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arXiv · 2406.12166

Universal polynomials for multi-singularity loci of maps

Toru Ohmoto
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Abstract

In the present paper, we prove the existence of universal polynomials which express multi-singularity loci classes of prescribed types for proper morphisms between smooth schemes over an algebraically closed field of characteristic zero -- we call them Thom polynomials for multi-singularity types of maps. It has been referred to as the Thom-Kazarian principle and unsolved for a long time. This result solidifies the foundation for a general enumerative theory of singularities of maps which is applicable to a broad range of problems in classical and modern algebraic geometry. In particular, it would contribute to a satisfactory answer to the rest of (an advanced form of) Hilbert's 15th problem and connect such classics to recent new interests in enumerations inspired by mathematical physics and other fields. A main feature of our proof is a striking use of algebro-geometric cohomology operations. Somewhat surprisingly, when trying to grasp a full perspective of classical enumerative geometry, we will inevitably encounter algebraic cobordism and motivic cohomology.

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Toru Ohmoto. 2024-06-18. Universal polynomials for multi-singularity loci of maps. https://arxiv.org/abs/2406.12166

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