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Betti numbers and pseudoeffective cones in 2-Fano varieties

Giosuè Emanuele Muratore

Abstract

The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher dimensional analogous properties of Fano varieties. We propose a definition of (weak) $k$-Fano variety and conjecture the polyhedrality of the cone of pseudoeffective $k$-cycles for those varieties in analogy with the case $k=1$. Then, we calculate some Betti numbers of a large class of $k$-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index $\ge n-2$, and also we complete the classification of weak 2-Fano varieties of Araujo and Castravet.

Betti numbers and pseudoeffective cones in 2-Fano varieties

Abstract

The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher dimensional analogous properties of Fano varieties. We propose a definition of (weak) -Fano variety and conjecture the polyhedrality of the cone of pseudoeffective -cycles for those varieties in analogy with the case . Then, we calculate some Betti numbers of a large class of -Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index , and also we complete the classification of weak 2-Fano varieties of Araujo and Castravet.

Paper Structure

This paper contains 8 sections, 15 theorems, 42 equations, 3 tables.

Table of Contents

  1. Introduction
  2. General facts about cycles
  3. Cycles on Fano Varieties
  4. Weighted projective spaces
  5. Rational homogeneous varieties
  6. Complete intersection of rational homogeneous varieties
  7. Other examples
  8. Fano manifolds of dimension $n$ and index $i_{X}>n-3$

Key Result

Theorem 1.3

Let $X$ be a $n$-dimensional 2-Fano variety with $i_{X}\ge n-2$. Then $\overline{\mathrm{Eff}}_{2}(X)$ is polyhedral. Also, $\overline{\mathrm{Eff}}_{3}(X)$ is polyhedral with the possible exception of the complete intersection of type $(2,2)$ in $\mathbb{\mathbb{P}}^{8}$. In particular, Conjecture

Theorems & Definitions (41)

  • Definition 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 31 more