  In the present paper, we consider upper bounds of higher linear syzygies i.e.
graded Betti numbers in the first linear strand of the minimal free resolutions
of projective varieties in arbitrary characteristic. For this purpose, we first
remind `Partial Elimination Ideals (PEIs)' theory and introduce a new framework
in which one can study the syzygies of embedded projective schemes well using
PEIs theory and the reduction method via inner projections.
  Next we establish fundamental inequalities which govern the relations between
the graded Betti numbers in the first linear strand of an algebraic set $X$ and
those of its inner projection $X_q$. Using these results, we obtain some
natural sharp upper bounds for higher linear syzygies of any nondegenerate
projective variety in terms of the codimension with respect to its own
embedding and classify what the extremal case and next-to-extremal case are.
This is a generalization of Castelnuovo and Fano's results on the number of
quadrics containing a given variety and another characterization of varieties
of minimal degree and del Pezzo varieties from the viewpoint of `syzygies'.
Note that our method could be also applied to get similar results for more
general categories (e.g. connected in codimension one algebraic sets).
