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Let $a_{i}(t) \in \mathbb{Z}[t]$. We shall denote these by $a_{i}$. The equation $y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}$ is the affine equation for the Weierstrass form of a family of elliptic curves. Under what conditions does this represent a K3 surface?

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If its projectivization is a nonsingular quartic then it is a K3 surface. So Jacobian Criterion to get some conditions maybe? – Matt Jun 14 '11 at 23:35
I think you're asking for the coefficients $a_i$ to be functions of a single variable, so that you get a one-parameter family of elliptic curves. – Scott Carnahan Jun 16 '11 at 10:38
up vote 3 down vote accepted

A good reference for this would be Abhinav Kumar's PhD thesis, which you can find here. In particular, look at Chapter 5, and Section 5.1. If an elliptic surface $y^2+a_1(t)xy+a_3(t)y = x^3+a_2(t)x^2+a_4(t)x+a_6(t)$ is K3, then the degree of $a_i(t)$ must be $\leq 2i$.

I hope this helps.

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