Is canonical model always with canonical singularity

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Let X be a smooth variety, take the proj of canonical ring of X and denote it by Y. Is Y always a canonical variety? I know it's true for general type variety. Thank in advance.

ag.algebraic-geometry singularity-theory resolution-of-singularities

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asked Aug 22 at 14:30

xin fu

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up vote
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Let X be a smooth variety, take the proj of canonical ring of X and denote it by Y. Is Y always a canonical variety? I know it's true for general type variety. Thank in advance.

ag.algebraic-geometry singularity-theory resolution-of-singularities

share|cite|improve this question

asked Aug 22 at 14:30

xin fu

1016

add a comment | 

up vote
6
down vote

favorite

1

up vote
6
down vote

favorite

1

1

Let X be a smooth variety, take the proj of canonical ring of X and denote it by Y. Is Y always a canonical variety? I know it's true for general type variety. Thank in advance.

ag.algebraic-geometry singularity-theory resolution-of-singularities

share|cite|improve this question

asked Aug 22 at 14:30

xin fu

1016

Let X be a smooth variety, take the proj of canonical ring of X and denote it by Y. Is Y always a canonical variety? I know it's true for general type variety. Thank in advance.

ag.algebraic-geometry singularity-theory resolution-of-singularities

share|cite|improve this question

asked Aug 22 at 14:30

xin fu

1016

share|cite|improve this question

share|cite|improve this question

asked Aug 22 at 14:30

xin fu

1016

asked Aug 22 at 14:30

xin fu

1016

asked Aug 22 at 14:30

xin fu

1016

1016

add a comment | 

add a comment | 

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I believe that $(Y,B)$ is always klt for some boundary $B$. In fact by Theorem 5.2 of https://projecteuclid.org/download/pdf_1/euclid.jdg/1090347529, after passing to a truncation of the pluricanonical rings, there is a klt pair of log general type $(Y',B')$ with isomorphic pluricanonical ring (more precisely $R(K_X)^(a)cong R(K_X'+B')^(b)$ for appropriate integers $a,b>0$).
But then $Y=rm Proj R(K_X)cong rm Proj R(K_Y'+B')$ and $Y'to Y$ is the log canonical model of $(Y',B')$ which has klt singularities as $(Y',B')$ has klt singularities.

I also believe that $Y$ may not be canonical. I think one can construct examples where $Y$ is not canonical by considering a surface $S$ of general type with a cyclic group $G$ of order $n$ acting on it such that $S/G$ has non-canonical singularities (eg. $n=3$ and locally $g(x,y)=(xi x, xi ^2 y)$ where $xi$ is a primitive third root of 1) and $E$ an elliptic curve where $G$ acts via a translation of order $n$. Then $X=(Stimes E)/G$ should be smooth but $Y=S/G$ is klt but not canonical (I did not check the details!).

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answered Aug 22 at 16:13

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I believe that $(Y,B)$ is always klt for some boundary $B$. In fact by Theorem 5.2 of https://projecteuclid.org/download/pdf_1/euclid.jdg/1090347529, after passing to a truncation of the pluricanonical rings, there is a klt pair of log general type $(Y',B')$ with isomorphic pluricanonical ring (more precisely $R(K_X)^(a)cong R(K_X'+B')^(b)$ for appropriate integers $a,b>0$).
But then $Y=rm Proj R(K_X)cong rm Proj R(K_Y'+B')$ and $Y'to Y$ is the log canonical model of $(Y',B')$ which has klt singularities as $(Y',B')$ has klt singularities.

I also believe that $Y$ may not be canonical. I think one can construct examples where $Y$ is not canonical by considering a surface $S$ of general type with a cyclic group $G$ of order $n$ acting on it such that $S/G$ has non-canonical singularities (eg. $n=3$ and locally $g(x,y)=(xi x, xi ^2 y)$ where $xi$ is a primitive third root of 1) and $E$ an elliptic curve where $G$ acts via a translation of order $n$. Then $X=(Stimes E)/G$ should be smooth but $Y=S/G$ is klt but not canonical (I did not check the details!).

share|cite|improve this answer

answered Aug 22 at 16:13

Hacon

1,542814

add a comment | 

up vote
10
down vote

I believe that $(Y,B)$ is always klt for some boundary $B$. In fact by Theorem 5.2 of https://projecteuclid.org/download/pdf_1/euclid.jdg/1090347529, after passing to a truncation of the pluricanonical rings, there is a klt pair of log general type $(Y',B')$ with isomorphic pluricanonical ring (more precisely $R(K_X)^(a)cong R(K_X'+B')^(b)$ for appropriate integers $a,b>0$).
But then $Y=rm Proj R(K_X)cong rm Proj R(K_Y'+B')$ and $Y'to Y$ is the log canonical model of $(Y',B')$ which has klt singularities as $(Y',B')$ has klt singularities.

I also believe that $Y$ may not be canonical. I think one can construct examples where $Y$ is not canonical by considering a surface $S$ of general type with a cyclic group $G$ of order $n$ acting on it such that $S/G$ has non-canonical singularities (eg. $n=3$ and locally $g(x,y)=(xi x, xi ^2 y)$ where $xi$ is a primitive third root of 1) and $E$ an elliptic curve where $G$ acts via a translation of order $n$. Then $X=(Stimes E)/G$ should be smooth but $Y=S/G$ is klt but not canonical (I did not check the details!).

share|cite|improve this answer

answered Aug 22 at 16:13

Hacon

1,542814

add a comment | 

up vote
10
down vote

up vote
10
down vote

I believe that $(Y,B)$ is always klt for some boundary $B$. In fact by Theorem 5.2 of https://projecteuclid.org/download/pdf_1/euclid.jdg/1090347529, after passing to a truncation of the pluricanonical rings, there is a klt pair of log general type $(Y',B')$ with isomorphic pluricanonical ring (more precisely $R(K_X)^(a)cong R(K_X'+B')^(b)$ for appropriate integers $a,b>0$).
But then $Y=rm Proj R(K_X)cong rm Proj R(K_Y'+B')$ and $Y'to Y$ is the log canonical model of $(Y',B')$ which has klt singularities as $(Y',B')$ has klt singularities.

I also believe that $Y$ may not be canonical. I think one can construct examples where $Y$ is not canonical by considering a surface $S$ of general type with a cyclic group $G$ of order $n$ acting on it such that $S/G$ has non-canonical singularities (eg. $n=3$ and locally $g(x,y)=(xi x, xi ^2 y)$ where $xi$ is a primitive third root of 1) and $E$ an elliptic curve where $G$ acts via a translation of order $n$. Then $X=(Stimes E)/G$ should be smooth but $Y=S/G$ is klt but not canonical (I did not check the details!).

share|cite|improve this answer

answered Aug 22 at 16:13

Hacon

1,542814

I believe that $(Y,B)$ is always klt for some boundary $B$. In fact by Theorem 5.2 of https://projecteuclid.org/download/pdf_1/euclid.jdg/1090347529, after passing to a truncation of the pluricanonical rings, there is a klt pair of log general type $(Y',B')$ with isomorphic pluricanonical ring (more precisely $R(K_X)^(a)cong R(K_X'+B')^(b)$ for appropriate integers $a,b>0$).
But then $Y=rm Proj R(K_X)cong rm Proj R(K_Y'+B')$ and $Y'to Y$ is the log canonical model of $(Y',B')$ which has klt singularities as $(Y',B')$ has klt singularities.

I also believe that $Y$ may not be canonical. I think one can construct examples where $Y$ is not canonical by considering a surface $S$ of general type with a cyclic group $G$ of order $n$ acting on it such that $S/G$ has non-canonical singularities (eg. $n=3$ and locally $g(x,y)=(xi x, xi ^2 y)$ where $xi$ is a primitive third root of 1) and $E$ an elliptic curve where $G$ acts via a translation of order $n$. Then $X=(Stimes E)/G$ should be smooth but $Y=S/G$ is klt but not canonical (I did not check the details!).

share|cite|improve this answer

answered Aug 22 at 16:13

Hacon

1,542814

share|cite|improve this answer

share|cite|improve this answer

answered Aug 22 at 16:13

Hacon

1,542814

answered Aug 22 at 16:13

Hacon

1,542814

answered Aug 22 at 16:13

Hacon

1,542814

1,542814

add a comment | 

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