Algebraic Varieties: Minimal Models and Finite Generation

The first graduate-level introduction to the finite generation theorem of the canonical ring, a major achievement of modern algebraic geometry. Largely self-contained, this text explains the basics of minimal model theory, covering all the progress of the last three decades and assuming only the basics in algebraic geometry.

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The finite generation theorem is a major achievement in modern algebraic geometry. Based on the minimal model theory, it states that the canonical ring of an algebraic variety defined over a field of characteristic 0 is a finitely generated graded ring. This graduate-level text is the first to explain this proof. It covers the progress on the minimal model theory over the last 30 years, culminating in the landmark paper on finite generation by Birkar—Cascini—Hacon—McKernan. Building up to this proof, the author presents important results and techniques that are now part of the standard toolbox of birational geometry, including Mori's bend-and-break method, vanishing theorems, positivity theorems, and Siu's analysis on multiplier ideal sheaves. Assuming only the basics in algebraic geometry, the text keeps prerequisites to a minimum with self-contained explanations of terminology and theorems.