Mixing
One-point compactification and algebraic geometry
$begingroup$
An affine curve $C:{(x,y)inmathbb{C}^2:Q(x,y)=0}$ can always be extended to a projective curve $tilde{C}:{[x:y:z]inmathbb{CP}^2:P(x,y,z)}$ where $P(X,Y,Z)$ is a homogenisation of the polynomial $Q$.
It is a well-known fact that elliptic curves have only one point at infinity as the homogenisation of $Q(X,Y)=X^3+aX+b-Y^2$ is the polynomial $P(X,Y,Z)=X^3+aXZ^2+bZ^3-Y^2Z$ and $P(X,Y,0)=X^3$ has only one solution so the only "point at infinity" is given by the homogeneous coordinates $[0:1:0]$.
But there are other curves which have two or more points at infinity. For example, the curve $X^2-Y^2=1$ has two points $[1:1:0]$ and $[-1:1:0]$ at infinity.
Given a topology $(X,tau)$, the one-point compactification is defined as $(X^*tau^*)$ where $X^*=Xcup{infty}$ with open sets which are $Ein tau$ or $E=(Xsetminus K)cup{infty} $ where $K$ is a compact subset of $X$.
I would like to ask if we do the one-point compactification on an elliptic curve, is it equivalent to adding the point at infinity?
Secondly, for points with two or more points at infinity, clearly the one-point compactification is not the same thing. Then, what topological differences is causing these two affine curves to behave differently?
general-topology algebraic-geometry projective-space compactification
asked Jan 26 at 13:30
darumadaruma
937612
$endgroup$
add a comment |
$begingroup$
An affine curve $C:{(x,y)inmathbb{C}^2:Q(x,y)=0}$ can always be extended to a projective curve $tilde{C}:{[x:y:z]inmathbb{CP}^2:P(x,y,z)}$ where $P(X,Y,Z)$ is a homogenisation of the polynomial $Q$.
It is a well-known fact that elliptic curves have only one point at infinity as the homogenisation of $Q(X,Y)=X^3+aX+b-Y^2$ is the polynomial $P(X,Y,Z)=X^3+aXZ^2+bZ^3-Y^2Z$ and $P(X,Y,0)=X^3$ has only one solution so the only "point at infinity" is given by the homogeneous coordinates $[0:1:0]$.
But there are other curves which have two or more points at infinity. For example, the curve $X^2-Y^2=1$ has two points $[1:1:0]$ and $[-1:1:0]$ at infinity.
Given a topology $(X,tau)$, the one-point compactification is defined as $(X^*tau^*)$ where $X^*=Xcup{infty}$ with open sets which are $Ein tau$ or $E=(Xsetminus K)cup{infty} $ where $K$ is a compact subset of $X$.
I would like to ask if we do the one-point compactification on an elliptic curve, is it equivalent to adding the point at infinity?
Secondly, for points with two or more points at infinity, clearly the one-point compactification is not the same thing. Then, what topological differences is causing these two affine curves to behave differently?
general-topology algebraic-geometry projective-space compactification
asked Jan 26 at 13:30
darumadaruma
937612
$endgroup$
1
$begingroup$
I feel like I should point out that curves are topological spaces with additional information. All (irreducible) curves with the same cardinality of points are homeomorphic as topological spaces (having the cofinite topology), and even the affine ones are compact, so "the one point compactification" isn't a compactification in this case, since the affine part of the curve won't be dense.
$endgroup$
– jgon
Jan 26 at 19:05
add a comment |
$begingroup$
An affine curve $C:{(x,y)inmathbb{C}^2:Q(x,y)=0}$ can always be extended to a projective curve $tilde{C}:{[x:y:z]inmathbb{CP}^2:P(x,y,z)}$ where $P(X,Y,Z)$ is a homogenisation of the polynomial $Q$.
It is a well-known fact that elliptic curves have only one point at infinity as the homogenisation of $Q(X,Y)=X^3+aX+b-Y^2$ is the polynomial $P(X,Y,Z)=X^3+aXZ^2+bZ^3-Y^2Z$ and $P(X,Y,0)=X^3$ has only one solution so the only "point at infinity" is given by the homogeneous coordinates $[0:1:0]$.
But there are other curves which have two or more points at infinity. For example, the curve $X^2-Y^2=1$ has two points $[1:1:0]$ and $[-1:1:0]$ at infinity.
Given a topology $(X,tau)$, the one-point compactification is defined as $(X^*tau^*)$ where $X^*=Xcup{infty}$ with open sets which are $Ein tau$ or $E=(Xsetminus K)cup{infty} $ where $K$ is a compact subset of $X$.
I would like to ask if we do the one-point compactification on an elliptic curve, is it equivalent to adding the point at infinity?
Secondly, for points with two or more points at infinity, clearly the one-point compactification is not the same thing. Then, what topological differences is causing these two affine curves to behave differently?
general-topology algebraic-geometry projective-space compactification
asked Jan 26 at 13:30
darumadaruma
937612
$endgroup$
An affine curve $C:{(x,y)inmathbb{C}^2:Q(x,y)=0}$ can always be extended to a projective curve $tilde{C}:{[x:y:z]inmathbb{CP}^2:P(x,y,z)}$ where $P(X,Y,Z)$ is a homogenisation of the polynomial $Q$.
It is a well-known fact that elliptic curves have only one point at infinity as the homogenisation of $Q(X,Y)=X^3+aX+b-Y^2$ is the polynomial $P(X,Y,Z)=X^3+aXZ^2+bZ^3-Y^2Z$ and $P(X,Y,0)=X^3$ has only one solution so the only "point at infinity" is given by the homogeneous coordinates $[0:1:0]$.
But there are other curves which have two or more points at infinity. For example, the curve $X^2-Y^2=1$ has two points $[1:1:0]$ and $[-1:1:0]$ at infinity.
Given a topology $(X,tau)$, the one-point compactification is defined as $(X^*tau^*)$ where $X^*=Xcup{infty}$ with open sets which are $Ein tau$ or $E=(Xsetminus K)cup{infty} $ where $K$ is a compact subset of $X$.
I would like to ask if we do the one-point compactification on an elliptic curve, is it equivalent to adding the point at infinity?
Secondly, for points with two or more points at infinity, clearly the one-point compactification is not the same thing. Then, what topological differences is causing these two affine curves to behave differently?
general-topology algebraic-geometry projective-space compactification
general-topology algebraic-geometry projective-space compactification
asked Jan 26 at 13:30
darumadaruma
937612
asked Jan 26 at 13:30
darumadaruma
937612
asked Jan 26 at 13:30
darumadaruma
937612
asked Jan 26 at 13:30
darumadaruma
937612
asked Jan 26 at 13:30
darumadaruma
937612
937612
1
$begingroup$
I feel like I should point out that curves are topological spaces with additional information. All (irreducible) curves with the same cardinality of points are homeomorphic as topological spaces (having the cofinite topology), and even the affine ones are compact, so "the one point compactification" isn't a compactification in this case, since the affine part of the curve won't be dense.
$endgroup$
– jgon
Jan 26 at 19:05
add a comment |
1
$begingroup$
I feel like I should point out that curves are topological spaces with additional information. All (irreducible) curves with the same cardinality of points are homeomorphic as topological spaces (having the cofinite topology), and even the affine ones are compact, so "the one point compactification" isn't a compactification in this case, since the affine part of the curve won't be dense.
$endgroup$
– jgon
Jan 26 at 19:05
1
1
$begingroup$
I feel like I should point out that curves are topological spaces with additional information. All (irreducible) curves with the same cardinality of points are homeomorphic as topological spaces (having the cofinite topology), and even the affine ones are compact, so "the one point compactification" isn't a compactification in this case, since the affine part of the curve won't be dense.
$endgroup$
– jgon
Jan 26 at 19:05
$begingroup$
I feel like I should point out that curves are topological spaces with additional information. All (irreducible) curves with the same cardinality of points are homeomorphic as topological spaces (having the cofinite topology), and even the affine ones are compact, so "the one point compactification" isn't a compactification in this case, since the affine part of the curve won't be dense.
$endgroup$
– jgon
Jan 26 at 19:05
add a comment |
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$begingroup$
I feel like I should point out that curves are topological spaces with additional information. All (irreducible) curves with the same cardinality of points are homeomorphic as topological spaces (having the cofinite topology), and even the affine ones are compact, so "the one point compactification" isn't a compactification in this case, since the affine part of the curve won't be dense.
$endgroup$
– jgon
Jan 26 at 19:05