A birational Nevanlinna constant and its consequences
Min Ru, Paul Vojta
Abstract
The purpose of this paper is to modify the notion of the Nevanlinna constant Nev(D) introduced by the first author (see [Ru15] and [Ru17]) for an effective Cartier divisor on a projective variety X. The modified notion is called the birational Nevanlinna constant and is denoted by Nev bir (D). The goal of Nev(D) and Nev bir (D) is to measure what is possible using the filtration method developed by Corvaja and Zannier and, independently, by Evertse and Ferretti. By computing Nev bir (D) using subsequent work of Autissier [Aut11], we establish a general result (see the General Theorem in Section 1), in both the arithmetic and complex cases, which extends the results of Evertse-Ferretti [EF08] and of Ru [Ru09] to general divisors. The notion Nev bir (D) originally came from applications involving Weil functions, but it also can be defined in terms of local effectivity of Cartier divisors after lifting by a proper birational map.