The Stacks project

Exercise 111.63.6. 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 $d$.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0G18. Beware of the difference between the letter 'O' and the digit '0'.