Analysis on some infinite modules, inner projection, and applications

Abstract

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

A bound on the degree of schemes defined by quadratic equations

“…This is an example where [8, Theorem 1] is sharp for p D 2. Furthermore, recent work of Han and Kwak shows that if X P r is a reduced scheme, then property N 2;p behaves well under inner projections (see [9]). Thanks to their result, we can improve Corollary 1.2 in the following way: Corollary 1.3.…”

On syzygies, degree, and geometric properties of projective schemes with property N3,p

“…Then the monomial T can be written as a product of two monomials N 1 and N 2 such that Example 3.6. In [15], the authors have shown that if a non-degenerate reduced scheme X ⊂ P n satisfies N 2,p for some p ≥ 1 then the inner projection from any smooth point of X satisfies at least property N 2,p−1 . So it is natural to ask whether the inner projection from any smooth point of X satisfies at least property N 3,p−1 when X satisfies N 3,p for some p ≥ 1.…”

“…Much attention has been paid to linear syzygies of quadratic schemes (d = 2) and their geometric interpretations (cf. [1,9,[15][16][17]). However, not very much is actually known about algebraic sets satisfying property N d,p , d ≥ 3.…”

On multisecants and the Green-Lazarsfeld index of projective varieties

“…(1) implies that del Pezzo varieties of codimension > 1 are characterized by the condition index(X) = c − 1. To the authors knowledge, the earliest proof of this fact is due to K. Han and S. Kwak [7]. They show that the condition index(X) = c − 1 implies h 0 (P r , I X (2)) ≥ c+1 2 − 1 which enables them to apply Fano's characterization of del Pezzo varieties.…”