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Concept# Algebraic variety

Summary

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.
Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility.
The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the

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Algebraic geometry is a branch of mathematics which classically studies zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly fro

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In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the

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In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus

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The aim of this course is to learn the basics of the modern scheme theoretic language of algebraic geometry.

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The aim of the course is to give an introduction to linear algebraic groups and to give an insight into a beautiful subject that combines algebraic geometry with group theory.

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This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves.

Quentin Arthur Frantisek Posva

This thesis is constituted of one article and three preprints that I wrote during my PhD thesis. Their common theme is the moduli theory of algebraic varieties. In the first article I study the Chow--Mumford line bundle for families of uniformly K-stable Fano pairs, and I show it is ample when the family has maximal variation. The three preprints deal with a generalization to positive characteristic of Kollár's gluing theory for stable varieties. I generalize this theory to surfaces and threefolds. Then I apply it to study the abundance conjecture for surfaces, the topology of lc centers on threefolds, existence of semi-resolutions for surfaces, and gluing theory for families of surfaces in mixed characteristic.

We establish p-adic versions of the Manin-Mumford conjecture, which states that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic spaces and formal groups over a p-adic field or its ring of integers, respectively. In particular, we show that the underlying rigidity results for algebraic functions generalize to suitable p-adic analytic functions. This leads us to uncover purely p-adic Manin-Mumford-type results for formal groups not coming from abelian schemes. Moreover, we observe that a version of the Tate-Voloch conjecture holds: torsion points either lie squarely on a subscheme or are uniformly bounded away from it in the p-adic distance.

The Uniformization Theorem due to Koebe and Poincaré implies that every compact Riemann surface of genus greater or equal to 2 can be endowed with a metric of constant curvature – 1. On the other hand, a compact Riemann surface is a complex algebraic curve and is therefore described by a polynomial equation with complex coefficients. The uniformization problem is then to link explicitly these two descriptions. In [BS05b], Peter Buser and Robert Silhol develop a new uniformization method for compact Riemann surfaces of genus two. Given such a surface S, the method describes a polynomial equation of an algebraic curve conformally equivalent to S. However, in this method appear a complex number τ BS and a function f BS which is holomorphic on the unit disk, both being characterized by some functional equations. This means that τ BS, f BS are given implicitly. P. Buser and R. Silhol then approximate them numerically by a complex number τ and a polynomial p using the approximation method developped in [BS05a]. In cases where the equation of the algebraic curve is known, they notice that these approximations are very good. In this thesis we prove a convergence theorem for the approximation method of P. Buser and R. Silhol, and we propose an adaptation of their method that allows to solve some of the numerical problems to which it is prone. Moreover, we generalize this uniformization method to hyperelliptic Riemann surfaces of genus greater than 2, and we give some examples of numerical uniformization in genus 3.