Galois group of exceptional curves on the generic del Pezzo surface

Abstract: We give a new proof that the Galois action on the $27$ lines on the generic cubic surface is maximal, i.e., given by the Weyl group $W(E_6)$. Our proof is uniform for all base fields $k$; in particular, the result is new when $k$ has characteristic $2$ or $3$. We show an analogous result for the generic del Pezzo surfaces of all degrees.

As a consequence, we deduce that over any number field or $\F_q(t)$, $100\%$ of cubic surfaces have no Brauer-Manin obstruction. Combined with the recent result that the Brauer-Manin obstruction is the only one for cubic surfaces over global fields of characteristic $p>0$ not equal to $3$, this implies that $100\%$ of cubic surfaces over $\F_q(t)$ with $3\nmid q$ satisfy the Hasse principle.

algebraic geometrynumber theory

Audience: researchers in the topic

Boston University Number Theory Seminar