Aspects of Enumerative Geometry with Quadratic Forms

  • Marc Levine

    Fakultät Mathematik, Universität Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany
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Using the motivic stable homotopy category over a field kk, a smooth variety XX over kk has an Euler characteristic χ(X/k)\chi(X/k) in the Grothendieck-Witt ring GW(k)\operatorname{GW}(k). The rank of χ(X/k)\chi(X/k) is the classical Z\mathbb{Z}-valued Euler characteristic, defined using singular cohomology or étale cohomology, and the signature of χ(X/k)\chi(X/k) under a real embedding σ:kR\sigma:k\to \mathbb{R} gives the topological Euler characteristic of the real points Xσ(R)X^\sigma(\mathbb{R}).

We develop tools to compute χ(X/k)\chi(X/k), assuming kk has characteristic 2\neq 2 and apply these to refine some classical formulas in enumerative geometry, such as formulas for the top Chern class of the dual, symmetric powers and tensor products of bundles, to identities for the Euler classes in Chow-Witt groups. We also refine the classical Riemann-Hurwitz formula to an identity in GW(k)\operatorname{GW}(k) and compute χ(X/k)\chi(X/k) for hypersurfaces in Pkn+1\mathbb{P}^{n+1}_k defined by a polynomial of the form i=0n+1aiXim\sum_{i=0}^{n+1}a_iX_i^m; this latter includes the case of an arbitrary quadric hypersurface.

This paper is a revision of [M. Levine,"Toward an enumerative geometry with quadratic forms", Preprint, arXiv:1703.03049v3].

Cite this article

Marc Levine, Aspects of Enumerative Geometry with Quadratic Forms. Doc. Math. 25 (2020), pp. 2179–2239

DOI 10.4171/DM/797