New directions in the Minimal Model Program
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New directions in the Minimal Model Program Paolo Cascini1 Received: 15 April 2020 / Accepted: 17 July 2020 © The Author(s) 2020
Abstract We survey some recents developments in the Minimal Model Program. After an elementary introduction to the program, we focus on its generalisations to the category of foliated varieties and the category of varieties defined over any algebraically closed field of positive characteristic.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . 2 Classical Minimal Model Program . . . . . . . . 2.1 K-stable varieties . . . . . . . . . . . . . . . 2.2 Boundedness and moduli spaces . . . . . . . 3 Minimal Model Program for foliations . . . . . . 4 Minimal Model Program in positive characteristic References . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction The Minimal Model Program (in short MMP), also known as Mori’s program, originated in the 80’s as an attempt to generalise some of the main results of the Italian school of algebraic geometry, led by Castelnuovo at the beginning of the last century. Its aim is to construct a minimal model, i.e. a good birational representative, for any complex projective variety. Over the last two decades, the Minimal Model Program, and more in general birational geometry, has witnessed an exceptional amount of work, with three main objectives: prove some of the main outstanding conjectures of Mori’s program, apply techniques from birational geometry to solve some open problems in other fields of mathematics and, finally, extend the main ideas of the program to different contexts, such as to the category of varieties defined over fields of positive characteristic, the category of Kähler vaireities or the category of foliated varieties. The goal of this note is to survey some of these achievements. It is by far not exhaustive and it does not cover many interesting aspects of the program, as it mainly focus on the author’s expertise (e.g. see [33,58] for other important recent directions).
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Paolo Cascini [email protected] Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
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P. Cascini
We begin by a general overview of the Minimal Model Program. Although we refer to [37] for an introduction to its main techniques, we provide here a gentle introduction to some of its main goals. To this end, as in [13], we use some basic notions of graph theory, aiming to explain some of the tools and objectives in birational geometry. In particular, recall that a directed graph is a set of vertices connected by oriented edges, i.e. ordered pairs of vertices, denoted by X → Y . A chain
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