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