Open Collections

Description

This is joint work with Tim Browning. The talk concerns the variety $X_1Y_1^2+X_2Y_2^2+X_3Y_3^2+X_4Y_4^2=0$ in $P^3 \times P^3$. The height of a point $(x,y)$ is given by $H(x)^3H(y)^2$, and Manin's conjecture predicts asymptotically $cB \log B$ points of height at most $B$. To obtain this one must exclude points on the subvariety $x_1x_2x_3x_4=0$; but in order to achieve the Peyre constant one must exclude an infinite number of subvarieties in which $x_1x_2x_3x_4$ is a square. By combining the circle method with lattice point counting techniques we are able to prove Manin's conjecture for this example, and the talk will give an overview of the various ingredients, and the way that they fit together.