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

\[ \varphi : \mathbf{P}^2 \longrightarrow \mathbf{P}^5 \]

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

\[ \varphi = \varphi _{\mathcal{O}_{\mathbf{P}^2}(2)} : \mathbf{P}^2 \longrightarrow \mathbf{P}(\Gamma (\mathbf{P}^2, \mathcal{O}_{\mathbf{P}^2}(2))) \]

In terms of homogeneous coordinates it is given by

\[ [a_0 : a_1 : a_2] \longmapsto [a_0^2 : a_0a_1 : a_0a_2 : a_1^2 : a_1a_2 : a_2^2] \]

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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