American Journal of Mathematics

abstract:

The purpose of this paper is to modify the notion of the Nevanlinna constant ${\rm Nev}(D)$ introduced by the first author for an effective Cartier divisor on a projective variety $X$. The modified notion is called the {\it birational Nevanlinna constant} and is denoted by ${\rm Nev}_{{\rm bir}}(D)$. The goal of ${\rm Nev}(D)$ and ${\rm Nev}_{{\rm bir}}(D)$ is to measure what is possible using the filtration method introduced by Faltings and W\"{u}stholz, and further developed by Corvaja and Zannier and, independently, by Evertse and Ferretti. By computing ${\rm Nev}_{{\rm bir}}(D)$ using subsequent work of Autissier, 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 and of Ru to general divisors. The notion ${\rm Nev}_{{\rm bir}}(D)$ 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.