Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fano fourfolds of K3 type

Marcello Bernardara, Enrico Fatighenti, Laurent Manivel, Fabio Tanturri

TL;DR

This work constructs and analyzes a comprehensive catalog of Fano fourfolds of K3 type (FK3) by realizing them as zero loci of general sections of completely reducible homogeneous bundles on products of flag varieties. It builds a large database (thousands of Fano fourfolds) and isolates 64 FK3 families with $h^{3,1}=1$, then studies their geometry, birational maps, and potential semiorthogonal decompositions, organizing them by origin (Cubic, GM, K3-origin, Rogue) and detailing explicit projections and SODs. The paper also develops and applies a toolbox—Cayley tricks, Eagon–Northcott resolutions, conic-bundle analysis, and derived-category techniques—to understand how these FK3s relate to cubic GM, and K3 geometries, and to identify possible HP dualities. A key goal is to illuminate a rich web of birational links among FK3s, assess rationality in several cases, and frame a general program for understanding FK3 varieties via derived categories and HP duality, while acknowledging nonuniqueness and incompleteness in the broader classification. Overall, the results provide a structured, geometry-driven framework for constructing and analyzing FK3 fourfolds and their associated K3 categories, with potential implications for moduli, rationality, and mirror-symmetric perspectives in higher-dimensional Fano geometry.

Abstract

We produce a list of 64 families of Fano fourfolds of K3 type, extracted from our database of at least 634 Fano fourfolds constructed as zero loci of general global sections of completely reducible homogeneous vector bundles on products of flag manifolds. We study the geometry of these Fano fourfolds in some detail, and we find the origin of their K3 structure by relating most of them either to cubic fourfolds, Gushel-Mukai fourfolds, or actual K3 surfaces. Their main invariants and some information on their rationality and on possible semiorthogonal decompositions for their derived categories are provided.

Fano fourfolds of K3 type

TL;DR

This work constructs and analyzes a comprehensive catalog of Fano fourfolds of K3 type (FK3) by realizing them as zero loci of general sections of completely reducible homogeneous bundles on products of flag varieties. It builds a large database (thousands of Fano fourfolds) and isolates 64 FK3 families with , then studies their geometry, birational maps, and potential semiorthogonal decompositions, organizing them by origin (Cubic, GM, K3-origin, Rogue) and detailing explicit projections and SODs. The paper also develops and applies a toolbox—Cayley tricks, Eagon–Northcott resolutions, conic-bundle analysis, and derived-category techniques—to understand how these FK3s relate to cubic GM, and K3 geometries, and to identify possible HP dualities. A key goal is to illuminate a rich web of birational links among FK3s, assess rationality in several cases, and frame a general program for understanding FK3 varieties via derived categories and HP duality, while acknowledging nonuniqueness and incompleteness in the broader classification. Overall, the results provide a structured, geometry-driven framework for constructing and analyzing FK3 fourfolds and their associated K3 categories, with potential implications for moduli, rationality, and mirror-symmetric perspectives in higher-dimensional Fano geometry.

Abstract

We produce a list of 64 families of Fano fourfolds of K3 type, extracted from our database of at least 634 Fano fourfolds constructed as zero loci of general global sections of completely reducible homogeneous vector bundles on products of flag manifolds. We study the geometry of these Fano fourfolds in some detail, and we find the origin of their K3 structure by relating most of them either to cubic fourfolds, Gushel-Mukai fourfolds, or actual K3 surfaces. Their main invariants and some information on their rationality and on possible semiorthogonal decompositions for their derived categories are provided.
Paper Structure (34 sections, 17 theorems, 120 equations, 2 tables)

This paper contains 34 sections, 17 theorems, 120 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Context and results
  3. Fano varieties of K3 type
  4. Fano fourfolds of K3 type of index bigger than one
  5. Fano fourfolds of K3 type of index 1
  6. FK3 in this paper: a recap
  7. Construction of the database
  8. What can the reader do with this list?
  9. Notation
  10. A few preliminaries
  11. Cayley tricks
  12. Ubiquity
  13. Torelli
  14. Conic bundles
  15. Eagon--Northcott complexes
  16. ...and 19 more sections

Key Result

Lemma 3.1

Let $X$ be a smooth projective variety with a globally generated line bundle $\mathcal{L}$, and consider $Y:=\mathscr{Z}(\mathbb{P}^m \times X,\mathcal{Q}_{\mathbb{P}^m}\boxtimes \mathcal{L})$. Then $Y \cong \mathop{\mathrm{Bl}}\nolimits_Z X$, where $Z=\mathscr{Z}(X,\mathcal{L}^{\oplus m+1})$ and $Y

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Lemma 3.1
  • proof
  • ...and 38 more