Bounds on the Picard rank of toric Fano varieties with minimal curve constraints

2022/02/03 by Roya Beheshti, Beheshti, Roya, Ben Wormleighton +1
Mathematics · Social Sciences · #14H10 (Secondary) #14J45 (Primary) 14M25 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.2202.01852

openalex publication_date 2022/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Picard rank of smooth toric Fano varieties possessing families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen-Fu-Hwang and prove a version of their statement that recovers the original conjecture in sufficiently high dimension. We also prove new cases of the original conjecture for high degrees in all dimensions. Our main tools come from toric Mori theory and the combinatorics of Fano polytopes.

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