Bigness of the tangent bundle of a Fano threefold with Picard number two
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913810811453440 |
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| author | Kim, Hosung Kim, Jeong-Seop Lee, Yongnam |
| author_facet | Kim, Hosung Kim, Jeong-Seop Lee, Yongnam |
| contents | In this paper, we study the positivity property of the tangent bundle $T_X$ of a Fano threefold $X$ with Picard number 2. We determine the bigness of the tangent bundle of the whole 36 deformation types. Our result shows that $T_X$ is big if and only if $(-K_X)^3\ge 34$. As a corollary, we prove that the tangent bundle is not big when $X$ has a standard conic bundle structure with non-empty discriminant. Our main methods are to produce irreducible effective divisors on $\mathbb{P}(T_X)$ constructed from the total dual VMRT associated to a family of rational curves. Additionally, we present some criteria to determine the bigness of $T_X$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2201_06351 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bigness of the tangent bundle of a Fano threefold with Picard number two Kim, Hosung Kim, Jeong-Seop Lee, Yongnam Algebraic Geometry 14J45, 14J30, 14E30 In this paper, we study the positivity property of the tangent bundle $T_X$ of a Fano threefold $X$ with Picard number 2. We determine the bigness of the tangent bundle of the whole 36 deformation types. Our result shows that $T_X$ is big if and only if $(-K_X)^3\ge 34$. As a corollary, we prove that the tangent bundle is not big when $X$ has a standard conic bundle structure with non-empty discriminant. Our main methods are to produce irreducible effective divisors on $\mathbb{P}(T_X)$ constructed from the total dual VMRT associated to a family of rational curves. Additionally, we present some criteria to determine the bigness of $T_X$. |
| title | Bigness of the tangent bundle of a Fano threefold with Picard number two |
| topic | Algebraic Geometry 14J45, 14J30, 14E30 |
| url | https://arxiv.org/abs/2201.06351 |