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

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 $T_X$ .