  In this paper we study holomorphic immersions of open Riemann surfaces into
C^n whose derivative lies in a conical algebraic subvariety A of C^n that is
smooth away from the origin. Classical examples of such A-immersions include
null curves in C^3 which are closely related to minimal surfaces in R^3, and
null curves in SL_2(C) that are related to Bryant surfaces. We establish a
basic structure theorem for the set of all A-immersions of a bordered Riemann
surface, and we prove several approximation and desingularization theorems.
Assuming that A is irreducible and is not contained in any hyperplane, we show
that every A-immersion can be approximated by A-embeddings; this holds in
particular for null curves in C^3. If in addition A-{0} is an Oka manifold,
then A-immersions are shown to satisfy the Oka principle, including the Runge
and the Mergelyan approximation theorems. Another version of the Oka principle
holds when A admits a smooth Oka hyperplane section. This lets us prove in
particular that every open Riemann surface is biholomorphic to a properly
embedded null curve in C^3.
