Regular immersions directed by algebraically elliptic cones

Abstract Let 𝑀 be an open Riemann surface and 𝐴 the punctured cone in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="double-struck">C</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>∖</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> \mathbb{C}^{n}\setminus\{0\} on a smooth projective variety 𝑌 in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">P</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> \mathbb{P}^{n-1} . Recently, Runge approximation theorems with interpolation for holomorphic immersions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo stretchy="false">→</m:mo> <m:msup> <m:mi mathvariant="double-struck">C</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> M\to\mathbb{C}^{n} , directed by 𝐴, have been proved under the assumption that 𝐴 is an Oka manifold. We prove analogous results in the algebraic setting, for regular immersions directed by 𝐴 from a smooth affine curve 𝑀 into <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">C</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> \mathbb{C}^{n} . The Oka property is naturally replaced by the stronger assumption that 𝐴 is algebraically elliptic, which it is if 𝑌 is uniformly rational. Under this assumption, a homotopy-theoretic necessary and sufficient condition for approximation and interpolation emerges. We show that this condition is satisfied in many cases of interest.