Let $X$, $Y$ be projective varieties with isomorphic homogeneous coordinate rings (as graded rings). What can we say about $X$ and $Y$? Are they necessarily isomorphic?
Projective varieties with isomorphic coordinate rings
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algebraic-geometry
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0I doubt if not. In fact projective variety can be identified with its coordinate ring, if it is a zeroes of some polynomial equations. – 2017-02-21
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0@Rick_Student The converse is false (isomorphic projective varieties may have non-isomorphic coordinate rings). – 2017-02-21