As well as the Hartshorne-Frankel conjecture on the ampleness of tangent bundle, it has been asked to characterize a smooth projective variety X whose tangent bundle TX attains certain positivity, e.g., nefness, k-ampleness, or bigness. But for the ampleness, the complete answers are not known even within the class of smooth Fano varieties, only partial answers are known in the case of lower dimension or lower Picard number, some of which rely on classification theorems. On the bigness of TX, the characterization has been done recently in the case of dimension 2 (Höring-Liu-Shao) and dimension 3 with Picard number 1 (Höring-Liu) using a special divisor on P(TX), called the total dual VMRT. In this talk, I will briefly review the classification of Fano threefolds and the theory of total dual VMRT. Then I will introduce some criteria to determine the bigness of TX, and announce a result on the bigness of TX in the case of dimension 3 with higher Picard number. This is joint work with Hosung Kim and Yongnam Lee.
Jeong-Seop Kim, Stability of Symmetric Powers of Vector Bundles on a Curve
For a stable vector bundle E on a smooth projective curve, it is known that the symmetric powers Sk E are semi-stable and are stable for all k > 0 in sufficiently general. Moreover, if E has rank 2, then Sk E is destabilized by a line subbundle if and only if the ruled surface PC(E) admits a k-section of zero self-intersection. In this talk, concentrating on the case of rank 2, we will find answers to the questions of which E has strictly semi-stable Sk E, and how many such E there are. Also, we will introduce relations between such E and the orthogonal bundles when k = 2, and Nori’s finite bundles when k ≥ 3.