Forms in many variables and differing degrees

Abstract

We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin–Peyre conjecture for a smooth and geometrically integral variety $X\subseteq {\mathbb{P}}^m$, provided only that its dimension is large enough in terms of its degree.