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I'm looking for a reference which deals with limits of families of algebraic varieties as the degree increases (or at least keywords from this subject).

For the kind of example I have in mind, consider the exponential function as a convergent power series e^x=1+x+x^2/2!+... on the interval [-1;1] say, and consider the varieties A0:={(x,y)\in\R^2 s.t. y=1}, A1:={(x,y)\in\R^2 s.t. y=1+x}, A2:={(x,y)\in\R^2 s.t. y=1+x+x^2/2!}... which converge (in the euclidean topology) to the graph of the exponential (as in the picture here).

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up vote 4 down vote accepted

See On the limit of families of algebraic subvarieties with unbounded volume, to appear in Astérisque.

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Informative text and references to the literature, thanks! –  Thomas Sauvaget Oct 29 '09 at 13:11

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