Tangent cones at infinity

TL;DR

This paper introduces new tangent cones at infinity for algebraic sets in complex space, establishing their properties and linking them to the geometric structure of the set, including conditions for linearity.

Contribution

It defines and analyzes tangent cones at infinity for algebraic sets, proving that certain cone properties imply the set is an affine linear subspace.

Findings

If the tangent cone at infinity has pure dimension, the set is affine linear.

Properties of tangent cones at infinity relate to the set's structure outside compact regions.

Results extend classical theorems of Whitney and Stutz to the setting at infinity.

Abstract

Let $X\subset\mathbb{C}^m$ be an unbounded pure $k$-dimensional algebraic set. We define the tangent cones $C_{4, \infty}(X)$ and $C_{5,\infty}(X)$ of $X$ at infinity. We establish some of their properties and relations. We prove that $X$ must be an affine linear subspace of $\mathbb{C}^m$ provided that $C_{5, \infty}(X)$ has pure dimension $k$. Also, we study the relation between the tangent cones at infinity and representations of $X$ outside a compact set as a branched covering. Our results can be seen as versions at infinity of results of Whitney and Stutz.