An arithmetic count of the lines on a smooth cubic surface

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.

THE REAL GRADED BRAUER GROUP

We introduce a version of the Brauer–Wall group for Real vector bundles of algebras (in the sense of Atiyah) and compare it to the topological analogue of the Witt group. For varieties over the reals, these invariants capture the topological parts of the Brauer–Wall and Witt groups.

Level bounds for exceptional quantum subgroups in rank two

There is a long-standing belief that the modular tensor categories [Formula: see text], for [Formula: see text] and finite-dimensional simple complex Lie algebras [Formula: see text], contain exceptional connected étale algebras (sometimes called quantum subgroups) at only finitely many levels [Formula: see text]. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here, we confirm this conjecture when [Formula: see text] has rank 2, contributing proofs and explicit bounds when [Formula: see text] is of type [Formula: see text] or [Formula: see text], adding to the previously known positive results for types [Formula: see text] and [Formula: see text].