Patakfalvi, ZsoltPosva, Quentin Arthur Frantisek2022-06-162022-06-162022-06-16202210.5075/epfl-thesis-9570https://infoscience.epfl.ch/handle/20.500.14299/188513This 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.enAlgebraic geometrymoduli theoryFano varietyK-stabilityChow--Mumford line bundleKollár's gluing theorystable varietypositive characteristicthesis::doctoral thesis