Bigness of the tangent bundle of a Fano threefold with Picard number two

TL;DR

This work determines when the tangent bundle of a Fano threefold with Picard number two is big. It develops and applies the total dual VMRT framework to produce explicit divisors on $\mathbb{P}(T_X)$ and to derive bigness criteria that are deformation-invariant. The authors prove that $T_X$ is big precisely when $(-K_X)^3 \ge 34$, and give detailed analyses showing non-bigness in all standard conic bundle cases with non-empty discriminant, along with comprehensive results for certain blow-ups. The results unify and extend known cases, and support a broader conjecture connecting $T_X$-positivity to the anti-canonical degree, while providing practical criteria for checking bigness in concrete geometries.

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 $T_X$.

Figures (2)

• Figure 1: The total dual VMRT $\breve\mathcal{C}|_x$ at $x$ when the VMRT $\mathcal{C}_x$ is finite.
• Figure 2: The restriction $\widetilde{f}^*\breve\mathcal{D}|_x\subseteq\mathbb{P}(f^*T_Q|_x)$ of $\widetilde{f}^*\breve\mathcal{D}\subseteq\mathbb{P}(f^*T_Q)$ over $x\in D$.

Theorems & Definitions (35)

• Corollary
• Proposition 2.1: cf. OSW
• Lemma 2.2
• proof
• Proposition 2.3: HLS
• Remark 2.4
• Remark 2.5
• Proposition 2.6
• proof
• Remark 2.7