Morphisms of Projective Varieties from the viewpoint of Minimal Model Theory

Originally a technical tool, the derived category of coherent sheaves over an algebraic variety has become over the last twenty years an important invariant in the birational study of algebraic varieties. Problems of birational invariance and of minimization of the derived category have appeared, inspired by Kontsevich's homological mirror symmetry conjecture and Mori's minimal model program. We present the main conjectures and their proofs in dimension 3 and for particular c...

This is the Foreword to the book ``Explicit birational geometry of 3-folds'', edited by A. Corti and M. Reid, CUP Jun 2000, ISBN: 0 521 63641 8, with papers by K. Altmann, A. Corti, A.R. Iano-Fletcher, J. Koll\'ar, A.V. Pukhlikov and M. Reid. One of the main achievements of algebraic geometry over the last 20 years is the work of Mori and others extending minimal models and the Enriques--Kodaira classification to 3-folds. This book is an integrated suite of papers centred a...

We present an addendum/erratum to the paper "Weyl Groups and Birational Transformations among Minimal Models" written by the author and published in 1995, adding the analysis of the "88-th" deformation type of a smooth Fano 3-fold with $B_2 = 4$ denoted as $n^o\ 13$, which was missing from the original classification table by Mori-Mukai and added later to the list of smooth Fano 3-folds with $B_2 \geq 2$, while correcting the mistake pointed out by Eric Jovinelly (also notice...

This paper is devoted to study the birational properties of the Albanese map. I generalize a theorem of Kawamata to tell when the Albanese map is surjective and when it is an algebraic fiber space.

This paper resolves several outstanding questions regarding the Minimal Model Program for klt threefolds in mixed characteristic. Namely termination for pairs which are not pseudo-effective, finiteness of minimal models and the Sarkisov Program.

We survey new results on finite groups of birational transformations of algebraic varieties.

In this paper, we prove various results on boundedness and singularities of Fano fibrations and of Fano type fibrations. A Fano fibration is a projective morphism $X\to Z$ of algebraic varieties with connected fibres such that $X$ is Fano over $Z$, that is, $X$ has "good" singularities and $-K_X$ is ample over $Z$. A Fano type fibration is similarly defined where $X$ is assumed to be close to being Fano over $Z$. This class includes many central ingredients of birational geom...

Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006. Does not include: rationality problems, techniques of the minimal model problem and much of the rest. Includes: Lecture 0: geometry and arithmetic of curves Lecture 1: Kodaira dimension and properties, rational connectendess, Lang's and Campana's conjectures. Lecture 2: Campana's program; Campana constellations framed in terms of b-divisors, to allow for a definit...

This document contains notes from the lectures of Corti, Koll\'ar, Lazarsfeld, and Musta\c{t}\u{a} at the workshop ``Minimal and canonical models in algebraic geometry" at MSRI, Berkeley, April 2007. The lectures give an overview of the recent advances on canonical and minimal models of algebraic varieties obtained by Hacon--McKernan and Birkar--Cascini--Hacon--McKernan.

This is a survey for the 2015 AMS Summer Institute on Algebraic Geometry about the Frobenius type techniques recently used extensively in positive characteristic algebraic geometry. We first explain the basic ideas through simple versions of the fundamental definitions and statements, and then we survey most of the recent algebraic geometry results obtained using these techniques.