Forms in many variables and differing degrees (1403.5937v2)

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 projective variety, provided only that its dimension is large enough in terms of its degree.