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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2205.12743 |
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| _version_ | 1866916926109777920 |
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| author | Krylov, Igor Okada, Takuzo Paemurru, Erik Park, Jihun |
| author_facet | Krylov, Igor Okada, Takuzo Paemurru, Erik Park, Jihun |
| contents | The $4 n^2$-inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type $cA_1$, and obtain a $2 n^2$-inequality for $cA_1$ points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree $1$ over $\mathbb{P}^1$ satisfying the $K^2$-condition, all of which have at most terminal $cA_1$ singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a $cA_1$ point which is not an ordinary double point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_12743 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $2 n^2$-inequality for $cA_1$ points and applications to birational rigidity Krylov, Igor Okada, Takuzo Paemurru, Erik Park, Jihun Algebraic Geometry The $4 n^2$-inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type $cA_1$, and obtain a $2 n^2$-inequality for $cA_1$ points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree $1$ over $\mathbb{P}^1$ satisfying the $K^2$-condition, all of which have at most terminal $cA_1$ singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a $cA_1$ point which is not an ordinary double point. |
| title | $2 n^2$-inequality for $cA_1$ points and applications to birational rigidity |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2205.12743 |