On tangent cones at infinity of algebraic varieties

TL;DR

This paper proves that geometric and algebraic tangent cones at infinity of complex algebraic varieties are equivalent, providing a geometric characterization and computational methods via Gr"obner bases.

Contribution

It establishes the equivalence of geometric and algebraic tangent cones at infinity for complex algebraic varieties and introduces computational techniques using Gr"obner bases.

Findings

Geometric and algebraic tangent cones at infinity coincide.

Explicit computation of tangent cones at infinity using Gr"obner bases.

A geometric characterization of tangent cones at infinity.

Abstract

In this paper, we establish the following version at infinity of Whitney's theorem [6, 7]: Geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of geometric tangent cones at infinity and using the global \L ojasiewicz inequality with explicit exponents for complex algebraic varieties. We also show that tangent cones at infinity of complex algebraic varieties can be computed using Gr\"obner bases.