Tangent cones at infinity

Let Xᵐ be an unbounded pure k-dimensional algebraic set. We define the tangent cones C4, ∞(X) and C5,∞(X) of X at infinity. We establish some of their properties and relations. We prove that X must be an affine linear subspace of Cᵐ provided that C5, ∞(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.