Topics in Birational Geometry: Seshadri constants, singularities and the Albanese morphism

This thesis consists of two preprints that I wrote and two preprints that I co-authored during my PhD. They are diverse in nature, but are all connected to the birational classification of projective varieties. We study Seshadri constants of adjoint divisors in arbitrary characteristic, develop a new method to bound them from below, and apply it to surfaces and threefolds. On surfaces we recover some known results. For a smooth threefold $X$ and an ample divisor $A$ on $X$, we show that if the Seshadri constant of $K_X+6A$ at a closed point is below $1/\sqrt{8}$, then the Seshadri constant is rational. In joint work with Jefferson Baudin, Zsolt Patakfalvi and Maciej Zdanowicz, we study different versions of rational singularities in positive characteristic. More specifically, we look for converse statements to the well-known result of Elkik that in characteristic zero, Kawamata log terminal singularities are rational. We obtain, e.g., that Qp-rational, Gorenstein, $F$-pure, threefold singularities are canonical. In joint work with Jefferson Baudin and Tatsuro Kawakami, we show a version of Grauert-Riemenschneider vanishing in arbitrary characteristic, and obtain analogs in positive characteristic of the aforementioned rationality of Kawamata log terminal singularities: e.g., we show that a Noetherian, excellent, normal scheme of dimension $3$ which is Cohen-Macaulay and of Kawamata log terminal type, has rational singularities. Finally, we study the Albanese morphism of a log Calabi-Yau pair over the complex numbers. We obtain a birational version of the weak Beauville-Bogomolov decomposition if the automorphism group of the pair contains a torus of maximal dimension.