On the classification of toric $2$-Fano manifolds: generic $\mathbb{P}^2$-bundles
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| Format: | Preprint |
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2026
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| _version_ | 1866914491413823488 |
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| author | Araujo, Carolina Beheshti, Roya Castravet, Ana-Maria Jabbusch, Kelly Makarova, Svetlana Mazzon, Enrica Viswanathan, Nivedita |
| author_facet | Araujo, Carolina Beheshti, Roya Castravet, Ana-Maria Jabbusch, Kelly Makarova, Svetlana Mazzon, Enrica Viswanathan, Nivedita |
| contents | In this paper, we advance the classification of toric 2-Fano manifolds by continuing the investigation of the minimal projective bundle dimension $m(X) \in \{1,\dots,\dim(X)\}$ introduced in our previous work. This invariant captures the minimal degree of a dominating family of rational curves on $X$ and admits a natural combinatorial interpretation in terms of centered primitive collections. We develop an approach that relates, via toric blowdowns and flips, a toric Fano manifold $X$ to a toric manifold $Y$ that admits a $\mathbb{P}^{m(X)}$-bundle structure on a big open subset. We then compare positivity of the second Chern characters of $X$ and $Y$, and show that the only toric 2-Fano manifold $X$ with $m(X) = 2$ is $X\cong \mathbb{P}^2$. In the example-driven Appendix B, we demonstrate that extending this strategy to the case $m(X)>2$ requires either a substantially more detailed analysis of the combinatorics of primitive collections or a fundamentally new approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_18054 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the classification of toric $2$-Fano manifolds: generic $\mathbb{P}^2$-bundles Araujo, Carolina Beheshti, Roya Castravet, Ana-Maria Jabbusch, Kelly Makarova, Svetlana Mazzon, Enrica Viswanathan, Nivedita Algebraic Geometry 14J45, 14M25 (Primary) 14H10, 14E05 (Secondary) In this paper, we advance the classification of toric 2-Fano manifolds by continuing the investigation of the minimal projective bundle dimension $m(X) \in \{1,\dots,\dim(X)\}$ introduced in our previous work. This invariant captures the minimal degree of a dominating family of rational curves on $X$ and admits a natural combinatorial interpretation in terms of centered primitive collections. We develop an approach that relates, via toric blowdowns and flips, a toric Fano manifold $X$ to a toric manifold $Y$ that admits a $\mathbb{P}^{m(X)}$-bundle structure on a big open subset. We then compare positivity of the second Chern characters of $X$ and $Y$, and show that the only toric 2-Fano manifold $X$ with $m(X) = 2$ is $X\cong \mathbb{P}^2$. In the example-driven Appendix B, we demonstrate that extending this strategy to the case $m(X)>2$ requires either a substantially more detailed analysis of the combinatorics of primitive collections or a fundamentally new approach. |
| title | On the classification of toric $2$-Fano manifolds: generic $\mathbb{P}^2$-bundles |
| topic | Algebraic Geometry 14J45, 14M25 (Primary) 14H10, 14E05 (Secondary) |
| url | https://arxiv.org/abs/2604.18054 |