# On a conjecture of Poonen and
Voloch I: Probabilistic models for counting rational
points on random Fano hypersurfaces

## Will Sawin

Poonen and Voloch have conjectured that almost every degree \(d\) Fano
hypersurface in \(\mathbf P^n\) defined over the field of rational numbers satisfies
the Hasse principle. In joint work with T. Browning and P. Le Boudec, we
establish this conjecture under the mild assumption that \(n > d+1\). This talk
is the first of two talks. Our goal in this first talk will be to introduce
our main results and to present the strategy of the proof.