Hartshorne 一章, 問題2.13

Let $Y$ be the image of the 2-uple embedding of $\mathbb{P}^2$ in $\mathbb{P}^5$. This is the Veronese surface. If $Z \subseteq Y$ is a closed curve (a curve is a variety of dimension 1), show that there exists a hypersurface $V \subseteq \mathbb{P}^5$ such that $V \cap Y = Z$.

Proof. The Veronese embedding is defined by $v_2\colon \mathbb{P}^2 \hookrightarrow \mathbb{P}^5$, $(x_0:x_1:x_2) \mapsto (x_0^2,x_1^2,x_2^2,x_0x_1,x_1x_2,x_2x_0)$. If $Z$ is defined by $f(x_0,x_1,x_2) = 0$, then $f^2 \in k[x_0^2,x_1^2,x_2^2,x_0x_1,x_1x_2,x_2x_0]$, and so defines a hypersurface $V \subseteq \mathbb{P}^5$. Thus $Z = v_2(Y) = V \cap Y$.