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connection between a theorem and Nakai-Moishezon criterion
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So, consider Nakai-Moishezon criterion given in Lucian Badescu, Algebraic Surfaces:
Theorem 1[Nakai- Moishezon criterion]: Let $V$ be a complete algebraic scheme over $k$ and let $L$ be an invertible $O_V$-module. Then $L$ is ample if and only if for every integral closed subscheme $W subseteq V$ of dimension $t>0$, we have $(L^-tcdot W)>0$
and consider:
Theorem 2[Jens Franke, Algebraic Geometry 2, lecture notes Summer Semester 2018. at University of Bonn, Theorem 6, www.github.com/~nicholas42]: Let $f:X to Spec(A)$ be a proper morphism. For a line bundle $L$ on $X$, the following are equivalent:
a) $L$ is ample
b) Some power $L^otimes k$ of $L$ is $very ample$ in the sense that $L$ is generated by global sections $l_0, ..., l_n$ and the morphisms $X to P_A^n (=Proj_A(A[x_0,...,x_n]))$ and the multiplicative epimorphism $O_X[x_0,..,x_n] to oplus_j=0^+infty L^otimes k_j$'
c) For some $k in mathbbN, L^otimes k$ is generated by global sections $l_0,..., l_n$ and the above morphism $X to P_A^n$ is finite (or equivalently, affine)
d) $L$ is the $cohomology killer$ in the following sense: If $M$ is a coherent $O_X$-module and $p>0$ then $H^p(X, M otimes_O_X L^otimes k) = 0$ for $k>>0$ (k enough big)
So, my logic tell me there is some connection between those two theorems(as both provide equivalent conditions to just saying that line bundle $L$ on $X$ is ample), any explanation why and how one is implying other will be much appreciated.
algebraic-geometry
asked Jul 23 at 12:52
nikola
557214
add a comment |Â
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0
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So, consider Nakai-Moishezon criterion given in Lucian Badescu, Algebraic Surfaces:
Theorem 1[Nakai- Moishezon criterion]: Let $V$ be a complete algebraic scheme over $k$ and let $L$ be an invertible $O_V$-module. Then $L$ is ample if and only if for every integral closed subscheme $W subseteq V$ of dimension $t>0$, we have $(L^-tcdot W)>0$
and consider:
Theorem 2[Jens Franke, Algebraic Geometry 2, lecture notes Summer Semester 2018. at University of Bonn, Theorem 6, www.github.com/~nicholas42]: Let $f:X to Spec(A)$ be a proper morphism. For a line bundle $L$ on $X$, the following are equivalent:
a) $L$ is ample
b) Some power $L^otimes k$ of $L$ is $very ample$ in the sense that $L$ is generated by global sections $l_0, ..., l_n$ and the morphisms $X to P_A^n (=Proj_A(A[x_0,...,x_n]))$ and the multiplicative epimorphism $O_X[x_0,..,x_n] to oplus_j=0^+infty L^otimes k_j$'
c) For some $k in mathbbN, L^otimes k$ is generated by global sections $l_0,..., l_n$ and the above morphism $X to P_A^n$ is finite (or equivalently, affine)
d) $L$ is the $cohomology killer$ in the following sense: If $M$ is a coherent $O_X$-module and $p>0$ then $H^p(X, M otimes_O_X L^otimes k) = 0$ for $k>>0$ (k enough big)
So, my logic tell me there is some connection between those two theorems(as both provide equivalent conditions to just saying that line bundle $L$ on $X$ is ample), any explanation why and how one is implying other will be much appreciated.
algebraic-geometry
asked Jul 23 at 12:52
nikola
557214
In theorem 1, I think you meant $(L^dim V-tcdot W)>0$. The second theorem is the more standard definition-theorem of ampleness. Theorem 1 is a non-trivial theorem and the proof is somewhat complicated and better read in a text book.
– Mohan
Jul 23 at 13:59
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
So, consider Nakai-Moishezon criterion given in Lucian Badescu, Algebraic Surfaces:
Theorem 1[Nakai- Moishezon criterion]: Let $V$ be a complete algebraic scheme over $k$ and let $L$ be an invertible $O_V$-module. Then $L$ is ample if and only if for every integral closed subscheme $W subseteq V$ of dimension $t>0$, we have $(L^-tcdot W)>0$
and consider:
Theorem 2[Jens Franke, Algebraic Geometry 2, lecture notes Summer Semester 2018. at University of Bonn, Theorem 6, www.github.com/~nicholas42]: Let $f:X to Spec(A)$ be a proper morphism. For a line bundle $L$ on $X$, the following are equivalent:
a) $L$ is ample
b) Some power $L^otimes k$ of $L$ is $very ample$ in the sense that $L$ is generated by global sections $l_0, ..., l_n$ and the morphisms $X to P_A^n (=Proj_A(A[x_0,...,x_n]))$ and the multiplicative epimorphism $O_X[x_0,..,x_n] to oplus_j=0^+infty L^otimes k_j$'
c) For some $k in mathbbN, L^otimes k$ is generated by global sections $l_0,..., l_n$ and the above morphism $X to P_A^n$ is finite (or equivalently, affine)
d) $L$ is the $cohomology killer$ in the following sense: If $M$ is a coherent $O_X$-module and $p>0$ then $H^p(X, M otimes_O_X L^otimes k) = 0$ for $k>>0$ (k enough big)
So, my logic tell me there is some connection between those two theorems(as both provide equivalent conditions to just saying that line bundle $L$ on $X$ is ample), any explanation why and how one is implying other will be much appreciated.
algebraic-geometry
asked Jul 23 at 12:52
nikola
557214
So, consider Nakai-Moishezon criterion given in Lucian Badescu, Algebraic Surfaces:
Theorem 1[Nakai- Moishezon criterion]: Let $V$ be a complete algebraic scheme over $k$ and let $L$ be an invertible $O_V$-module. Then $L$ is ample if and only if for every integral closed subscheme $W subseteq V$ of dimension $t>0$, we have $(L^-tcdot W)>0$
and consider:
Theorem 2[Jens Franke, Algebraic Geometry 2, lecture notes Summer Semester 2018. at University of Bonn, Theorem 6, www.github.com/~nicholas42]: Let $f:X to Spec(A)$ be a proper morphism. For a line bundle $L$ on $X$, the following are equivalent:
a) $L$ is ample
b) Some power $L^otimes k$ of $L$ is $very ample$ in the sense that $L$ is generated by global sections $l_0, ..., l_n$ and the morphisms $X to P_A^n (=Proj_A(A[x_0,...,x_n]))$ and the multiplicative epimorphism $O_X[x_0,..,x_n] to oplus_j=0^+infty L^otimes k_j$'
c) For some $k in mathbbN, L^otimes k$ is generated by global sections $l_0,..., l_n$ and the above morphism $X to P_A^n$ is finite (or equivalently, affine)
d) $L$ is the $cohomology killer$ in the following sense: If $M$ is a coherent $O_X$-module and $p>0$ then $H^p(X, M otimes_O_X L^otimes k) = 0$ for $k>>0$ (k enough big)
So, my logic tell me there is some connection between those two theorems(as both provide equivalent conditions to just saying that line bundle $L$ on $X$ is ample), any explanation why and how one is implying other will be much appreciated.
algebraic-geometry
asked Jul 23 at 12:52
nikola
557214
asked Jul 23 at 12:52
nikola
557214
asked Jul 23 at 12:52
nikola
557214
asked Jul 23 at 12:52
nikola
557214
557214
In theorem 1, I think you meant $(L^dim V-tcdot W)>0$. The second theorem is the more standard definition-theorem of ampleness. Theorem 1 is a non-trivial theorem and the proof is somewhat complicated and better read in a text book.
– Mohan
Jul 23 at 13:59
add a comment |Â
In theorem 1, I think you meant $(L^dim V-tcdot W)>0$. The second theorem is the more standard definition-theorem of ampleness. Theorem 1 is a non-trivial theorem and the proof is somewhat complicated and better read in a text book.
– Mohan
Jul 23 at 13:59
In theorem 1, I think you meant $(L^dim V-tcdot W)>0$. The second theorem is the more standard definition-theorem of ampleness. Theorem 1 is a non-trivial theorem and the proof is somewhat complicated and better read in a text book.
– Mohan
Jul 23 at 13:59
In theorem 1, I think you meant $(L^dim V-tcdot W)>0$. The second theorem is the more standard definition-theorem of ampleness. Theorem 1 is a non-trivial theorem and the proof is somewhat complicated and better read in a text book.
– Mohan
Jul 23 at 13:59
add a comment |Â
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In theorem 1, I think you meant $(L^dim V-tcdot W)>0$. The second theorem is the more standard definition-theorem of ampleness. Theorem 1 is a non-trivial theorem and the proof is somewhat complicated and better read in a text book.
– Mohan
Jul 23 at 13:59