The dimension growth conjecture

Abstract

For any n≥3, let V⊂ℙⁿ⁻¹ be an irreducible variety of degree d whose ideal is generated by forms defined over the rationals. In this degree of generality one might still ask whether anything meaningful can be said about the corresponding counting function N _v (B), as defined in (1.6). In contrast to the preceding chapter, where precise asymptotic formulae were sought for N _U (B) for suitable open subsets U⊆V, the aim of the present chapter is to seek general upper bounds for the full counting function N _V (B), making as few assumptions on V as possible.