On Surfaces of Maximal Sectional Regularity

Brodmann, Markus; Lee, Wanseok; Park, Euisung; Schenzel, Peter

doi:10.11650/tjm/7753

Cited by 3 publications

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References 18 publications

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“…But it is unknown if X is a singular variety. In this direction, the authors in [2] study various cohomological and homological properties of X when it is a surface of maximal sectional regularity. In particular, it is shown that such a surface achieves the regularity bound in (1.1).…”

Section: Introductionmentioning

confidence: 99%

Projective varieties of maximal sectional regularity

Brodmann

Lee

Park

et al. 2017

Journal of Pure and Applied Algebra

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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

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Section: Introductionmentioning

confidence: 99%

Projective varieties of maximal sectional regularity

Brodmann

Lee

Park

et al. 2017

Journal of Pure and Applied Algebra

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On zero-dimensional linear sections of surfaces of maximal sectional regularity

Lee

Park

2022

J. Algebra Appl.

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Let [Formula: see text] be an [Formula: see text]-dimensional nondegenerate irreducible projective variety of degree [Formula: see text] and codimension [Formula: see text]. For [Formula: see text] and a [Formula: see text]-dimensional linear subspace [Formula: see text] satisfying [Formula: see text], [Formula: see text] is defined as the possibly maximal length of the scheme theoretic intersection [Formula: see text]. Then it is well known that [Formula: see text] if [Formula: see text] is a curve. Also it was generalized by Noma [Multisecant lines to projective varieties, Projective Varieties with Unexpected Properties (Walter de Gruyter, GmbH and KG, Berlin, 2005), pp. 349–359] that [Formula: see text] when [Formula: see text] is locally Cohen–Macaulary. On the other hand, the possible values of [Formula: see text] are unknown if [Formula: see text] is not locally Cohen–Macaulay. In this paper, we construct surfaces [Formula: see text] of maximal sectional regularity (which are not locally Cohen–Macaulay) and of degree [Formula: see text] for every [Formula: see text] such that [Formula: see text] for all [Formula: see text].

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On hyperquadrics containing projective varieties

Park¹

2020

Forum Mathematicum

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AbstractClassical Castelnuovo Lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension c is at most {{{c+1}\choose{2}}} and the equality is attained if and only if the variety is of minimal degree. Also G. Fano’s generalization of Castelnuovo Lemma implies that the next case occurs if and only if the variety is a del Pezzo variety. Recently, these results are extended to the next case in [E. Park, On hypersurfaces containing projective varieties, Forum Math. 27 2015, 2, 843–875]. This paper is intended to complete the classification of varieties satisfying at least {{{c+1}\choose{2}}-3} linearly independent quadratic equations. Also we investigate the zero set of those quadratic equations and apply our results to projective varieties of degree {\geq 2c+1}.

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On Surfaces of Maximal Sectional Regularity

Cited by 3 publications

References 18 publications

Projective varieties of maximal sectional regularity

Projective varieties of maximal sectional regularity

On zero-dimensional linear sections of surfaces of maximal sectional regularity

On hyperquadrics containing projective varieties

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