Timeline for Discriminant locus of elliptic K3 surfaces

6 events

when toggle format	what		by	license	comment
Oct 9, 2018 at 6:26	vote	accept	Davide Cesare Veniani
Oct 8, 2018 at 20:46	history	edited	Ariyan Javanpeykar		Added one tag.
Oct 8, 2018 at 14:46	comment	added	Ariyan Javanpeykar		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 $f:X\to \mathbb{P}^1$, then there is a non-constant morphism $\mathbb{P}^1 \setminus D \to \mathcal{M}$ induced by the Jacobian of $X\setminus f^{-1}D\to \mathbb{P}^1\setminus D$, where $\mathcal{M}$ is the moduli of elliptic curves.
Oct 8, 2018 at 14:24	answer	added	Noam D. Elkies		timeline score: 9
Oct 8, 2018 at 12:51	comment	added	Ariyan Javanpeykar		Since the moduli space of polarized K3 surfaces is hyperbolic, the inequality $s\geq 3$ 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 $\mathbb{C}-\{0,1\}$. The total space of this fibration is not K3 though.
Oct 8, 2018 at 12:47	history	asked	Davide Cesare Veniani	CC BY-SA 4.0