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Let $M$ be an open Riemann surface and $A$ be the punctured cone in$\mathbb{C}^n\setminus\{0\}$ on a smooth projective variety $Y$ in$\mathbb{P}^{n-1}$. Recently, Runge approximation theorems with interpolationfor holomorphic immersions $M\to\mathbb{C}^n$, directed by $A$, have beenproved under the assumption that $A$ is an Oka manifold. We prove analogousresults in the algebraic setting, for regular immersions directed by $A$ from asmooth affine curve $M$ into $\mathbb{C}^n$. The Oka property is naturallyreplaced by the stronger assumption that $A$ is algebraically elliptic, whichit is if $Y$ is uniformly rational. Under this assumption, a homotopy-theoreticnecessary and sufficient condition for approximation and interpolation emerges.We show that this condition is satisfied in many cases of interest.

Regular immersions directed by algebraically elliptic cones