Uniqueness theorems for meromorphic mappings with few hyperplanes

Let f,g be linearly nondegenerate meromorphic mappings of Cm into CPn. Let be hyperplanes in CPn in general position, such that

(a)

f−1(Hj)=g−1(Hj), for all 1?j?q,

(b)

dim(f−1(Hi)∩f−1(Hj))?m−2 for all 1?i<j?q, and

(c)

f=g on .

It is well known that if q?3n+2, then f≡g. In this paper we show that for every nonnegative integer c there exists positive integer N(c) depending only on c in an explicit way such that the above result remains valid if q?(3n+2−c) and n?N(c). Furthermore, we also show that the coefficient of n in the formula of q can be replaced by a number which is strictly smaller than 3 for all n?0. At the same time, a big number of recent uniqueness theorems are generalized considerably.