We know that if $X,Y$ are affine varieties with $A(X)\simeq A(Y)$, then they are isomorphic (where $A(X)$ is the coordinate ring of $X$).
I'm trying to verify if there is an analogous result for projective varieties, namely:
If $X,Y$ are projective varieties with $S(X)\simeq S(Y)$, can we say $X\simeq Y$?
(where $S(X)$ is the homogeneous coordinate ring)
My first idea was to consider the affine cones $C(X), C(Y)$, whose coordinate rings are $S(X), S(Y)$. Then $S(X)\simeq S(Y)$ is equivalent to $C(X)\simeq C(Y)$.
Now I'm stuck because I don't know how to prove (or disprove) that $C(X)\simeq C(Y)\Leftrightarrow X\simeq Y$.