An arithmetic count of the lines meeting four lines in 𝐏³

Abstract

We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field k k , this enrichment counts the number of lines meeting four lines defined over k k in P k 3 \mathbf {P}^3_k , with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in A 1 \mathbf {A}^1 -homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms.

A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections

“…This builds on results of Finahin-Kharlamov [FK13], J.L. Kass and the second-named author [KW17], M. Levine [Lev19] [Lev17], S. McKean [McK19], Okonek-Teleman [OT14], S. Pauli [Pau19], J. Solomon [Sol06], P.Srinivasan and the second-named author [SW18], and M. Wendt [Wen18].…”

“…It seems that proving this would take us too far afield, however. If V admits a non-degenerate section, then the results of §8 (showing that the Euler number can be computed in terms of Scheja-Storch forms) imply that this is true, arguing as in [SW18,Lemma 4]. This holds in most cases below.…”

On quadratically enriched excess and residual intersections

“…By this we mean that we will give each of the points of Γ a weight in GW(k) associated to its field of definition and the quadrics Q i and compute the sum of these weights in GW(k). See also [KW21], [SW21], [Lev19], [McK21], [Pau20b], [Pau20c] and [BW20] [CDH20c] for other arithmetic or quadratically enriched counts. In this context, the following definition of a relative orientation of a vector bundle is useful.…”

“…define a pushforward π : GW V Z (X) ∼ = GW T X/k Z (X) → GW(k), and the Euler number of V , with respect to the relative orientation given by (18), is n(V ) := π * [K(V, σ)]. Since V contains an odd dimensional summand, n(V ) is a multiple of the hyperbolic form, and since its classical Euler number is 6, we have n(V ) = 3( 1 + −1 ) (see for example [SW21,Proposition 19]). On the other hand, I/J = kx ⊕ kx 2 is rank 2, whence so is π * [I/J], preventing a possible equality i * c = [i * (I/J)] = [K(V, σ)] in GW V Z (X).…”

$\mathbb{A}^{1}$-Local Degree via Stacks

“…Many classical enumerative problems in algebraic geometry were originally only solved over C. In [10], Kass and Wickelgren develop a toolkit for solving enumerative geometry problems over arbitrary fields (possibly of characteristic not 2). Their program, which has expanded to several other articles ( [11], [14], [21], [13], [15], [9]) and is a source of ongoing research, gives arithmetic enrichments of many enumerative problems which can be solved by counting zeroes of a suitable vector bundle. These authors do this by defining an enriched Euler class of that vector bundle, provided that the vector bundle satisfies certain hypotheses.…”