Number Theory Seminar

Speaker: Julia Brandes (Gothenburg)

Title: On the number of linear spaces on hypersurfaces with a prescribed discriminant

Abstract: For a given form $F \in \Z[x_1,\dots,x_s]$ we apply the circle method in order to give an asymptotic estimate of the number of $m$-tuples $\mathbf x_1, \dots, \mathbf x_m$ on the hypersurface $F(X) = 0$ having $\det ( \mathbf x_1, \dots, \mathbf x_m)^t \, (\mathbf x_1, \dots, \mathbf x_m)) = b$. As a corollary, we obtain a count of rational linear spaces contained in the hypersurface $F(\mathbf x) = 0$ having dimension exactly $m$, thus addressing a weakness of previous results.