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Exercise 110.65.3 (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

\[ (*)\quad F(\lambda x + \mu y + \nu z) = 0 \text{ in } \mathbf{C}[\lambda , \mu , \nu ] \]

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

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

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

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

  5. 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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