  The aim of this paper is to start a systematic investigation of the
arithmetic degree of projective schemes as introduced by D. Bayer and D.
Mumford. One main theme concerns itself with the behaviour of this arithmetic
degree under hypersurface sections. The notion of arithmetic degree involves
the new concept of length-multiplicity of embedded primary ideals. Therefore it
is much harder to control the arithmetic degree under a hypersurface section
than in the case for the classical degree theory. Nevertheless it has important
and interesting applications. We describe such applications to the
Castelnuovo-Mumford regularity and to Bezout-type theorems.
