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On Casagrande-Druel Fano varieties with Lefschetz defect 2

Pier Roberto Pastorino

TL;DR

This work analyzes the Lefschetz defect δ_X of smooth Fano varieties, focusing on the δ=2 case via two explicit constructions, Construction A (Casagrande–Druël approach) and its generalization Construction B. It proves that Construction A accounts for the vast majority of δ=2 smooth Fano 3-folds and constructs 147 distinct δ=2 4-fold families with ρ≥4, while Construction B covers all but one of the 3-fold families. The authors connect these constructions with Maruyama’s elementary transformations, study recursion to generate higher-dimensional examples, and provide an exhaustive classification of δ=2 Fano 3- and 4-folds arising from these constructions, including detailed numerical invariants and numerous non-toric examples. The results highlight structural patterns: most δ=2 3-folds come from A and most 4-folds from A, with B offering a broader, though more delicate, unifying perspective, and they lay groundwork for future exploration of δ=2 Fano 4-folds beyond Construction A. Overall, the paper delivers a near-complete, construction-based taxonomy of Casagrande–Druël Fano varieties with Lefschetz defect 2 and expands the known landscape of δ=2 Fano 4-folds with explicit invariants.

Abstract

The larger the Lefschetz defect delta(X) of a smooth complex Fano variety X, the more information we can deduce about the geometry of X. The structure of varieties with delta(X) greater than 2 is known. In this paper, we study the case delta(X)=2. In particular, we focus on Fano varieties with delta(X)=2 arising from the so called Casagrande-Druel construction, which we refer to as Construction A. We show that among the 19 families of Fano 3-folds with delta(X)=2 classified by Mori and Mukai, 15 arise from such construction. Moreover, we construct all Fano 4-folds with Picard number greater than 3 and delta(X)=2 admitting such a structure, obtaining 147 distinct families in total. This completes the classification of all Casagrande-Druel Fano 4-folds with delta(X)=2. To broaden the scope, we also study a generalized version of Construction A, which we call Construction B, and we show that 18 out of the 19 families of Fano 3-folds with delta(X)=2 arise from it.

On Casagrande-Druel Fano varieties with Lefschetz defect 2

TL;DR

This work analyzes the Lefschetz defect δ_X of smooth Fano varieties, focusing on the δ=2 case via two explicit constructions, Construction A (Casagrande–Druël approach) and its generalization Construction B. It proves that Construction A accounts for the vast majority of δ=2 smooth Fano 3-folds and constructs 147 distinct δ=2 4-fold families with ρ≥4, while Construction B covers all but one of the 3-fold families. The authors connect these constructions with Maruyama’s elementary transformations, study recursion to generate higher-dimensional examples, and provide an exhaustive classification of δ=2 Fano 3- and 4-folds arising from these constructions, including detailed numerical invariants and numerous non-toric examples. The results highlight structural patterns: most δ=2 3-folds come from A and most 4-folds from A, with B offering a broader, though more delicate, unifying perspective, and they lay groundwork for future exploration of δ=2 Fano 4-folds beyond Construction A. Overall, the paper delivers a near-complete, construction-based taxonomy of Casagrande–Druël Fano varieties with Lefschetz defect 2 and expands the known landscape of δ=2 Fano 4-folds with explicit invariants.

Abstract

The larger the Lefschetz defect delta(X) of a smooth complex Fano variety X, the more information we can deduce about the geometry of X. The structure of varieties with delta(X) greater than 2 is known. In this paper, we study the case delta(X)=2. In particular, we focus on Fano varieties with delta(X)=2 arising from the so called Casagrande-Druel construction, which we refer to as Construction A. We show that among the 19 families of Fano 3-folds with delta(X)=2 classified by Mori and Mukai, 15 arise from such construction. Moreover, we construct all Fano 4-folds with Picard number greater than 3 and delta(X)=2 admitting such a structure, obtaining 147 distinct families in total. This completes the classification of all Casagrande-Druel Fano 4-folds with delta(X)=2. To broaden the scope, we also study a generalized version of Construction A, which we call Construction B, and we show that 18 out of the 19 families of Fano 3-folds with delta(X)=2 arise from it.
Paper Structure (51 sections, 23 theorems, 58 equations, 4 figures, 4 tables)

This paper contains 51 sections, 23 theorems, 58 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Notation and conventions.
  4. Acknowledgements.
  5. Construction A
  6. Setting
  7. Conditions for being Fano
  8. Construction B
  9. Setting
  10. Conditions for being Fano
  11. Maruyama's elementary transformation
  12. Setting
  13. Relation to Construction A
  14. Relation to Construction B
  15. Recursion
  16. ...and 36 more sections

Key Result

Theorem 1.1

Let $X$ be a smooth Fano variety with Lefschetz defect $\delta_X\ge 4$. Then $X\simeq S\times T$, where $S$ is a del Pezzo surface with $\rho_S=\delta_X+1$.

Figures (4)

  • Figure 1: Alternative factorization of Construction A.
  • Figure 2: Maruyama's elementary transformation diagram.
  • Figure 3: Maruyama's elementary transformation in the setting of Construction B.
  • Figure 4: Iteration of two constructions of type A.

Theorems & Definitions (51)

  • Theorem 1.1: Cas12, Theorem 3.3
  • Theorem 1.2: Cas14, Theorem 5.22
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 41 more