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Discriminant locus of elliptic K3 surfaces
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Given a complex elliptic K3 surface $picolon Xrightarrow mathbb P^1$, its discriminant locus is the divisor $$D = sum_i = 1^s n_i P_i$$ on $mathbb P^1$ such that $n_i$ is equal to the Euler-Poincaré characteristic of the fiber $pi^-1(P_i)$, where the sum runs over the points $P_i in mathbb P^1$ such that $pi^-1(P_i)$ is singular. It is well known that $deg D = 24$.
Conversely, given an effective divisor $D$ of degree $24$ on $mathbb P^1$, when is it the discriminant locus of a complex elliptic K3 surface?
I am particularly curious about the minimal possible $s$. The maximal Euler-Poincaré characteristic of a singular fiber is $20$, so $s geq 2$. But in case, say, $n_1 = 20$, then the fibration is of type $I_14^*,I_1,I_1,I_1,I_1$ (see Schütt-Schweizer), so indeed $s = 5$. Are smaller $s$ possible?
k3-surfaces elliptic-surfaces
asked 2 hours ago
DCV
321112
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up vote
3
down vote
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Given a complex elliptic K3 surface $picolon Xrightarrow mathbb P^1$, its discriminant locus is the divisor $$D = sum_i = 1^s n_i P_i$$ on $mathbb P^1$ such that $n_i$ is equal to the Euler-Poincaré characteristic of the fiber $pi^-1(P_i)$, where the sum runs over the points $P_i in mathbb P^1$ such that $pi^-1(P_i)$ is singular. It is well known that $deg D = 24$.
Conversely, given an effective divisor $D$ of degree $24$ on $mathbb P^1$, when is it the discriminant locus of a complex elliptic K3 surface?
I am particularly curious about the minimal possible $s$. The maximal Euler-Poincaré characteristic of a singular fiber is $20$, so $s geq 2$. But in case, say, $n_1 = 20$, then the fibration is of type $I_14^*,I_1,I_1,I_1,I_1$ (see Schütt-Schweizer), so indeed $s = 5$. Are smaller $s$ possible?
k3-surfaces elliptic-surfaces
asked 2 hours ago
DCV
321112
3
Since the moduli space of polarized K3 surfaces is hyperbolic, the inequality $sgeq 3$ holds. An example of an elliptic fibration with $s=3$ is provided by the Legendre elliptic curve $y^2= x(x-1)(x-lambda)$ over $mathbbC-0,1$. The total space of this fibration is not K3 though.
– Ariyan Javanpeykar
2 hours ago
Sorry, I actually miswrote something in my comment. I meant to say that the moduli "space" of elliptic curves is hyperbolic. (Although the moduli space of polarized K3 surfaces is hyperbolic as well, it is the moduli space of elliptic curves which plays a role here.) Indeed, if $D$ is the support of the discriminant divisor of the elliptic surface $f:Xto mathbbP^1$, then there is a non-constant morphism $mathbbP^1 setminus D to mathcalM$ induced by the Jacobian of $Xsetminus f^-1Dto mathbbP^1setminus D$, where $mathcalM$ is the moduli of elliptic curves.
– Ariyan Javanpeykar
30 mins ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Given a complex elliptic K3 surface $picolon Xrightarrow mathbb P^1$, its discriminant locus is the divisor $$D = sum_i = 1^s n_i P_i$$ on $mathbb P^1$ such that $n_i$ is equal to the Euler-Poincaré characteristic of the fiber $pi^-1(P_i)$, where the sum runs over the points $P_i in mathbb P^1$ such that $pi^-1(P_i)$ is singular. It is well known that $deg D = 24$.
Conversely, given an effective divisor $D$ of degree $24$ on $mathbb P^1$, when is it the discriminant locus of a complex elliptic K3 surface?
I am particularly curious about the minimal possible $s$. The maximal Euler-Poincaré characteristic of a singular fiber is $20$, so $s geq 2$. But in case, say, $n_1 = 20$, then the fibration is of type $I_14^*,I_1,I_1,I_1,I_1$ (see Schütt-Schweizer), so indeed $s = 5$. Are smaller $s$ possible?
k3-surfaces elliptic-surfaces
asked 2 hours ago
DCV
321112
Given a complex elliptic K3 surface $picolon Xrightarrow mathbb P^1$, its discriminant locus is the divisor $$D = sum_i = 1^s n_i P_i$$ on $mathbb P^1$ such that $n_i$ is equal to the Euler-Poincaré characteristic of the fiber $pi^-1(P_i)$, where the sum runs over the points $P_i in mathbb P^1$ such that $pi^-1(P_i)$ is singular. It is well known that $deg D = 24$.
Conversely, given an effective divisor $D$ of degree $24$ on $mathbb P^1$, when is it the discriminant locus of a complex elliptic K3 surface?
I am particularly curious about the minimal possible $s$. The maximal Euler-Poincaré characteristic of a singular fiber is $20$, so $s geq 2$. But in case, say, $n_1 = 20$, then the fibration is of type $I_14^*,I_1,I_1,I_1,I_1$ (see Schütt-Schweizer), so indeed $s = 5$. Are smaller $s$ possible?
k3-surfaces elliptic-surfaces
k3-surfaces elliptic-surfaces
asked 2 hours ago
DCV
321112
asked 2 hours ago
DCV
321112
asked 2 hours ago
DCV
321112
asked 2 hours ago
DCV
321112
asked 2 hours ago
DCV
321112
321112
3
Since the moduli space of polarized K3 surfaces is hyperbolic, the inequality $sgeq 3$ holds. An example of an elliptic fibration with $s=3$ is provided by the Legendre elliptic curve $y^2= x(x-1)(x-lambda)$ over $mathbbC-0,1$. The total space of this fibration is not K3 though.
– Ariyan Javanpeykar
2 hours ago
Sorry, I actually miswrote something in my comment. I meant to say that the moduli "space" of elliptic curves is hyperbolic. (Although the moduli space of polarized K3 surfaces is hyperbolic as well, it is the moduli space of elliptic curves which plays a role here.) Indeed, if $D$ is the support of the discriminant divisor of the elliptic surface $f:Xto mathbbP^1$, then there is a non-constant morphism $mathbbP^1 setminus D to mathcalM$ induced by the Jacobian of $Xsetminus f^-1Dto mathbbP^1setminus D$, where $mathcalM$ is the moduli of elliptic curves.
– Ariyan Javanpeykar
30 mins ago
add a comment |Â
3
Since the moduli space of polarized K3 surfaces is hyperbolic, the inequality $sgeq 3$ holds. An example of an elliptic fibration with $s=3$ is provided by the Legendre elliptic curve $y^2= x(x-1)(x-lambda)$ over $mathbbC-0,1$. The total space of this fibration is not K3 though.
– Ariyan Javanpeykar
2 hours ago
Sorry, I actually miswrote something in my comment. I meant to say that the moduli "space" of elliptic curves is hyperbolic. (Although the moduli space of polarized K3 surfaces is hyperbolic as well, it is the moduli space of elliptic curves which plays a role here.) Indeed, if $D$ is the support of the discriminant divisor of the elliptic surface $f:Xto mathbbP^1$, then there is a non-constant morphism $mathbbP^1 setminus D to mathcalM$ induced by the Jacobian of $Xsetminus f^-1Dto mathbbP^1setminus D$, where $mathcalM$ is the moduli of elliptic curves.
– Ariyan Javanpeykar
30 mins ago
3
3
Since the moduli space of polarized K3 surfaces is hyperbolic, the inequality $sgeq 3$ holds. An example of an elliptic fibration with $s=3$ is provided by the Legendre elliptic curve $y^2= x(x-1)(x-lambda)$ over $mathbbC-0,1$. The total space of this fibration is not K3 though.
– Ariyan Javanpeykar
2 hours ago
Since the moduli space of polarized K3 surfaces is hyperbolic, the inequality $sgeq 3$ holds. An example of an elliptic fibration with $s=3$ is provided by the Legendre elliptic curve $y^2= x(x-1)(x-lambda)$ over $mathbbC-0,1$. The total space of this fibration is not K3 though.
– Ariyan Javanpeykar
2 hours ago
Sorry, I actually miswrote something in my comment. I meant to say that the moduli "space" of elliptic curves is hyperbolic. (Although the moduli space of polarized K3 surfaces is hyperbolic as well, it is the moduli space of elliptic curves which plays a role here.) Indeed, if $D$ is the support of the discriminant divisor of the elliptic surface $f:Xto mathbbP^1$, then there is a non-constant morphism $mathbbP^1 setminus D to mathcalM$ induced by the Jacobian of $Xsetminus f^-1Dto mathbbP^1setminus D$, where $mathcalM$ is the moduli of elliptic curves.
– Ariyan Javanpeykar
30 mins ago
Sorry, I actually miswrote something in my comment. I meant to say that the moduli "space" of elliptic curves is hyperbolic. (Although the moduli space of polarized K3 surfaces is hyperbolic as well, it is the moduli space of elliptic curves which plays a role here.) Indeed, if $D$ is the support of the discriminant divisor of the elliptic surface $f:Xto mathbbP^1$, then there is a non-constant morphism $mathbbP^1 setminus D to mathcalM$ induced by the Jacobian of $Xsetminus f^-1Dto mathbbP^1setminus D$, where $mathcalM$ is the moduli of elliptic curves.
– Ariyan Javanpeykar
30 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
The minimal $s$ is $3$.
It is attained by several elliptic K3's,
including $y^2 = x^3 + (t^2-t)^4$ which has IV* fibers at
$t = 0, 1, infty$ and no other singular fibers.
The comment by Ariyan Javanpeykar gives one argument that $s$ can be
no smaller.
(This uses characteristic zero; in positive characteristic
$s$ can be as small as $1$, e.g. in characteristic 2
the elliptic K3 surface $y^2 + y = x^3 + t^9$ has only one
reducible fiber, at $t = infty$.)
answered 52 mins ago
Noam D. Elkies
54.8k9194279
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
The minimal $s$ is $3$.
It is attained by several elliptic K3's,
including $y^2 = x^3 + (t^2-t)^4$ which has IV* fibers at
$t = 0, 1, infty$ and no other singular fibers.
The comment by Ariyan Javanpeykar gives one argument that $s$ can be
no smaller.
(This uses characteristic zero; in positive characteristic
$s$ can be as small as $1$, e.g. in characteristic 2
the elliptic K3 surface $y^2 + y = x^3 + t^9$ has only one
reducible fiber, at $t = infty$.)
answered 52 mins ago
Noam D. Elkies
54.8k9194279
add a comment |Â
up vote
4
down vote
The minimal $s$ is $3$.
It is attained by several elliptic K3's,
including $y^2 = x^3 + (t^2-t)^4$ which has IV* fibers at
$t = 0, 1, infty$ and no other singular fibers.
The comment by Ariyan Javanpeykar gives one argument that $s$ can be
no smaller.
(This uses characteristic zero; in positive characteristic
$s$ can be as small as $1$, e.g. in characteristic 2
the elliptic K3 surface $y^2 + y = x^3 + t^9$ has only one
reducible fiber, at $t = infty$.)
answered 52 mins ago
Noam D. Elkies
54.8k9194279
add a comment |Â
up vote
4
down vote
up vote
4
down vote
The minimal $s$ is $3$.
It is attained by several elliptic K3's,
including $y^2 = x^3 + (t^2-t)^4$ which has IV* fibers at
$t = 0, 1, infty$ and no other singular fibers.
The comment by Ariyan Javanpeykar gives one argument that $s$ can be
no smaller.
(This uses characteristic zero; in positive characteristic
$s$ can be as small as $1$, e.g. in characteristic 2
the elliptic K3 surface $y^2 + y = x^3 + t^9$ has only one
reducible fiber, at $t = infty$.)
answered 52 mins ago
Noam D. Elkies
54.8k9194279
The minimal $s$ is $3$.
It is attained by several elliptic K3's,
including $y^2 = x^3 + (t^2-t)^4$ which has IV* fibers at
$t = 0, 1, infty$ and no other singular fibers.
The comment by Ariyan Javanpeykar gives one argument that $s$ can be
no smaller.
(This uses characteristic zero; in positive characteristic
$s$ can be as small as $1$, e.g. in characteristic 2
the elliptic K3 surface $y^2 + y = x^3 + t^9$ has only one
reducible fiber, at $t = infty$.)
answered 52 mins ago
Noam D. Elkies
54.8k9194279
answered 52 mins ago
Noam D. Elkies
54.8k9194279
answered 52 mins ago
Noam D. Elkies
54.8k9194279
answered 52 mins ago
Noam D. Elkies
54.8k9194279
54.8k9194279
add a comment |Â
add a comment |Â
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3
Since the moduli space of polarized K3 surfaces is hyperbolic, the inequality $sgeq 3$ holds. An example of an elliptic fibration with $s=3$ is provided by the Legendre elliptic curve $y^2= x(x-1)(x-lambda)$ over $mathbbC-0,1$. The total space of this fibration is not K3 though.
– Ariyan Javanpeykar
2 hours ago
Sorry, I actually miswrote something in my comment. I meant to say that the moduli "space" of elliptic curves is hyperbolic. (Although the moduli space of polarized K3 surfaces is hyperbolic as well, it is the moduli space of elliptic curves which plays a role here.) Indeed, if $D$ is the support of the discriminant divisor of the elliptic surface $f:Xto mathbbP^1$, then there is a non-constant morphism $mathbbP^1 setminus D to mathcalM$ induced by the Jacobian of $Xsetminus f^-1Dto mathbbP^1setminus D$, where $mathcalM$ is the moduli of elliptic curves.
– Ariyan Javanpeykar
30 mins ago