A note on the stable equivalence problem

2013/08/12 by Poloni, Pierre-Marie
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1308.2652

Abstract

We provide counterexamples to the stable equivalence problem in every dimension d≥2. That means that we construct hypersurfaces H₁, H₂⊂ℂ^d+1 whose cylinders H₁×ℂ and H₂×ℂ are equivalent hypersurfaces in ℂ^d+2, although H₁ and H₂ themselves are not equivalent by an automorphism of ℂ^d+1. We also give, for every d≥2, examples of two non-isomorphic algebraic varieties of dimension d which are biholomorphic.

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• Affine varieties with equivalent cylinders
• Embeddings of hypersurfaces in affine spaces
• The Stable Equivalence and Cancellation Problems
• Real algebraic surfaces biholomorphically equivalent but not algebraically equivalent
• Degenerate real hypersurfaces in ℂ² with few automorphisms