The ampleness of canonical sheaves and the proof of “$X simeq mathrm{Proj}left(bigoplus_k H^0(X,...
$begingroup$
In Bondal and Orlov's paper, ''Reconstruction of a variety from the derived category and groups of autoequivalences'' (http://www.mi-ras.ru/~orlov/papers/Compositio2001.pdf), I think that they use the following result:
Let X be the smooth projective variety, $omega_X$ is the canonical
sheaf of $X$. If $omega_X$ is ample,
$$X simeqmathrm{Proj}left(bigoplus_k H^0(X, omega_X^k)right). $$
I am looking for a complete proof of this fact. I have read some literature, but the proof is omitted in them. Is this a simple fact? Could you tell me the proof or the literature on which it is listed?
algebraic-geometry category-theory homology-cohomology sheaf-theory derived-categories
asked Jan 10 at 15:41
RuiSenRuiSen
61
$endgroup$
add a comment |
$begingroup$
In Bondal and Orlov's paper, ''Reconstruction of a variety from the derived category and groups of autoequivalences'' (http://www.mi-ras.ru/~orlov/papers/Compositio2001.pdf), I think that they use the following result:
Let X be the smooth projective variety, $omega_X$ is the canonical
sheaf of $X$. If $omega_X$ is ample,
$$X simeqmathrm{Proj}left(bigoplus_k H^0(X, omega_X^k)right). $$
I am looking for a complete proof of this fact. I have read some literature, but the proof is omitted in them. Is this a simple fact? Could you tell me the proof or the literature on which it is listed?
algebraic-geometry category-theory homology-cohomology sheaf-theory derived-categories
asked Jan 10 at 15:41
RuiSenRuiSen
61
$endgroup$
1
$begingroup$
If $R=oplus_{kgeq 0} R_k$ with $R_0=K$, the base field and $R$ finitely generated $K$-algebra, then consider $S=oplus_{n|k} R_ksubset R$ for some integer $n>0$. One easily checks that there is an induced isomorphism $mathrm{Proj},Rtomathrm{Proj}, S$. In your case, $omega_X$ is ample, so for a large enough $n$, $omega_X^n$ is very ample. Rest should be clear.
$endgroup$
– Mohan
Jan 10 at 16:38
add a comment |
$begingroup$
In Bondal and Orlov's paper, ''Reconstruction of a variety from the derived category and groups of autoequivalences'' (http://www.mi-ras.ru/~orlov/papers/Compositio2001.pdf), I think that they use the following result:
Let X be the smooth projective variety, $omega_X$ is the canonical
sheaf of $X$. If $omega_X$ is ample,
$$X simeqmathrm{Proj}left(bigoplus_k H^0(X, omega_X^k)right). $$
I am looking for a complete proof of this fact. I have read some literature, but the proof is omitted in them. Is this a simple fact? Could you tell me the proof or the literature on which it is listed?
algebraic-geometry category-theory homology-cohomology sheaf-theory derived-categories
asked Jan 10 at 15:41
RuiSenRuiSen
61
$endgroup$
In Bondal and Orlov's paper, ''Reconstruction of a variety from the derived category and groups of autoequivalences'' (http://www.mi-ras.ru/~orlov/papers/Compositio2001.pdf), I think that they use the following result:
Let X be the smooth projective variety, $omega_X$ is the canonical
sheaf of $X$. If $omega_X$ is ample,
$$X simeqmathrm{Proj}left(bigoplus_k H^0(X, omega_X^k)right). $$
I am looking for a complete proof of this fact. I have read some literature, but the proof is omitted in them. Is this a simple fact? Could you tell me the proof or the literature on which it is listed?
algebraic-geometry category-theory homology-cohomology sheaf-theory derived-categories
algebraic-geometry category-theory homology-cohomology sheaf-theory derived-categories
asked Jan 10 at 15:41
RuiSenRuiSen
61
asked Jan 10 at 15:41
RuiSenRuiSen
61
asked Jan 10 at 15:41
RuiSenRuiSen
61
asked Jan 10 at 15:41
RuiSenRuiSen
61
asked Jan 10 at 15:41
RuiSenRuiSen
61
61
1
$begingroup$
If $R=oplus_{kgeq 0} R_k$ with $R_0=K$, the base field and $R$ finitely generated $K$-algebra, then consider $S=oplus_{n|k} R_ksubset R$ for some integer $n>0$. One easily checks that there is an induced isomorphism $mathrm{Proj},Rtomathrm{Proj}, S$. In your case, $omega_X$ is ample, so for a large enough $n$, $omega_X^n$ is very ample. Rest should be clear.
$endgroup$
– Mohan
Jan 10 at 16:38
add a comment |
1
$begingroup$
If $R=oplus_{kgeq 0} R_k$ with $R_0=K$, the base field and $R$ finitely generated $K$-algebra, then consider $S=oplus_{n|k} R_ksubset R$ for some integer $n>0$. One easily checks that there is an induced isomorphism $mathrm{Proj},Rtomathrm{Proj}, S$. In your case, $omega_X$ is ample, so for a large enough $n$, $omega_X^n$ is very ample. Rest should be clear.
$endgroup$
– Mohan
Jan 10 at 16:38
1
1
$begingroup$
If $R=oplus_{kgeq 0} R_k$ with $R_0=K$, the base field and $R$ finitely generated $K$-algebra, then consider $S=oplus_{n|k} R_ksubset R$ for some integer $n>0$. One easily checks that there is an induced isomorphism $mathrm{Proj},Rtomathrm{Proj}, S$. In your case, $omega_X$ is ample, so for a large enough $n$, $omega_X^n$ is very ample. Rest should be clear.
$endgroup$
– Mohan
Jan 10 at 16:38
$begingroup$
If $R=oplus_{kgeq 0} R_k$ with $R_0=K$, the base field and $R$ finitely generated $K$-algebra, then consider $S=oplus_{n|k} R_ksubset R$ for some integer $n>0$. One easily checks that there is an induced isomorphism $mathrm{Proj},Rtomathrm{Proj}, S$. In your case, $omega_X$ is ample, so for a large enough $n$, $omega_X^n$ is very ample. Rest should be clear.
$endgroup$
– Mohan
Jan 10 at 16:38
add a comment |
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$begingroup$
If $R=oplus_{kgeq 0} R_k$ with $R_0=K$, the base field and $R$ finitely generated $K$-algebra, then consider $S=oplus_{n|k} R_ksubset R$ for some integer $n>0$. One easily checks that there is an induced isomorphism $mathrm{Proj},Rtomathrm{Proj}, S$. In your case, $omega_X$ is ample, so for a large enough $n$, $omega_X^n$ is very ample. Rest should be clear.
$endgroup$
– Mohan
Jan 10 at 16:38