Local contributions in $\mathbb{A}^1$-enumerative geometry

Abstract

B\'ezout's theorem is a fundamental result in enumerative geometry: over an algebraically closed field, the intersection of $n$ general hypersurfaces in $\mb{P}^n$ of degrees $d_1,\ldots,d_n$ consists of $d_1\cdots d_n$ points, provided that one counts these intersection points with the appropriate multiplicity.B\'ezout's theorem implies other classical theorems from enumerative geometry, such as the count of the circles of Apollonius. Given three circles in general position, there are eight circles that are simultaneously tangent to the original three. From a B\'ezout-theoretic perspective, this is because the space of circles tangent to a given circle is a quadratic cone in $\mb{P}^3$, and the intersection of three quadratic cones in $\mb{P}^3$ consists of $2^3=8$ points.We prove versions of B\'ezout's theorem and the circles of Apollonius over non-algebraically closed fields. Our work follows the general theme of the $\mb{A}^1$-enumerative geometry program as initiated by Kass--Wickelgren, Levine, and others. We express a global \textit{fixed count} as a sum of \textit{local contributions}, where the local contributions depend on the objects being enumerated. Working over a field $k$, both the fixed count and local contributions are valued in the Grothendieck--Witt ring $\GW(k)$ of isomorphism classes of non-degenerate symmetric bilinear forms over $k$. By taking relevant invariants of bilinear forms over $k$, we obtain a weighted count of intersection points over $k$.While there have been significant developments in the literature on computing fixed counts in $\mb{A}^1$-enumerative geometry, the story of local contributions is nonplussing. Our work on B\'ezout's theorem and the circles of Apollonius focuses especially on describing the local contributions geometrically. We pose the \textit{geometricity problem}, which asks whether one can construct a geometric taxonomy for local contributions in $\mb{A}^1$-enumerative geometry. We show how B\'ezout's theorem answers a na\"ive version of the geometricity problem, and we use the circles of Apollonius to explain why B\'ezout's theorem does not answer a more interesting version of geometricity.For B\'ezout's theorem, we use the $\mb{A}^1$-degree to associate a bilinear form (up to isomorphism) to each intersection point, with the rank of the quadratic form given by the intersection multiplicity. At transverse intersection points, this bilinear form is determined by the gradient vectors of the hypersurfaces. At non-transverse intersection points, one can use a deformation to express the bilinear form as a direct sum over transverse intersections. Using an Euler class from motivic homotopy theory, we show that the direct sum of these ``intersection forms'' is hyperbolic of rank $d_1\cdots d_n$.Our $\mb{A}^1$-enumerative version of B\'ezout's theorem gives the global fixed count for the circles of Apollonius. We also show that the B\'ezout-theoretic local contribution can be viewed as a universal, albeit unsatisfactory, local contribution in $\mb{A}^1$-enumerative geometry. By giving a geometric description of the local contributions associated to the circles of Apollonius, we illustrate the shortcomings of B\'ezout's theorem as a universal local contribution.