# Mathematics Colloquia and Seminars

For analytic curves of infinite genus, to get a theory parallel to algebraic geometry one needs to restrict attention to holomorphic sections satisfying some conditions on growth at infinity''. Each such condition effectively attaches an ideal point'' to the curve; this process is similar to compactification. We discuss the algebraic geometry on curves with such ideal points''. Conditions on the lengths of handles'' of the curve are found which ensure the "standard theorems" hold. It turns out that these conditions give no restriction on the density of ideal points on the curve. In particular, such curves may have a dense set of ideal points; these curves have no smooth points at all, and have a purely fractal nature. (Such foam'' curves live near the periphery'' of the corresponding moduli space; one needs to study these curves too, since they may be included in the support of natural measures arising on the moduli spaces.)