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Exceptional Fano 3-folds from rational curves

Jaime Cuadros Valle, Joe Lope Vicente

TL;DR

This work constructs new exceptional Fano 3-folds with klt singularities by perturbing invertible polynomials in weighted projective spaces, placing them on the boundary of the K-moduli and linking their geometry to Sasaki-Einstein structures. The authors develop a framework based on Type I–III Thom-Sebastiani sums with a cycle block, analyze both quasismooth and non-quasismooth cases, and establish klt properties using Newton polyhedra and stack-weighted Tangent Cones. They prove exceptionality for two families (Type A and Type B) under explicit numerical bounds Id < min{wi wj}, and provide concrete examples that realize these exceptional Fano 3-folds, indicating rich boundary phenomena for moduli of K-stable Fano varieties. The results connect algebro-geometric stability with differential-geometric KE metrics and offer a path to understanding boundary points via weighted combinatorics and deformation theory.

Abstract

We show exceptionality of certain families of non-quasismooth weighted hypersurfaces. In particular these admit Kähler-Einstein metrics. Our examples are produced by the monomials generating the complex deformations of orbifolds whose corresponding $S^1$-Seifert bundles are smooth rational homology 7-spheres admitting Sasaki-Einstein metrics. From our construction, it follows that these exceptional Fano hypersurfaces describe elements in the boundary of the K-moduli of $\mathbb{Q}$-Fano 3-folds.

Exceptional Fano 3-folds from rational curves

TL;DR

This work constructs new exceptional Fano 3-folds with klt singularities by perturbing invertible polynomials in weighted projective spaces, placing them on the boundary of the K-moduli and linking their geometry to Sasaki-Einstein structures. The authors develop a framework based on Type I–III Thom-Sebastiani sums with a cycle block, analyze both quasismooth and non-quasismooth cases, and establish klt properties using Newton polyhedra and stack-weighted Tangent Cones. They prove exceptionality for two families (Type A and Type B) under explicit numerical bounds Id < min{wi wj}, and provide concrete examples that realize these exceptional Fano 3-folds, indicating rich boundary phenomena for moduli of K-stable Fano varieties. The results connect algebro-geometric stability with differential-geometric KE metrics and offer a path to understanding boundary points via weighted combinatorics and deformation theory.

Abstract

We show exceptionality of certain families of non-quasismooth weighted hypersurfaces. In particular these admit Kähler-Einstein metrics. Our examples are produced by the monomials generating the complex deformations of orbifolds whose corresponding -Seifert bundles are smooth rational homology 7-spheres admitting Sasaki-Einstein metrics. From our construction, it follows that these exceptional Fano hypersurfaces describe elements in the boundary of the K-moduli of -Fano 3-folds.
Paper Structure (9 sections, 16 theorems, 75 equations)

This paper contains 9 sections, 16 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted hypersurfaces
  4. Monomials of interest
  5. Singularities of the minimal model program
  6. Exceptionality.
  7. klt weighted Fano 3-folds
  8. Exceptionality of klt Fano 3-folds
  9. Examples

Key Result

Lemma 2.1

A hypersurface defined by the weighted homogeneous polynomial $f$ of degree $d$ is well-formed in the well-formed weighted projective space $Y=\mathbb{P}\left(w_0, \ldots, w_n\right)$ if $\operatorname{gcd}\left(w_0, \ldots, \hat{w}_i, \ldots, \hat{w}_j, \ldots, w_n\right) \mid d$ for distinct $i, j

Theorems & Definitions (34)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • Remark 2.1
  • Lemma 2.3
  • proof
  • Theorem 2.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 24 more