A projective scheme $X$ is called `quadratic' if $X$ is scheme-theoretically cut out by homogeneous equations of degree 2. Furthermore, we say $X$ satisfies `property $\textbf{N}_{2,p}$' if it is quadratic and the quadratic ideal has only linear syzygies up to first $p$-th steps. In the present paper, we compare the linear syzygies of the inner projections with those of $X$ and obtain a theorem on `embedded linear syzygies' as one of our main results. This is the natural projection-analogue of `restricting linear syzygies' in the linear section case, \cite{EGHP1}. As an immediate corollary, we show that the inner projections of $X$ satisfy property $\textbf{N}_{2,p-1}$ for any reduced scheme $X$ with property $\textbf{N}_{2,p}$. Moreover, we also obtain the neccessary lower bound $(\codim X)\cdot p -\frac{p(p-1)}{2}$, which is sharp, on the number of quadrics vanishing on $X$ in order to satisfy $\textbf{N}_{2,p}$ and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme $X$. These results admit an interesting `syzygetic' rigidity theorem on property $\textbf{N}_{2,p}$ which leads the classifications of extremal and next to extremal cases. For these results we develope the elimination mapping cone theorem for infinitely generated graded modules and improve the partial elimination ideal theory initiated by M. Green. This new method allows us to treat a wider class of projective schemes which can not be covered by the Koszul cohomology techniques, because these are not projectively normal in general.Comment: 22 pages, minor changes (example 3.12 corrected, references updated, etc.), to appear in Trans. of Amer. Math. So

Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X . Let W k be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci W k for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1, 3, 3), and curves. An intermediate result provides a lower bound on the dimension of any GL n orbit of a homogeneous form.

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 Xq. 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).

In this paper we study singularities of third secant varieties of Veronese embedding v d (P n ), which corresponds to the variety of symmetric tensors of border rank at most three in (C n+1 ) ⊗d .