Space of nonsingular cubics
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The general nonsingular projective cubic is of the form
$$F_lambda = Y^2 Z - X (X - Z) (X - lambda Z), qquad lambda ne 0,1$$
where $F_lambda$ and $F_mu$ are isomorphic if they have the same modulus
$$J(lambda) = frac {27} 4 frac {(1 - lambda + lambda^2)^2} {lambda^2 (1 - lambda)}$$
From this description, I expect the space of nonsingular cubics to be itself a curve (it has a single parameter!), but what kind of curve is it? So far, I can only tell that $mathbb A^1 - { 0,1 }$ is a 6-fold cover of it.
algebraic-curves
asked Dec 23 '18 at 21:39
pyonpyon
26719
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add a comment |
$begingroup$
The general nonsingular projective cubic is of the form
$$F_lambda = Y^2 Z - X (X - Z) (X - lambda Z), qquad lambda ne 0,1$$
where $F_lambda$ and $F_mu$ are isomorphic if they have the same modulus
$$J(lambda) = frac {27} 4 frac {(1 - lambda + lambda^2)^2} {lambda^2 (1 - lambda)}$$
From this description, I expect the space of nonsingular cubics to be itself a curve (it has a single parameter!), but what kind of curve is it? So far, I can only tell that $mathbb A^1 - { 0,1 }$ is a 6-fold cover of it.
algebraic-curves
asked Dec 23 '18 at 21:39
pyonpyon
26719
$endgroup$
$begingroup$
Do you know about the $j$-invariant?
$endgroup$
– Lubin
Dec 23 '18 at 21:56
$begingroup$
@Lubin: Nope. I'm reading about it right now.
$endgroup$
– pyon
Dec 23 '18 at 21:56
add a comment |
$begingroup$
The general nonsingular projective cubic is of the form
$$F_lambda = Y^2 Z - X (X - Z) (X - lambda Z), qquad lambda ne 0,1$$
where $F_lambda$ and $F_mu$ are isomorphic if they have the same modulus
$$J(lambda) = frac {27} 4 frac {(1 - lambda + lambda^2)^2} {lambda^2 (1 - lambda)}$$
From this description, I expect the space of nonsingular cubics to be itself a curve (it has a single parameter!), but what kind of curve is it? So far, I can only tell that $mathbb A^1 - { 0,1 }$ is a 6-fold cover of it.
algebraic-curves
asked Dec 23 '18 at 21:39
pyonpyon
26719
$endgroup$
The general nonsingular projective cubic is of the form
$$F_lambda = Y^2 Z - X (X - Z) (X - lambda Z), qquad lambda ne 0,1$$
where $F_lambda$ and $F_mu$ are isomorphic if they have the same modulus
$$J(lambda) = frac {27} 4 frac {(1 - lambda + lambda^2)^2} {lambda^2 (1 - lambda)}$$
From this description, I expect the space of nonsingular cubics to be itself a curve (it has a single parameter!), but what kind of curve is it? So far, I can only tell that $mathbb A^1 - { 0,1 }$ is a 6-fold cover of it.
algebraic-curves
algebraic-curves
asked Dec 23 '18 at 21:39
pyonpyon
26719
asked Dec 23 '18 at 21:39
pyonpyon
26719
asked Dec 23 '18 at 21:39
pyonpyon
26719
asked Dec 23 '18 at 21:39
pyonpyon
26719
asked Dec 23 '18 at 21:39
pyonpyon
26719
26719
$begingroup$
Do you know about the $j$-invariant?
$endgroup$
– Lubin
Dec 23 '18 at 21:56
$begingroup$
@Lubin: Nope. I'm reading about it right now.
$endgroup$
– pyon
Dec 23 '18 at 21:56
add a comment |
$begingroup$
Do you know about the $j$-invariant?
$endgroup$
– Lubin
Dec 23 '18 at 21:56
$begingroup$
@Lubin: Nope. I'm reading about it right now.
$endgroup$
– pyon
Dec 23 '18 at 21:56
$begingroup$
Do you know about the $j$-invariant?
$endgroup$
– Lubin
Dec 23 '18 at 21:56
$begingroup$
Do you know about the $j$-invariant?
$endgroup$
– Lubin
Dec 23 '18 at 21:56
$begingroup$
@Lubin: Nope. I'm reading about it right now.
$endgroup$
– pyon
Dec 23 '18 at 21:56
$begingroup$
@Lubin: Nope. I'm reading about it right now.
$endgroup$
– pyon
Dec 23 '18 at 21:56
add a comment |
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$begingroup$
Do you know about the $j$-invariant?
$endgroup$
– Lubin
Dec 23 '18 at 21:56
$begingroup$
@Lubin: Nope. I'm reading about it right now.
$endgroup$
– pyon
Dec 23 '18 at 21:56