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Mukai models of Fano varieties

Arend Bayer, Alexander Kuznetsov, Emanuele Macrì

TL;DR

The paper provides a complete, self-contained proof of Mukai's classification for prime Fano varieties of coindex 3 and genus in {6,7,8,9,10,12}, showing they arise as iterated cones over Mukai varieties M_g via Gushel embeddings and Mukai bundles. It builds a bridge from very general K3 hyperplane sections to ambient Fano varieties by extending the Mukai bundle from a K3 surface to the whole variety, using a robust extension principle and a careful analysis of linear sections, cones, and Schubert nondegeneracy. The results unify the K3–Fano correspondence across dimensions, provide explicit models for each genus, and include a genus-7 special case using Lazarsfeld bundles; they also establish a framework for higher-dimensional extensions and discuss remaining open questions related to Brill–Noether generality and characteristic p. Overall, the work solidifies Mukai’s vision of realizing Fano varieties as linear sections of homogeneous spaces, with precise control over singularities, embeddings, and extensions.

Abstract

We give a self-contained and simplified proof of Mukai's classification of prime Fano threefolds of index 1 and genus $g \ge 6$ with at most Gorenstein factorial terminal singularities, and of its extension to higher-dimension.

Mukai models of Fano varieties

TL;DR

The paper provides a complete, self-contained proof of Mukai's classification for prime Fano varieties of coindex 3 and genus in {6,7,8,9,10,12}, showing they arise as iterated cones over Mukai varieties M_g via Gushel embeddings and Mukai bundles. It builds a bridge from very general K3 hyperplane sections to ambient Fano varieties by extending the Mukai bundle from a K3 surface to the whole variety, using a robust extension principle and a careful analysis of linear sections, cones, and Schubert nondegeneracy. The results unify the K3–Fano correspondence across dimensions, provide explicit models for each genus, and include a genus-7 special case using Lazarsfeld bundles; they also establish a framework for higher-dimensional extensions and discuss remaining open questions related to Brill–Noether generality and characteristic p. Overall, the work solidifies Mukai’s vision of realizing Fano varieties as linear sections of homogeneous spaces, with precise control over singularities, embeddings, and extensions.

Abstract

We give a self-contained and simplified proof of Mukai's classification of prime Fano threefolds of index 1 and genus with at most Gorenstein factorial terminal singularities, and of its extension to higher-dimension.

Paper Structure

This paper contains 29 sections, 49 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. History
  4. Open questions
  5. Conventions
  6. Preliminaries
  7. Zero loci
  8. Brill--Noether property
  9. Gushel embeddings for prime K3 surfaces
  10. Construction of the Gushel map
  11. Injectivity and Schubert divisors
  12. Prime K3 surfaces of genus 6, 8, 9, 10, or 12
  13. Mukai varieties of genus 9, 10, or 12
  14. Containment in the Mukai subvarieties
  15. K3 surfaces of genus 6, 8, 9, 10, or 12
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $X$ be a Fano variety of dimension $n \ge 3$ with at most factorial terminal singularities over an algebraically closed field of characteristic zero such that for $g \in \{6,7,8,9,10,12\}$. The above representation of $X$ is canonical and unique up to automorphisms of ${\mathrm{M}}_g$.

Theorems & Definitions (108)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 98 more