Forgive me to ask an elementary question, because I really need the answer to this today (I already asked this in Stackexchange.)

Let $S$ be the rational quartic scroll in $\mathbb{P}^5$ ($S$ is the image of the embedding $\mathbb{P}^1\times\mathbb{P}^1\rightarrow\mathbb{P}^5$ via $|\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(1,2)|$).

1/ Why does every point on $\mathbb{P}^5$ lie on a unique secant line or tangent line to $S$?

 

2/ Suppose a cubic fourfold $X\subset \mathbb{P}^5$ contains $S$. Why does $X$ necessarily contain two skewed 2-planes?

It's fine if you just leave me a reference. Thank you very much.

Forgive me to ask an elementary question, because I really need the answer to this today (I already asked this in Stackexchange.)

Let $S$ be the rational quartic scroll in $\mathbb{P}^5$ ($S$ is the image of the embedding $\mathbb{P}^1\times\mathbb{P}^1\rightarrow\mathbb{P}^5$ via $|\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(1,2)|$).

1/ Why does every point on $\mathbb{P}^5$ lie on a unique secant line or tangent line to $S$?

2/ Suppose a cubic fourfold $X\subset \mathbb{P}^5$ contains $S$. Why does $X$ necessarily contain two skewed 2-planes?

It's fine if you just leave me a reference. Thank you very much.