Curves and divisors in weighted projective planes
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Let us consider the weighted projective plane $mathbb{P}(q_0,q_1,q_2)=mathrm{Proj}(mathbb{C}[x_0,x_1,x_2])$ where $x_i$ has weight $q_i$ for every $iin {0,1,2}$. Let $fin mathbb{C}[x_0,x_1,x_2]$ be a homogeneous polynomial of degree $d$. Then $f$ defines a curve $C$ in $mathbb{P}(q_0,q_1,q_2)$.
On the other hand, $mathrm{Pic}(mathbb{P}(q_0,q_1,q_2))$ is generated by $mathcal{O}_{mathbb{P}(q_0,q_1,q_2)}(n)$ for some $n$.
In the case $q_0=q_1=q_2=1$ we have that $n=1$ and $mathcal{O}_{mathbb{P}^2}(C)simeq mathcal{O}_{mathbb{P}^2}(d)$. Can we generalize this for different values of $q_0,q_1,q_2$?
algebraic-geometry projective-space divisors-algebraic-geometry
asked Dec 18 '18 at 16:07
H. JacksonH. Jackson
31318
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add a comment |
$begingroup$
Let us consider the weighted projective plane $mathbb{P}(q_0,q_1,q_2)=mathrm{Proj}(mathbb{C}[x_0,x_1,x_2])$ where $x_i$ has weight $q_i$ for every $iin {0,1,2}$. Let $fin mathbb{C}[x_0,x_1,x_2]$ be a homogeneous polynomial of degree $d$. Then $f$ defines a curve $C$ in $mathbb{P}(q_0,q_1,q_2)$.
On the other hand, $mathrm{Pic}(mathbb{P}(q_0,q_1,q_2))$ is generated by $mathcal{O}_{mathbb{P}(q_0,q_1,q_2)}(n)$ for some $n$.
In the case $q_0=q_1=q_2=1$ we have that $n=1$ and $mathcal{O}_{mathbb{P}^2}(C)simeq mathcal{O}_{mathbb{P}^2}(d)$. Can we generalize this for different values of $q_0,q_1,q_2$?
algebraic-geometry projective-space divisors-algebraic-geometry
asked Dec 18 '18 at 16:07
H. JacksonH. Jackson
31318
$endgroup$
add a comment |
$begingroup$
Let us consider the weighted projective plane $mathbb{P}(q_0,q_1,q_2)=mathrm{Proj}(mathbb{C}[x_0,x_1,x_2])$ where $x_i$ has weight $q_i$ for every $iin {0,1,2}$. Let $fin mathbb{C}[x_0,x_1,x_2]$ be a homogeneous polynomial of degree $d$. Then $f$ defines a curve $C$ in $mathbb{P}(q_0,q_1,q_2)$.
On the other hand, $mathrm{Pic}(mathbb{P}(q_0,q_1,q_2))$ is generated by $mathcal{O}_{mathbb{P}(q_0,q_1,q_2)}(n)$ for some $n$.
In the case $q_0=q_1=q_2=1$ we have that $n=1$ and $mathcal{O}_{mathbb{P}^2}(C)simeq mathcal{O}_{mathbb{P}^2}(d)$. Can we generalize this for different values of $q_0,q_1,q_2$?
algebraic-geometry projective-space divisors-algebraic-geometry
asked Dec 18 '18 at 16:07
H. JacksonH. Jackson
31318
$endgroup$
Let us consider the weighted projective plane $mathbb{P}(q_0,q_1,q_2)=mathrm{Proj}(mathbb{C}[x_0,x_1,x_2])$ where $x_i$ has weight $q_i$ for every $iin {0,1,2}$. Let $fin mathbb{C}[x_0,x_1,x_2]$ be a homogeneous polynomial of degree $d$. Then $f$ defines a curve $C$ in $mathbb{P}(q_0,q_1,q_2)$.
On the other hand, $mathrm{Pic}(mathbb{P}(q_0,q_1,q_2))$ is generated by $mathcal{O}_{mathbb{P}(q_0,q_1,q_2)}(n)$ for some $n$.
In the case $q_0=q_1=q_2=1$ we have that $n=1$ and $mathcal{O}_{mathbb{P}^2}(C)simeq mathcal{O}_{mathbb{P}^2}(d)$. Can we generalize this for different values of $q_0,q_1,q_2$?
algebraic-geometry projective-space divisors-algebraic-geometry
algebraic-geometry projective-space divisors-algebraic-geometry
asked Dec 18 '18 at 16:07
H. JacksonH. Jackson
31318
asked Dec 18 '18 at 16:07
H. JacksonH. Jackson
31318
asked Dec 18 '18 at 16:07
H. JacksonH. Jackson
31318
asked Dec 18 '18 at 16:07
H. JacksonH. Jackson
31318
asked Dec 18 '18 at 16:07
H. JacksonH. Jackson
31318
31318
add a comment |
add a comment |
1 Answer
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The class group of Weil divisors on a weighted projective plane is isomorphic to the group of reflexive sheaves of rank 1 (with operation of tensor product followed by double dualization), and is a free abelian group of rank 1 generated by the reflexive sheaf $mathcal{O}(1)$. Under this isomorphism $mathcal{O}(C) cong mathcal{O}(d)$.
answered Dec 18 '18 at 17:45
SashaSasha
5,088139
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1 Answer
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1 Answer
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$begingroup$
The class group of Weil divisors on a weighted projective plane is isomorphic to the group of reflexive sheaves of rank 1 (with operation of tensor product followed by double dualization), and is a free abelian group of rank 1 generated by the reflexive sheaf $mathcal{O}(1)$. Under this isomorphism $mathcal{O}(C) cong mathcal{O}(d)$.
answered Dec 18 '18 at 17:45
SashaSasha
5,088139
$endgroup$
add a comment |
$begingroup$
The class group of Weil divisors on a weighted projective plane is isomorphic to the group of reflexive sheaves of rank 1 (with operation of tensor product followed by double dualization), and is a free abelian group of rank 1 generated by the reflexive sheaf $mathcal{O}(1)$. Under this isomorphism $mathcal{O}(C) cong mathcal{O}(d)$.
answered Dec 18 '18 at 17:45
SashaSasha
5,088139
$endgroup$
add a comment |
$begingroup$
The class group of Weil divisors on a weighted projective plane is isomorphic to the group of reflexive sheaves of rank 1 (with operation of tensor product followed by double dualization), and is a free abelian group of rank 1 generated by the reflexive sheaf $mathcal{O}(1)$. Under this isomorphism $mathcal{O}(C) cong mathcal{O}(d)$.
answered Dec 18 '18 at 17:45
SashaSasha
5,088139
$endgroup$
The class group of Weil divisors on a weighted projective plane is isomorphic to the group of reflexive sheaves of rank 1 (with operation of tensor product followed by double dualization), and is a free abelian group of rank 1 generated by the reflexive sheaf $mathcal{O}(1)$. Under this isomorphism $mathcal{O}(C) cong mathcal{O}(d)$.
answered Dec 18 '18 at 17:45
SashaSasha
5,088139
answered Dec 18 '18 at 17:45
SashaSasha
5,088139
answered Dec 18 '18 at 17:45
SashaSasha
5,088139
answered Dec 18 '18 at 17:45
SashaSasha
5,088139
5,088139
add a comment |
add a comment |
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