Let $X$ be a projective variety, and $Y$ its image under a veronese map. What is the relation between the homogeneous coordinate rings?

Question

So this is a question from Harris' Algebraic Geometry book, exercise 2.10. Now, I've thought about this and made some progress. The most obvious is that they're not equal. The simple case to analyse would be something like $\nu_2: \mathbb{P}^1 \to \mathbb{P}^2$ given by $\nu_2([x_0 : x_1]) = [x_0^2 : x_0 x_1 : x_1^2]$, where $\nu_2(\mathbb{P}^2) = V(x_0 x_2 - x_1^2)$. So we have the coordinate rings are

$$S(X) = \Bbbk[x_0, x_1] \quad \text{and} \quad S(\nu_2(\mathbb{P}^1)) = \Bbbk[x_0, x_1, x_2] / (x_0 x_2 - x_1^2),$$

which are definitely not isomorphic (the 2nd is not even a UFD). However, I cannot see any kind of "relationship" between these rings, as Harris seems to suggest there is. Even in the more general case, let's say our variety $X$ can be written as the zero locus of homogeneous polynomials $f_1, \dots, f_m$ of degree $d$. Then under the Veronese embedding $\nu_d$, we have that $\nu_d(X)$ is the zero locus of the polynomials defining the Veronese surface (i.e. $x_I x_J - x_K x_L$ where $I, J, K, L$ are multi-indices such that $I + J = K + L$ component wise) plus a bunch of linear forms as well. So we have

$$S(X) = \Bbbk[x_0, \dots, x_n] / (f_1, \dots, f_m) \quad \text{and} \quad S(\nu_d(X)) = \frac{\Bbbk[\dots, x_I, \dots]}{ (x_I x_J - x_K x_L \mid I + J = K + L) + (L_1, \dots, L_m)}$$

where $L_i$ is a linear form on $\mathbb{P}^N$ such that $L_i \circ \nu_d = f_i$. Now, these certainly appear to be not isomorphic in general, although I don't have the best reasoning as to why.

However, I still can't see any kind of "relationship" between these rings, even in this general case. There is a post here that says that these rings obviously must have the same Krull dimension, but honestly I don't even see that. Am I going about this the wrong way?

Answer 1

It's easier to see this relationship when you write the image of the Veronese subring differently. Recall that the Veronese map $\nu_d: \Bbb P^n \to \Bbb P^N$ sends sends $[x_0:\cdots:x_n]\to [\cdots:x^I:\cdots]$ as $I$ ranges over multi-indexes of non-negative integers summing to $d$ so that the induced map on homogeneous coordinate rings is $\varphi_d: k[w_0,\cdots,w_N]\to k[x_0,\cdots,x_n]$ by mapping $w_i \mapsto x^I$. By the first isomorphism theorem, this means $k[w_0,\cdots,w_N]/I(\nu_d(\Bbb P^n))$ is isomorphic to the image of $\varphi_d$, which is exactly $k[x_0,\cdots,x_n]^{(d)}$, the subring where the degree of every monomial is divisible by $d$. This extends directly: for any variety $X\subset\Bbb P^n$ with homogeneous coordinate ring $S$, the image of $X$ under the $d$-uple Veronese has homogeneous coordinate ring $S^{(d)}$.

To see that this preserves Krull dimensions, we may note that a ring and its Veronese subring have the same Proj (ref).