Tangent cones at infinity

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