Projective varieties of maximal sectional regularity

Abstract

We study projective varieties X ⊂ P r of dimension n ≥ 2, of codimension c ≥ 3 and of degree d ≥ c + 3 that are of maximal sectional regularity, i.e. varieties for which the Castelnuovo-Mumford regularity reg(C) of a general linear curve section is equal to d − c + 1, the maximal possible value (see [10]). As one of the main results we classify all varieties of maximal sectional regularity. If X is a variety of maximal sectional regularity, then either (a) it is a divisor on a rational normal (n + 1)-fold scroll Y ⊂ P n+3 or else (b) there is an n-dimensional linear subspace F ⊂ P r such that X ∩ F ⊂ F is a hypersurface of degree d − c + 1. Moreover, suppose that n = 2 or the characteristic of the ground field is zero. Then in case (b) we obtain a precise description of X as a birational linear projection of a rational normal n-fold scroll.H i (P r , I X (m − i)) = 0 for all i ≥ 1.The m-regularity condition implies the (m + 1)-regularity condition, so that one defines the Castelnuovo-Mumford regularity reg(X) of X as the least integer m such that X is m-regular. It is well known that if X is m-regular then its homogeneous ideal is generated by forms of degree ≤ m. This algebraic implication of m-regularity has an elementary geometric consequence that any (m + 1)-secant line to X should be contained in X. We say that a linear space L ⊂ P r is k-secant to X if length(X ∩ L) := dim k (O P r /I X + I L ) ≥ k.A well known conjecture due to Eisenbud and Goto (see [6]) says thatObviously this conjecture implies the following conjecture (1.2) X has no proper k-secant line if k > d − c + 1. So far the conjecture (1.1) has been proved only for irreducible but not necessarily smooth curves by Gruson-Lazarsfeld-Peskine[10] and for smooth complex surfaces by H. Pinkham[20] and R. Lazarsfeld[14]. Moreover, in [10] the curves in P r whose regularity takes the maximally possible value d − r + 2 are completely classified: they are either

On Surfaces of Maximal Sectional Regularity

“…For the purposes of the present paper, we notice in particular the following result (see also [BLPS1,Theorem 6.3]): 2.3. Corollary.…”

“…(B) We say that the finite morphism f : X ′ −→ X is almost non-singular if its singular locus Sing(f ) is a finite set. Now, we have the following result (see [BLPS1,Theorem 4.1]). 2.7.…”

“…Let X ⊂ P r be a non-degenerate irreducible projective variety of dimension n ≥ 2, codimension c ≥ 3 and degree d ≥ c + 3. We recall the following classification result on varieties of maximal sectional regularity, which was established in [BLPS1,Theorem 7.1] 2.1. Theorem.…”

“…Preview of results. Section 2: We give a few preliminaries, which mainly rely on results established in [BLPS1]. We also establish a bound for the Castelnuovo-Mumford regularity of varieties which are almost non-singular projections of varieties satisfying the Green-Lazarsfeld property N 2,p (see Theorem 2.9).…”

On hypersurfaces containing projective varieties

“…First, suppose that C is a smooth rational curve of degree d = c + 3 having a 4-secant line. Thus C is a curve of maximal regularity and hence X is a surface of maximal sectional regularity due to [BLPS1]. Note that Next, assume that C is the image of an isomorphic projection of a linearly normal curveC ⊂ P 2+c of arithmetic genus 1 from a point.…”