The minimal projective bundle dimension and toric \(2\)-Fano manifolds

Motivated by the problem of classifying toric \(2\)-Fano manifolds, we introduce a new invariant for smooth projective toric varieties, the minimal projective bundle dimension. This invariant \(m(X)\in\{1, \dots,\dim(X)\}\) captures the minimal degree of a dominating family of rational curves on \(X...

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Bibliographic Details
Published in:arXiv.org 2023-03
Main Authors: Araujo, Carolina, Beheshti, Roya, Ana-Maria Castravet, Jabbusch, Kelly, Makarova, Svetlana, Mazzon, Enrica, Viswanathan, Nivedita, Reynolds, Will
Format: Article
Language:English
Subjects:
Classification
Invariants
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Summary:Motivated by the problem of classifying toric \(2\)-Fano manifolds, we introduce a new invariant for smooth projective toric varieties, the minimal projective bundle dimension. This invariant \(m(X)\in\{1, \dots,\dim(X)\}\) captures the minimal degree of a dominating family of rational curves on \(X\) or, equivalently, the minimal length of a centrally symmetric primitive relation for the fan of \(X\). We classify smooth projective toric varieties with \(m(X)\geq \dim(X)-2\), and show that projective spaces are the only \(2\)-Fano manifolds among smooth projective toric varieties with \(m(X)\in\{1, \dim(X)-2,\dim(X)-1,\dim(X)\}\).
ISSN:2331-8422