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Mathematics > Algebraic Geometry

arXiv:2202.01852 (math)
[Submitted on 3 Feb 2022 (v1), last revised 30 Sep 2022 (this version, v4)]

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

Authors:Roya Beheshti, Ben Wormleighton
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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.
Comments: 11 pages; v.2. thanks to Cinzia Casagrande for noticing an error that has been corrected with no changes to the results; v.3. minor typos fixed; v.4. final version; comments welcome!
Subjects: Algebraic Geometry (math.AG)
MSC classes: 14J45 (Primary) 14M25, 14H10 (Secondary)
Cite as: arXiv:2202.01852 [math.AG]
  (or arXiv:2202.01852v4 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.2202.01852

Submission history

From: Ben Wormleighton [view email]
[v1] Thu, 3 Feb 2022 21:24:08 UTC (22 KB)
[v2] Fri, 11 Feb 2022 17:55:01 UTC (15 KB)
[v3] Mon, 14 Feb 2022 16:07:28 UTC (15 KB)
[v4] Fri, 30 Sep 2022 14:40:23 UTC (18 KB)
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