For smooth projective varieties $Y \subset \mathbb{P}^n$ we know that the coordinate ring $S(Y)$ may or may not be integrally closed. If $S(Y)$ is normal, we say $Y$ is projectievly normal. This notion measures normality of the affine cone $C(Y) \subset \mathbb{A}^{n+1}$ over $Y$.

A standard example of non-projectively normal variety is the twisted quartic in $\mathbb{P}^1 \hookrightarrow \mathbb{P}^3$, given by the embedding $$ [u:v]\mapsto [u^4:u^3v:uv^3:v^4] $$ This embedding is obtained via projecting the rational normal curve in $\mathbb{P}^1 \hookrightarrow \mathbb{P}^4$ given by $$ [u:v]\mapsto [u^4:u^3v:u^2v^2:uv^3:v^4] $$ on a suitable hyperplane in $\mathbb{P}^4$.

Call $Y_1$ the twisted quartic in $\mathbb{P}^3$ and $Y_2$ the rational normal curve in $\mathbb{P}^4$. We have that $S(Y_2)$ is the integral closure of $S(Y_1)$. Therefore, $C(Y_2)$ is the normalization of $C(Y_1)$.


My question is the following. How do I visualize the map $C(Y_2) \rightarrow C(Y_1)$?


Using the equations in this question, taking the jacobian matrix and checking for where the rank drops below 2, it looks like the cone $C(Y_1)$ is smooth away from the vertex. Thus, $C(Y_1)$ is already $R_1$. Is then the normalization happening just at level of structure sheaves, to ensure the $S_2$ property?


I kind of came up with some way of visualizing the map in a cohomological way instead of a geometric way.

By restricting the sheaf $\mathcal{O}_{\mathbb{P}^3}(1)$ to $Y_1$ we get $\mathcal{O}_{Y_1}(1)\cong \mathcal{O}_{\mathbb{P}^1}(4)$, however the restricting of global sections $H^0(\mathbb{P}^3,\mathcal{O}_{\mathbb{P}^3}(1))\to H^0(Y_1,\mathcal{O}_{Y_1}(1))$ is not surjective because we miss a section $u^2v^2$ by dimension counting. Now this means that the map $$ \mathrm{Spec}\bigoplus_{m\geq 0} H^0(Y_1,\mathcal{O}_{Y_1}(m))\to \mathrm{Spec}\bigoplus_{m\geq 0} H^0(\mathbb{P}^3,\mathcal{O}_{\mathbb{P}^3}(m)) $$ is not a closed imersion. This map is exactly $C(Y_2)\to C(\mathbb{P}^3)=\mathbb{A}^4$, and the image of this map is $C(Y_1)$.

This is also why I found online that the surjectiveness for the restriction of global sections here $H^0(\mathbb{P}^N,\mathcal{O}_{\mathbb{P}^N}(m))\to H^0(X,\mathcal{O}_{X}(m))$ gives an equivalent definition of projective normality.


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