1
vote
0answers
42 views

Construction of projective space of a graded ring

Let $S=k[x_0,x_1,x_2]$, $k$ an algebraically closed field and $ S_d=k[x_0^d,x_0^{d-1}x_1,\ldots,x_2^{d-1}x_1, x_2^d]$. $\mathbb{P}S_d $ is identified with the projective space $\mathbb{P}^{N_d} $, ...
7
votes
2answers
207 views

Coordinate ring in projective space. What are they?

When $X$ is an algebraic variety of affine $n$-space, then the coordinate ring of $X$ are polynomials restricted to $X$. But when $X$ is a variety of projective $n$ space, what are the elements ...
5
votes
1answer
190 views

rational functions on projective n space

How to prove that the field of rational functions on whole of projective n space is constant functions. By rational function I mean quotients of homogeneous polynomials of same degree ...
8
votes
1answer
196 views

What does projective space classify?

Let $A$ be a ring and let $\mathbb{P}^n = \operatorname{Proj} \mathbb{Z} [x_0, \ldots, x_n]$. Question. What does $\mathbb{P}^n$ classify? In other words, is there some kind of algebraic structure ...
4
votes
3answers
206 views

what is the definition of a line in $\mathbb{P}^n(k)$ + how to compute the hilbert polynomial of two intersecting lines?

(1) I have never studied any projective/affine geometry or algebraic curves. I'd like to see a clear definition of a line in the projective space $\mathbb{P}^n(k)$, since I need it for my algebraic ...