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In the Satake compactification of abelian surfaces we have the following degeneration of a family of abelian surfaces in $\mathbf{H}_2$

$lim_{t \to \infty}\begin{pmatrix} it & b \\\ b & \tau\end{pmatrix} = \tau.$

Since we have that $M_2$ is an open of $A_2$, it is natural to look for a family of genus 2 curves depending on $t$ which gives the previous family of period matrices.

Can you describe explicitely such a family of genus 2 curves?

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up vote 3 down vote accepted

The analytic solution is easy enough to describe: Compute the gradients of the Theta function at the six odd 2-torsion points, and projectivize these gradients. You now have six points on the projective line. This are the 6 Weierstrass points of the curve, alternatively there is the "Rosenhaon normal form", which expresses the Weierstrass points in terms of (quotients and products of) values of the theta function at even 2-torsion point.

In light of these two construction, and since theta functions involve non algebraic functions, I doubt the existence of a an algebraic expression for the family of curves you want.

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