Graded mapping cone theorem, multisecants and syzygies

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.