  Let $X$ be a reduced closed subscheme in $\mathbb P^n$. As a slight
generalization of property $\textbf{N}_p$ due to Green-Lazarsfeld, we can say
that $X$ satisfies property $\textbf{N}_{2,p}$ scheme-theoretically if there is
an ideal $I$ generating the ideal sheaf $\mathcal I_{X/\P^n}$ such that $I$ is
generated by quadrics and there are only linear syzygies up to $p$-th step (cf.
\cite{EGHP1}, \cite{EGHP2}, \cite{V}). Recently, many algebraic and geometric
results have been proved for projective varieties satisfying property
$\textbf{N}_{2,p}$(cf. \cite{CKP}, \cite{EGHP1}, \cite{EGHP2} \cite {KP}). In
this case, the Castelnuovo regularity and normality can be obtained by the
blowing-up method as $\reg(X)\le e+1$ where $e$ is the codimension of a smooth
variety $X$ (cf. \cite{BEL}). On the other hand, projection methods have been
very useful and powerful in bounding Castelnuovo regularity, normality and
other classical invariants in geometry(cf. \cite{BE}, \cite{K}, \cite{KP},
\cite{L} \cite {R}).
  In this paper, we first prove the graded mapping cone theorem on partial
eliminations as a general algebraic tools and give some applications. Then, we
bound the length of zero dimensional intersection of $X$ and a linear space $L$
in terms of graded Betti numbers and deduce a relation between $X$ and its
projections with respect to the geometry and syzygies in the case of projective
schemes satisfying property $\textbf{N}_{2,p}$ scheme-theoretically. In
addition, we give not only interesting information on the regularity of fibers
and multiple loci for the case of $\textbf{N}_{d,p}, d\ge 2$ but also geometric
structures for projections according to moving the center.
