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Applications to \({\mathbb {A}}^1\)-enumerative geometry of the \({\mathbb {A}}^1\)-degree

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Abstract

These are lecture notes from the conference Arithmetic Topology at the Pacific Institute of Mathematical Sciences on applications of Morel’s \({\mathbb {A}}^1\)-degree to questions in enumerative geometry. Additionally, we give a new dynamic interpretation of the \({\mathbb {A}}^1\)-Milnor number inspired by the first-named author’s enrichment of dynamic intersection numbers.

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Notes

  1. A critical point of f is a point where the partials \(\partial _i f\) vanish and a critical point is said to be isolated if there is an open neighborhood around that point not containing other critical points.

  2. A hypersurface of affine (respectively projective) space is the zero locus of a (respectively homogenous) polynomial, and a point x on a scheme X is said to be an isolated singularity if there is a Zariski open neighborhood U of x such that the only singular point of U is x.

  3. The condition that p is an isolated zero of \({\text {grad}}f\) is implied by p being an isolated singularity of X if the characteristic of k is 0.

  4. The reference proves the claim for k algebraically closed. The stated result follows by showing that the coefficients of an algebraic power series lie in a finite extension of k. Moreover, by [18, 19] a perfect extension of a tamely ramified extension of k((t)) lies in \(\cup _{k \subseteq L, n} L((t^{1/n}))\), even without the assumption on the characteristic.

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Acknowledgements

Kirsten Wickelgren was partially supported by NSF CAREER Grant DMS-2001890. Sabrina Pauli gratefully acknowledges support by the RCN Frontier Research Group Project No. 250399 Motivic Hopf Equations. We also wish to thank Joe Rabinoff.

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  1. Department of Mathematics, University of Oslo, Oslo, Norway

    Sabrina Pauli

  2. Department of Mathematics, Duke University, Durham, NC, 27708, USA

    Kirsten Wickelgren

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Pauli, S., Wickelgren, K. Applications to \({\mathbb {A}}^1\)-enumerative geometry of the \({\mathbb {A}}^1\)-degree. Res Math Sci 8, 24 (2021). https://doi.org/10.1007/s40687-021-00255-6

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