Sijong Kwak

For a smooth projective variety X ⊂ ℙr embedded by the complete linear system, Property Np 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 Np on higher normality and defining equations of a smooth variety embedded by a sub-linear s

Abstract Let X be a nondegenerate integral subscheme of dimension n and degree d in ℙ N defined over the complex number field ℂ. X is said to be k -regular if H i (ℙ N , ℐ X ( k – i )) = 0 for all i ≧ 1, where ℐ X is the sheaf of ideals of ℐ ℙ N and Castelnuovo-Mumford regularity reg( X ) of X is defined as the least such k . There is a well-known conjecture concerning k -regularity: reg( X ) ≦ deg( X ) – codim( X ) + 1. This regularity conjecture including the classification of borderline examp

Let $X \subseteq \mathbb{P}^r$ be a non-degenerate smooth projective variety of dimension $n$, codimension $e$, and degree $d$ defined over an algebraically closed field of characteristic zero. In this paper, we first show that $\text{reg} (\mathcal{O}_X) \leq d-e$, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCulloug

We intend to give a classification of smooth nondegenerate projective varieties admitting extremal or next to extremal curvilinear secant subspaces. Gruson, Lazarsfeld and Peskine classified all projective integral curves with extremal secant lines. On the other hand, if a locally Cohen-Macaulay variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Superscript n Baseline subset-of double-struck upper P Superscript n plus e">