Classification of complex algebraic curves under blow-spherical equivalence

TL;DR

This paper provides a complete classification of complex algebraic curves under blow-spherical equivalence, analyzing properties preserved under such transformations and establishing key invariants like degree.

Contribution

It offers the first comprehensive classification of complex algebraic curves under blow-spherical homeomorphisms, including properties preserved and invariants like degree.

Findings

Degree of curves is preserved under blow-spherical homeomorphisms at infinity.

Complex algebraic sets blow-spherical homeomorphic to Euclidean space are affine linear.

Complete classifications of complex algebraic curves under blow-spherical homeomorphisms.

Abstract

This article is devoted to studying complex algebraic sets under (global) blow-spherical equivalence. The main results of this article are complete classifications of complex algebraic curves. Firstly, we present a complete classification of complex algebraic curves under blow-spherical homeomorphisms at infinity and, then, we present a complete classification of complex algebraic curves under (global) blow-spherical homeomorphisms. We also present several properties of the blow-spherical equivalence. For instance, we prove that the degree of curves is preserved under blow-spherical homeomorphisms at infinity. Another property presented here is a Bernstein-type result which says that a pure dimensional complex algebraic set which is blow-spherical homeomorphic at infinity to a Euclidean space must be an affine linear subspace.