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In this paper, by introducing the notion of "\textit{distributive constant}"of a family of hypersurfaces with respect to a projective variety, we prove asecond main theorem in Nevanlinna theory for meromorphic mappings witharbitrary families of hypersurfaces in projective varieties. Our second maintheorem generalizes and improves previous results for meromorphic mappings withhypersurfaces, in particular for algebraically degenerate mappings and for thefamilies of hypersurfaces in subgeneral position. The analogous results for theholomorphic curves with finite growth index from a complex disc into a projectvariety, and for meromorphic mappings on a complete K\"{a}hler manifold arealso given. For the last aim, we will prove a Schmidt's subspace theorem for anarbitrary families of homogeneous polynomials, which is the counterpart inNumber theory of our second main theorem. Our these results are generalizationsand improvements of all previous results.

Generalizations of degeneracy second main theorem and Schmidt's subspace theorem