Exercise 109.63.6. Let $k$ be an algebraically closed field. Consider the $2$-uple embedding

In terms of the material/notation in the lectures this is the morphism

In terms of homogeneous coordinates it is given by

It is a closed immersion (please just use this). Let $I \subset k[T_0, \ldots , T_5]$ be the homogeneous ideal of $\varphi (\mathbf{P}^2)$, i.e., the elements of the homogeneous part $I_ d$ are the homogeneous polynomials $F(T_0, \ldots , T_5)$ of degree $d$ which restrict to zero on the closed subscheme $\varphi (\mathbf{P}^2)$. Compute $\dim _ k I_ d$ as a function of $d$.

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