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) $k$-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.

18 pages Replace Theorem 1.3. Insered hyperref support. Typo in statement 3.3. Removed very general form proof of theorem 1.4. Reference for the notation of SG(r,s). Revised Lemma 3.12 (now 3.13), results unchanged. Insered Remark 3.8 and Remark 3.13

References in corpus (3)

• A note on Fano manifolds whose second Chern character is positive
• On Küchle manifolds with Picard number greater than 1
• On Varieties of Lines on Linear Sections of Grassmannians

Cited by in corpus (1)

• A Recursive Formula for Osculating Curves