The purpose of this paper has twofold. The first is to establish a secondmain theorem with truncated counting functions for algebraically nondegeneratemeromorphic mappings into an arbitrary projective variety intersecting a familyof hypersurfaces in subgeneral position. In our result, the truncation level ofthe counting functions is estimated explicitly. Our result is an extension ofthe classical second main theorem of H. Cartan, also is a generalization of therecent second main theorem of M. Ru and improves some recent results. Thesecond purpose of this paper is to give another proof for the second maintheorem for the special case where the projective variety is a projectivespace, by which the truncation level of the counting functions is estimatedbetter than that of the general case.
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