Betti numbers and pseudoeffective cones in 2-Fano varieties

TL;DR

This paper explores higher-dimensional properties of 2-Fano varieties, proposing a new definition for (weak) $k$-Fano varieties, conjecturing polyhedrality of certain cones, and verifying this in specific cases through Betti number calculations.

Contribution

It introduces a new definition of (weak) $k$-Fano varieties and proves the polyhedrality conjecture for specific classes by calculating Betti numbers.

Findings

Conjecture holds for 2-Fano varieties of index ≥ n-2.

Complete classification of weak 2-Fano varieties of Araujo and Castravet.

Betti numbers computed for a large class of $k$-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) $k$-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.