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BIGNESS OF THE TANGENT BUNDLE OF A FANO THREEFOLD WITH PICARD NUMBER TWO

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  27 December 2024

HOSUNG KIM Open the ORCID record for HOSUNG KIM [Opens in a new window] ,
JEONG-SEOP KIM Open the ORCID record for JEONG-SEOP KIM [Opens in a new window]  and
YONGNAM LEE Open the ORCID record for YONGNAM LEE [Opens in a new window]
HOSUNG KIM
Affiliation:
Department of Mathematics Changwon National University 20 Changwondaehak-ro, Uichang-gu Changwon-si, Gyeongsangnam-do, 51140 Korea [email protected]
JEONG-SEOP KIM*
Affiliation:
School of Mathematics Korea Institute for Advanced Study (KIAS) 85 Hoegiro, Dongdaemun-gu Seoul, 02455 Korea
YONGNAM LEE
Affiliation:
Center for Complex Geometry Institute for Basic Science (IBS) 5 Expo-ro, Yuseong-gu Daejeon, 34126, and Department of Mathematical Sciences KAIST 291 Daehak-ro, Yuseong-gu Daejeon, 34141 Korea [email protected]
*
[email protected]
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Abstract

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 $T_X$.

Keywords

Fano threefoldconic bundletotal dual VMRTtangent bundle

MSC classification

Primary: 14J45: Fano varieties 14J30: $3$-folds 14E30: Minimal model program (Mori theory, extremal rays)
Type
Article
Information
Nagoya Mathematical Journal , Volume 257 , March 2025 , pp. 104 - 133
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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