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Publication# On the projectivity of some moduli spaces of varieties

Abstract

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.

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Related concepts (6)

Algebraic variety

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

Moduli space

In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or

Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive