Tangent cones 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.