Exercise 111.65.3 (Cubic hypersurfaces). Let $F \in \mathbf{C}[T_0, \ldots , T_ n]$ be homogeneous of degree $3$. Given $3$ vectors $x, y, z \in \mathbf{C}^{n + 1}$ consider the condition

What is the dimension of the space of all choices of $x, y, z$?

How many equations on the coordinates of $x$, $y$, and $z$ is condition (*)?

What is the expected dimension of the space of all triples $x, y, z$ such that (*) is true?

What is the dimension of the space of all triples such that $x, y, z$ are linearly dependent?

Conclude that on a hypersurface of degree $3$ in $\mathbf{P}^ n$ we expect to find a linear subspace of dimension $2$ provided $n \geq a$ where it is up to you to find $a$.

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