Projectivization of a vector space: projective geometry definition vs algebraic geometry definition

Let $V$ be a vector space. In a course of projective geometry I was told that $$ \mathbb{P}V=\{l\subseteq V:l \text{ is a line in }V\}. $$ Studying algebraic geometry I have seen that the projectivization of the vector space $V$ is defined as $$ \mathbb{P}V=\mathrm{Proj}\left(\bigoplus_{k=0}^{\infty}\mathrm{Sym}^{k}(V^{\vee})\right). $$ The problem is that I don't really understand how this spaces are related to each other. To be more precise, given a line $l\subseteq V$, which is the homogeneous prime ideal of $\bigoplus_{k=0}^{\infty}\mathrm{Sym}^{k}(V^{\vee})$ corresponding to $l$, and given a point $p\in \mathrm{Proj}\left(\bigoplus_{k=0}^{\infty}\mathrm{Sym}^{k}(V^{\vee})\right)$ (do we need it to be closed?), which is the line of $V$ corresponding to $p$?

Solution 1:

The elements of $V^\vee$ can be identified with the homogeneous linear polynomials on the coordinates of $V$. Indeed, a typical element of $V^\vee$ is a mapping $V \to k$ of the form $$ (x_0, \dots, x_n) \mapsto a_0 x_0 + \dots + a_n x_n,$$ for certain $a_0, \dots, a_n \in k$.

Similarly, we can argue that the elements of ${\rm Sym}^k (V^\vee)$ can be identified with the homogeneous polynomials of degree $k$ on the coordinates of $V$.

So the ring $\oplus_{k=0}^\infty {\rm Sym}^k (V^\vee)$ is nothing other than the polynomial ring, $$\oplus_{k=0}^\infty {\rm Sym}^k (V^\vee) = k[x_0, \dots, x_n].$$ This ring is graded by the degree of the polynomials, and the ring multiplication is the natural mulplication of polynomials.

The points in ${\rm Proj} \left( \oplus_{k=0}^\infty {\rm Sym}^k (V^\vee) \right)$ correspond to the homogeneous prime ideals of $k[x_0, \dots, x_n]$ that do not contain all of the elements of the "irrelevant ideal" $(x_0, \dots, x_n)$. A line $l \subseteq V$ is a closed point in the scheme ${\rm Proj} \left( \oplus_{k=0}^\infty {\rm Sym}^k (V^\vee) \right)$: it corresponds to a homogeneous prime ideal that is maximal within the class of homogenous prime ideals descibed.

For example, the line $$ l = \{ (t, c_1t, c_2t, \dots, c_n t) : t \in k \} \subset V,$$ corresponds to the homogeneous prime ideal, $$ \mathfrak p = (x_1 - c_1 x_0, \ x_2 - c_2 x_0, \ \dots, \ x_n - c_n x_0).$$ This $\mathfrak p$ is simply the ideal generated by all homogeneous polynomials vanishing on the line $l$.