TITLE: Fano variety of lines on the Segre and the Grassmannian
QUESTION [8 upvotes]: Does every $\mathbb{P}^{19}\subset \mathbb{P}(\mathbb{C}^5\otimes\mathbb{C}^5)$
intersect the Segre variety of rank one matrices in at least a $\mathbb{P}^1$?
A naive dimension count suggests this is possible. The intersection is a
$3$-fold and I would be happy for any qualitative information about it.

REPLY [5 votes]: This is a comment, but I cannot comment as my reputation is too low :/
I'm very sorry: why is this the top class?
I.e. why we do not get additionally $pr^*_{Grass}[c_2(S^\vee)^2c_1(S^\vee)^2]$ (giving $30=5*6$) and $c_2(S^\vee)^3$ (giving $10$)? Did I miscompute that the top class is $(c_2(S^\vee)+c_1(T^\vee)c_1(S^\vee)+c_1(T^\vee)^2)^5$ or I have an error elswhere?