For a smooth projective variety X H P r embedded by the complete linear system, Property N p has been studied for a long time ([5], [11], [12], [7] etc.). On the other hand, Castelnuovo-Mumford regularity conjecture and related problems have been focused for a projective variety which is not necessarily linearly normal ([2], [13], [15], [17], [20] etc.).This paper aims to explain the influence of Property N p on higher normality and defining equations of a smooth variety embedded by a sub-linear system. Also we prove a claim about Property N p of surface scrolls which is a generalization of Green's work in [11] about Property N p for curves.
Let X be a reduced closed subscheme in P n . As a slight generalization of property N p due to Green-Lazarsfeld, one says that X satisfies property N 2,p scheme-theoretically if there is an ideal I generating the ideal sheaf I X/P n such that I is generated by quadrics and there are only linear syzygies up to p-th step (cf. Eisenbud et al. (2005) [8], Vermeire (2001) [20]). Recently, many algebraic and geometric results have been proved for projective varieties satisfying property N 2,p (cf. Choi, Kwak, and Park (2008) [6], Eisenbud et al. (2005) [8], Kwak and Park (2005) [15]). In this case, the Castelnuovo regularity and normality can be obtained by the blowing-up method as reg( X) e + 1 where e is the codimension of a smooth variety X (cf. Bertram, Ein, and Lazarsfeld (2003) [3]). On the other hand, projection methods have been very useful and powerful in bounding Castelnuovo regularity, normality and other classical invariants in geometry (cf. Beheshti and Eisenbud (2010) [2], Kwak (1998) [14], Kwak and Park (2005) [15], Lazarsfeld (1987) [16].We first prove the graded mapping cone theorem on partial eliminations as a general algebraic tool to study syzygies of the noncomplete embedding of X. For applications, we give an optimal bound on the length of zero-dimensional intersections of X and a linear space L in terms of graded Betti numbers. We also deduce several theorems about the relationship between X and its projections with respect to the geometry and syzygies for a projective scheme X satisfying property N 2,p scheme-theoretically. In addition, we give not only interesting information on the regularity of fibers of the projection for the case of N d,p , d 2, but also geometric structures for projections according to moving the center.
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
For a reduced, irreducible projective variety X of degree d and codimension e in P N the Castelnuovo-Mumford regularity regX is defined as the least k such that X is k-regular, i.e.,There is a long standing conjecture about k-regularity (see [5]): regX ≤ d−e+1. Here we show that regX ≤ (d−e+1)+10 for any smooth fivefold and regX ≤ (d−e+1)+20 for any smooth sixfold by extending methods used in [10]. Furthermore, we give a bound for the regularity of a reduced, connected and equidimensional locally Cohen-Macaulay curve or surface in terms of degree d, codimension e and an arithmetic genus ρ a (see Theorem 4.1).
For a projective variety X of codimension 2 in P n+2 defined over the complex number field C, it is traditionally said that X has no apparent (k + 1)-ple points if the (k + 1)-secant lines of X do not fill up the ambient projective space P n+2 , equivalently, the locus of (k + 1)-ple points of a generic projection of X to P n+1 is empty. We show that a smooth threefold in P 5 has no apparent triple points if and only if it is contained in a quadric hypersurface. We also obtain an enumerative formula counting the quadrisecant lines of X passing through a general point of P 5 and give necessary cohomological conditions for smooth threefolds in P 5 without apparent quadruple points. This work is intended to generalize the work of F. Severi [fSe] and A. Aure [Au], where it was shown that a smooth surface in P 4 has no triple points if and only if it is either a quintic elliptic scroll or contained in a hyperquadric. Furthermore we give open questions along these lines.
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