of finding physical…