An arithmetic count of the lines on a smooth cubic surface

Abstract

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$ , generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$ . In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^*/(L^*)^2$ . There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$ , \[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \] where $\operatorname {Tr}_{L/k}$ denotes the trace $\operatorname {GW}(L) \to \operatorname {GW}(k)$ . Taking the rank and signature recovers the results over ${\mathbf {C}}$ and ${\mathbf {R}}$ . To do this, we develop an elementary theory of the Euler number in $\mathbf {A}^1$ -homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

An enriched degree of the Wronski

“…(c.f. [OT14], [KW21], [Mor12, §4.3]) Suppose E → X is a vector bundle of rank n over a smooth, projective n-scheme over a field k. Then we say E is relatively oriented over X if there is an isomorphism…”

Euler Classes: Six-Functors Formalism, Dualities, Integrality and Linear Subspaces of Complete Intersections

“…For a relatively oriented vector bundle V equipped with a section with only isolated zeros, an Euler number was defined in [49, Section 4] as a sum of local indices: The index can be computed explicitly with a formula of Scheja and Storch [67] or of Eisenbud and Levine/Khimshiashvili [29] (see §§2.4 and 2.3) and is also a local degree [48] (this is discussed further in §7). For example, when x is a simple zero of with , the index is given by a well-defined Jacobian of , illustrating the relation with the Poincaré–Hopf formula for topological vector bundles.…”

“…(For the definition of the Jacobian, see the beginning of §6.2.) In [49, Section 4, Corollary 36], it was shown that when and are in a family over of sections with only isolated zeros, where L is a field extension with odd. We strengthen this result by equating and ; this is the main result of §2.…”

“…Theorem 1.1 strengthens [15], removing its hypothesis (2) entirely. It also simplifies the proofs of [49, Theorem 1] and [72, Theorems 1 and 2]: it is no longer necessary to show that the sections of certain vector bundles with nonisolated isolated zeros are of codimension , as in [49, Lemmas 54, 56, and 57] and in [72, Lemma 1], because is independent of .…”

“…Note that if is smooth, this just means that the locally constant functions and on X agree, and that we are given an isomorphism . Hence we recover the definition from [49, Definition 17].…”