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Degeneration of certain functions such as theta functions or Green's functions in the moduli space $\overline{\mathcal{M}_g}$ of stable curves of genus $g$ has been studied quite alot. The idea is to take a path $M_t$ with $0\leq t\leq 1$ in $\overline{\mathcal{M}_g}$ such that $M_0$ lies on the normal crossings divisor $\overline{\mathcal{M}_g} - \mathcal{M}_g$ and study the behaviour of the function on $M_t$ as $t\to 0$.

My question is if there an algebraic analogue of this.

Basically I have a function which is only defined on the algebraic variety $\mathcal{M}_{g,\overline{\mathbf{Q}}}$. What would a good algebraic analogue of degeneration be?

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Degenerations are just maps from Spec k[[t]] (or quotients thereof) to the moduli space. Most of what can be done analytically can be (and has been) done algebraically. –  Felipe Voloch Sep 26 '11 at 15:55

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