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Fano threefolds

Yuri Prokhorov

TL;DR

This work consolidates the modern viewpoint on classifying complex Fano threefolds, with a focus on the Picard number one case. It develops a robust toolkit—log pair singularities, adjunction, multiplier ideals, vanishing theorems, and Mori theory—to establish the existence of smooth divisors in the anticanonical system and to illuminate the structure of threefolds via their anticanonical models. The notes then provide a detailed, multi-faceted classification of Fano threefolds of index 2 (del Pezzo threefolds) including their projective models, graded algebras, and higher Picard ranks, followed by analysis of base points in the anticanonical system and the geometry of hyperelliptic Fano threefolds. Collectively, the work offers a comprehensive, self-contained synthesis of methods and results that underpin the birational geometry and moduli of Fano threefolds, with explicit models in weighted projective spaces and Grassmannians. The approaches equip researchers with concrete criteria for constructing, recognizing, and distinguishing Fano threefolds across deformation types and birational configurations.

Abstract

The goal of these lecture notes is to present the modern point of view on the classification of Fano threefolds. We tried to offer a self-consistent treatment of the topics covered. \par\medskip\noindent These notes have been published in two versions: a Russian edition in \textit{Lektsionnye Kursy NOTs} \textbf{31}. Steklov Inst. Math., Moscow (ISBN 978-5-98419-085-5), doi: \href{https://doi.org/10.4213/lkn31}{10.4213/book1907}, and an English translation in the \textit{Proc. Steklov Inst. Math.}, \textbf{328}, Suppl. 1 (2025), doi: \href{https://doi.org/10.1134/S0081543825020014}{10.1134/S0081543825020014}

Fano threefolds

TL;DR

This work consolidates the modern viewpoint on classifying complex Fano threefolds, with a focus on the Picard number one case. It develops a robust toolkit—log pair singularities, adjunction, multiplier ideals, vanishing theorems, and Mori theory—to establish the existence of smooth divisors in the anticanonical system and to illuminate the structure of threefolds via their anticanonical models. The notes then provide a detailed, multi-faceted classification of Fano threefolds of index 2 (del Pezzo threefolds) including their projective models, graded algebras, and higher Picard ranks, followed by analysis of base points in the anticanonical system and the geometry of hyperelliptic Fano threefolds. Collectively, the work offers a comprehensive, self-contained synthesis of methods and results that underpin the birational geometry and moduli of Fano threefolds, with explicit models in weighted projective spaces and Grassmannians. The approaches equip researchers with concrete criteria for constructing, recognizing, and distinguishing Fano threefolds across deformation types and birational configurations.

Abstract

The goal of these lecture notes is to present the modern point of view on the classification of Fano threefolds. We tried to offer a self-consistent treatment of the topics covered. \par\medskip\noindent These notes have been published in two versions: a Russian edition in \textit{Lektsionnye Kursy NOTs} \textbf{31}. Steklov Inst. Math., Moscow (ISBN 978-5-98419-085-5), doi: \href{https://doi.org/10.4213/lkn31}{10.4213/book1907}, and an English translation in the \textit{Proc. Steklov Inst. Math.}, \textbf{328}, Suppl. 1 (2025), doi: \href{https://doi.org/10.1134/S0081543825020014}{10.1134/S0081543825020014}

Paper Structure

This paper contains 86 sections, 182 theorems, 569 equations, 3 figures, 12 tables.

Table of Contents

  1. Notation, preliminary concepts and the simplest properties of Fano varieties
  2. Notation
  3. Definition of Fano varieties
  4. Simplest properties
  5. Index of Fano varieties
  6. Divisors on Fano varieties
  7. Riemann--Roch Theorem on Fano varieties
  8. Fano varieties of large index
  9. Blowups
  10. Grassmannians and homogeneous spaces
  11. Fano varieties and finite morphisms
  12. Fano varieties in weighted projective spaces
  13. Fano varieties and birational morphisms
  14. The existence of a smooth divisor
  15. Main result and outline of the proof
  16. ...and 71 more sections

Key Result

Theorem 1

Let $X$ be a nonsingular Fano threefold of Picard number $\uprho(X)=1$ over an algebraically closed field of characteristic $0$ and let $\iota(X)$ be its Fano index, i.e. the maximal natural number that divides the canonical class in the group $\operatorname{Pic}(X)$(see Definition def:index). Then

Figures (3)

  • Figure 1: The Mori cone $\overline{\operatorname{NE}}(\overline{X})$
  • Figure 2: The Mori cone $\overline{\operatorname{NE}}(X)$ (a transversal section)
  • Figure 3: Atiyah--Kulikov flop

Theorems & Definitions (362)

  • Theorem 1
  • Definition 1.1
  • Example 1.2
  • Theorem 1.3
  • proof
  • Corollary 1.4
  • proof
  • Corollary 1.5
  • proof
  • Theorem 1.6: Kollar-1996-RC
  • ...and 352 more