Fano manifolds with big tangent bundle that are covered by “lines”

By definition the anticanonical bundle of a Fano manifold $X$ is ample, but in general the tangent bundle $T_X$ does not have any nice properties. In fact, recent examples obtained in joint work with Jie Liu and Feng Shao indicate that the tangent bundle is almost never big. Unfortunately so far we do not have many tools to study this property in an abstract setting. In this talk I will speak about work in progress with Jie Liu where we tackle the first case, i.e. when $T_X$ is big and $X$ admits a family of minimal rational curves of degree two.