Smooth projective curves of genus 0
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Is there a classification of smooth projective curves of genus $0$ over $mathbb{Q}$?
I know that if the curve has a rational point, then it is isomorphic to $mathbb{P}^1$.
The curve must embed as a degree $2$ curve in $mathbb{P}^2$, so it has a point over some quadratic extension of $mathbb{Q}$. This means the curve is a quadratic twist of $mathbb{P}^1$.
algebraic-geometry algebraic-curves
asked Nov 13 at 13:03
User
33718
add a comment |
up vote
0
down vote
favorite
Is there a classification of smooth projective curves of genus $0$ over $mathbb{Q}$?
I know that if the curve has a rational point, then it is isomorphic to $mathbb{P}^1$.
The curve must embed as a degree $2$ curve in $mathbb{P}^2$, so it has a point over some quadratic extension of $mathbb{Q}$. This means the curve is a quadratic twist of $mathbb{P}^1$.
algebraic-geometry algebraic-curves
asked Nov 13 at 13:03
User
33718
Doesn't Cassels say in his book on elliptic curves: "Fact. A genus 0 curve is equivalent to a line or conic"?
– Richard Martin
Nov 13 at 13:07
1
Your classification is correct. Are you asking for a stronger classification in some sense?
– Samir Canning
Nov 13 at 14:57
I would like a list of isomorphism classes of curves. I know each curve can be embedded as a conic in $mathbb{P}^2$, but it isn't obvious (to me, at least) when two conics are isomorphic.
– User
Nov 13 at 16:30
These kind of varieties are called Brauer-Severi varieties. The question is equivalent to asking for a classification of quadratic forms in three variables over $mathbb{Q}$. I am not sure how useful is this for you.
– random123
Nov 15 at 8:01
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Is there a classification of smooth projective curves of genus $0$ over $mathbb{Q}$?
I know that if the curve has a rational point, then it is isomorphic to $mathbb{P}^1$.
The curve must embed as a degree $2$ curve in $mathbb{P}^2$, so it has a point over some quadratic extension of $mathbb{Q}$. This means the curve is a quadratic twist of $mathbb{P}^1$.
algebraic-geometry algebraic-curves
asked Nov 13 at 13:03
User
33718
Is there a classification of smooth projective curves of genus $0$ over $mathbb{Q}$?
I know that if the curve has a rational point, then it is isomorphic to $mathbb{P}^1$.
The curve must embed as a degree $2$ curve in $mathbb{P}^2$, so it has a point over some quadratic extension of $mathbb{Q}$. This means the curve is a quadratic twist of $mathbb{P}^1$.
algebraic-geometry algebraic-curves
algebraic-geometry algebraic-curves
asked Nov 13 at 13:03
User
33718
asked Nov 13 at 13:03
User
33718
asked Nov 13 at 13:03
User
33718
asked Nov 13 at 13:03
User
33718
asked Nov 13 at 13:03
User
33718
33718
Doesn't Cassels say in his book on elliptic curves: "Fact. A genus 0 curve is equivalent to a line or conic"?
– Richard Martin
Nov 13 at 13:07
1
Your classification is correct. Are you asking for a stronger classification in some sense?
– Samir Canning
Nov 13 at 14:57
I would like a list of isomorphism classes of curves. I know each curve can be embedded as a conic in $mathbb{P}^2$, but it isn't obvious (to me, at least) when two conics are isomorphic.
– User
Nov 13 at 16:30
These kind of varieties are called Brauer-Severi varieties. The question is equivalent to asking for a classification of quadratic forms in three variables over $mathbb{Q}$. I am not sure how useful is this for you.
– random123
Nov 15 at 8:01
add a comment |
Doesn't Cassels say in his book on elliptic curves: "Fact. A genus 0 curve is equivalent to a line or conic"?
– Richard Martin
Nov 13 at 13:07
1
Your classification is correct. Are you asking for a stronger classification in some sense?
– Samir Canning
Nov 13 at 14:57
I would like a list of isomorphism classes of curves. I know each curve can be embedded as a conic in $mathbb{P}^2$, but it isn't obvious (to me, at least) when two conics are isomorphic.
– User
Nov 13 at 16:30
These kind of varieties are called Brauer-Severi varieties. The question is equivalent to asking for a classification of quadratic forms in three variables over $mathbb{Q}$. I am not sure how useful is this for you.
– random123
Nov 15 at 8:01
Doesn't Cassels say in his book on elliptic curves: "Fact. A genus 0 curve is equivalent to a line or conic"?
– Richard Martin
Nov 13 at 13:07
Doesn't Cassels say in his book on elliptic curves: "Fact. A genus 0 curve is equivalent to a line or conic"?
– Richard Martin
Nov 13 at 13:07
1
1
Your classification is correct. Are you asking for a stronger classification in some sense?
– Samir Canning
Nov 13 at 14:57
Your classification is correct. Are you asking for a stronger classification in some sense?
– Samir Canning
Nov 13 at 14:57
I would like a list of isomorphism classes of curves. I know each curve can be embedded as a conic in $mathbb{P}^2$, but it isn't obvious (to me, at least) when two conics are isomorphic.
– User
Nov 13 at 16:30
I would like a list of isomorphism classes of curves. I know each curve can be embedded as a conic in $mathbb{P}^2$, but it isn't obvious (to me, at least) when two conics are isomorphic.
– User
Nov 13 at 16:30
These kind of varieties are called Brauer-Severi varieties. The question is equivalent to asking for a classification of quadratic forms in three variables over $mathbb{Q}$. I am not sure how useful is this for you.
– random123
Nov 15 at 8:01
These kind of varieties are called Brauer-Severi varieties. The question is equivalent to asking for a classification of quadratic forms in three variables over $mathbb{Q}$. I am not sure how useful is this for you.
– random123
Nov 15 at 8:01
add a comment |
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Doesn't Cassels say in his book on elliptic curves: "Fact. A genus 0 curve is equivalent to a line or conic"?
– Richard Martin
Nov 13 at 13:07
1
Your classification is correct. Are you asking for a stronger classification in some sense?
– Samir Canning
Nov 13 at 14:57
I would like a list of isomorphism classes of curves. I know each curve can be embedded as a conic in $mathbb{P}^2$, but it isn't obvious (to me, at least) when two conics are isomorphic.
– User
Nov 13 at 16:30
These kind of varieties are called Brauer-Severi varieties. The question is equivalent to asking for a classification of quadratic forms in three variables over $mathbb{Q}$. I am not sure how useful is this for you.
– random123
Nov 15 at 8:01