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 $p$?


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 $l$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.