On the Abundance Problem for 33-folds in characteristic p>5p>5

In this article we prove two cases of the abundance conjecture for $3$-folds in characteristic $p>5$: $(i)$ $(X, \Delta)$ is KLT and $\kappa(X, K_X+\Delta)=1$, and $(ii)$ $(X, 0)$ is KLT, $K_X\equiv 0$ and $X$ is not uniruled.Comment: With an Appendix by Christopher Hacon. Some reference updated and minor changes. This is the final versio

Doctor of Philosophy

dissertationIn this dissertation, we prove a characteristic p>0 analogue of the log terminal inversion of adjunction and show the equality of the two technical terms F-Different and Different conjectured by Schwede. We also prove a special version of the (relative) Kawamata-Viehweg vanishing theorem for 3-folds, normality of minimal log canonical centers, Kodaira's Canonical Bundle formula for family of rational curves, and the Adjunction Formula on Q-factorial 3-folds in characteristic p>5

BOUNDEDNESS OF LOG-PLURICANONICAL MAPS FOR SURFACES OF LOG-GENERAL TYPE IN POSITIVE CHARACTERISTIC

In this article we prove the following boundedness result: Fix a DCC set I ⊆ [0, 1]. Let D be the set of all log pairs (X, Δ) satisfying the following properties: (i) X is a projective surface defined over an algebraically closed field, (ii) (X, Δ) is log canonical and the coefficients of Δ are in I, and (iii) Kₓ + Δ is big. Then there is a positive integer N = N(I) depending only on the set I such that the linear system |⎿m(Kₓ +Δ)⏌| defines a birational map onto its image for all m ≥ N and (X, Δ) ∈ D