Regularity of affine cones
Let $k$ be an algebraically closed field and $mathbb{P}^n_k$ be the projective space. Let $Y$ be a closed subvariety in $mathbb{P}^n_k$ of dimension $geq 1$ and $R$ be its homogeneous coordinate ring. Assume that $Y=mathtt{Proj}(R)$ is normal. How to prove that the affine cone $mathtt{Spec}(R)$ is regular at codimension $1$ too?
The question comes from the Exercise 8.4(b) of Hartshorne's book Algebraic Geometry. There we assume $Y$ is complete intersection and normal, and we want to show $Y$ is projective normal which means that $R$ is normal. Since $mathtt{Spec}(R)subset mathbb{A}_k^{n+1}$ is also complete intersection, we know $R$ is Cohen-Macaulay. Then $R$ is normal iff it is regular at codimension $1$. But lots of codimension $1$ points of $mathtt{Spec}(R)$ seems to have litte relationship with $Y$ and I don't know how to show they are regular.
algebraic-geometry
asked Jan 5 at 23:17
wuzxwuzx
1113
add a comment |
Let $k$ be an algebraically closed field and $mathbb{P}^n_k$ be the projective space. Let $Y$ be a closed subvariety in $mathbb{P}^n_k$ of dimension $geq 1$ and $R$ be its homogeneous coordinate ring. Assume that $Y=mathtt{Proj}(R)$ is normal. How to prove that the affine cone $mathtt{Spec}(R)$ is regular at codimension $1$ too?
The question comes from the Exercise 8.4(b) of Hartshorne's book Algebraic Geometry. There we assume $Y$ is complete intersection and normal, and we want to show $Y$ is projective normal which means that $R$ is normal. Since $mathtt{Spec}(R)subset mathbb{A}_k^{n+1}$ is also complete intersection, we know $R$ is Cohen-Macaulay. Then $R$ is normal iff it is regular at codimension $1$. But lots of codimension $1$ points of $mathtt{Spec}(R)$ seems to have litte relationship with $Y$ and I don't know how to show they are regular.
algebraic-geometry
asked Jan 5 at 23:17
wuzxwuzx
1113
add a comment |
Let $k$ be an algebraically closed field and $mathbb{P}^n_k$ be the projective space. Let $Y$ be a closed subvariety in $mathbb{P}^n_k$ of dimension $geq 1$ and $R$ be its homogeneous coordinate ring. Assume that $Y=mathtt{Proj}(R)$ is normal. How to prove that the affine cone $mathtt{Spec}(R)$ is regular at codimension $1$ too?
The question comes from the Exercise 8.4(b) of Hartshorne's book Algebraic Geometry. There we assume $Y$ is complete intersection and normal, and we want to show $Y$ is projective normal which means that $R$ is normal. Since $mathtt{Spec}(R)subset mathbb{A}_k^{n+1}$ is also complete intersection, we know $R$ is Cohen-Macaulay. Then $R$ is normal iff it is regular at codimension $1$. But lots of codimension $1$ points of $mathtt{Spec}(R)$ seems to have litte relationship with $Y$ and I don't know how to show they are regular.
algebraic-geometry
asked Jan 5 at 23:17
wuzxwuzx
1113
Let $k$ be an algebraically closed field and $mathbb{P}^n_k$ be the projective space. Let $Y$ be a closed subvariety in $mathbb{P}^n_k$ of dimension $geq 1$ and $R$ be its homogeneous coordinate ring. Assume that $Y=mathtt{Proj}(R)$ is normal. How to prove that the affine cone $mathtt{Spec}(R)$ is regular at codimension $1$ too?
The question comes from the Exercise 8.4(b) of Hartshorne's book Algebraic Geometry. There we assume $Y$ is complete intersection and normal, and we want to show $Y$ is projective normal which means that $R$ is normal. Since $mathtt{Spec}(R)subset mathbb{A}_k^{n+1}$ is also complete intersection, we know $R$ is Cohen-Macaulay. Then $R$ is normal iff it is regular at codimension $1$. But lots of codimension $1$ points of $mathtt{Spec}(R)$ seems to have litte relationship with $Y$ and I don't know how to show they are regular.
algebraic-geometry
algebraic-geometry
asked Jan 5 at 23:17
wuzxwuzx
1113
asked Jan 5 at 23:17
wuzxwuzx
1113
asked Jan 5 at 23:17
wuzxwuzx
1113
asked Jan 5 at 23:17
wuzxwuzx
1113
asked Jan 5 at 23:17
wuzxwuzx
1113
1113
add a comment |
add a comment |
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