Curves of infinite genus I Riemann--Roch theorem for small degree

Abstract

The most useful and interesting line bundles over algebraic curves of a very high genus have the ratio δof the degree to the genus close to half-integer values, usually δ\approx 0, δ\approx 1/2, or δ\approx 1; the numeric properties are very different in these three cases. This leads to three different theories for curves of infinite genus. 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. The theory of holomorphic functions on curves with such ``ideal points'' is developed (the variant presented in the first part of the series is tuned to the case δ\approx 0). Conditions on the ``lengths of handles'' of the curve are found which ensure the geometry to be parallel to algebraic geometry. 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 g = \infty 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.)