The minimal projective bundle dimension and toric $2$-Fano manifolds
Autor: | Araujo, Carolina, Beheshti, Roya, Castravet, Ana-Maria, Jabbusch, Kelly, Makarova, Svetlana, Mazzon, Enrica, Viswanathan, Nivedita, Reynolds, Will |
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Rok vydání: | 2023 |
Předmět: | |
Druh dokumentu: | Working Paper |
Popis: | 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 $m(X)\in\{1, \dim(X)-2,\dim(X)-1,\dim(X)\}$. Comment: 32 pages, 1 figure. Comments welcome. V2: expanded Remark 2.11, added a citation |
Databáze: | arXiv |
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