On varieties of almost minimal degree I: Secant loci of rational normal scrolls

M. Brodmann, Euisung Park

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

To provide a geometrical description of the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let X̃ ⊂ P{doubel-struck}Kr+1 be a variety of minimal degree and of codimension at least 2, and consider Xp=πp(X̃)⊂PKr where p ε P{doubel-struck}Kr+1\X̃. By Brodmann and Schenzel (2007) [1], it turns out that the cohomological and local properties of Xp are governed by the secant locus σp(X̃) of X̃ with respect to p.Along these lines, the present paper is devoted to giving a geometric description of the secant stratification of X̃, that is of the decomposition of P{doubel-struck}Kr+1 via the types of secant loci. We show that there are at most six possibilities for the secant locus σ p(X̃), and we precisely describe each stratum of the secant stratification of X̃, each of which turns out to be a quasi-projective variety.As an application, we obtain a different geometrical description of non-normal del Pezzo varieties X ⊂ P{doubel-struck}Kr, first classified by Fujita (1985) [3, Theorem 2.1(a)] by providing a complete list of pairs (X̃,p), where X̃ ⊂ P{doubel-struck}Kr+1 is a variety of minimal degree, p ε P{doubel-struck}Kr+1\X̃ and Xp=X ⊂ P{doubel-struck}Kr.

Original languageEnglish
Pages (from-to)2033-2043
Number of pages11
JournalJournal of Pure and Applied Algebra
Volume214
Issue number11
DOIs
Publication statusPublished - 2010 Nov 1

Fingerprint

Chord or secant line
Locus
Projective Variety
Stratification
Codimension
Local Properties
del operator
Exceed
Projection
Decompose
Line
Theorem

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

On varieties of almost minimal degree I : Secant loci of rational normal scrolls. / Brodmann, M.; Park, Euisung.

In: Journal of Pure and Applied Algebra, Vol. 214, No. 11, 01.11.2010, p. 2033-2043.

Research output: Contribution to journalArticle

@article{c3327bca5188491da98214d1e8f886c7,
title = "On varieties of almost minimal degree I: Secant loci of rational normal scrolls",
abstract = "To provide a geometrical description of the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let X̃ ⊂ P{doubel-struck}Kr+1 be a variety of minimal degree and of codimension at least 2, and consider Xp=πp(X̃)⊂PKr where p ε P{doubel-struck}Kr+1\X̃. By Brodmann and Schenzel (2007) [1], it turns out that the cohomological and local properties of Xp are governed by the secant locus σp(X̃) of X̃ with respect to p.Along these lines, the present paper is devoted to giving a geometric description of the secant stratification of X̃, that is of the decomposition of P{doubel-struck}Kr+1 via the types of secant loci. We show that there are at most six possibilities for the secant locus σ p(X̃), and we precisely describe each stratum of the secant stratification of X̃, each of which turns out to be a quasi-projective variety.As an application, we obtain a different geometrical description of non-normal del Pezzo varieties X ⊂ P{doubel-struck}Kr, first classified by Fujita (1985) [3, Theorem 2.1(a)] by providing a complete list of pairs (X̃,p), where X̃ ⊂ P{doubel-struck}Kr+1 is a variety of minimal degree, p ε P{doubel-struck}Kr+1\X̃ and Xp=X ⊂ P{doubel-struck}Kr.",
author = "M. Brodmann and Euisung Park",
year = "2010",
month = "11",
day = "1",
doi = "10.1016/j.jpaa.2010.02.009",
language = "English",
volume = "214",
pages = "2033--2043",
journal = "Journal of Pure and Applied Algebra",
issn = "0022-4049",
publisher = "Elsevier",
number = "11",

}

TY - JOUR

T1 - On varieties of almost minimal degree I

T2 - Secant loci of rational normal scrolls

AU - Brodmann, M.

AU - Park, Euisung

PY - 2010/11/1

Y1 - 2010/11/1

N2 - To provide a geometrical description of the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let X̃ ⊂ P{doubel-struck}Kr+1 be a variety of minimal degree and of codimension at least 2, and consider Xp=πp(X̃)⊂PKr where p ε P{doubel-struck}Kr+1\X̃. By Brodmann and Schenzel (2007) [1], it turns out that the cohomological and local properties of Xp are governed by the secant locus σp(X̃) of X̃ with respect to p.Along these lines, the present paper is devoted to giving a geometric description of the secant stratification of X̃, that is of the decomposition of P{doubel-struck}Kr+1 via the types of secant loci. We show that there are at most six possibilities for the secant locus σ p(X̃), and we precisely describe each stratum of the secant stratification of X̃, each of which turns out to be a quasi-projective variety.As an application, we obtain a different geometrical description of non-normal del Pezzo varieties X ⊂ P{doubel-struck}Kr, first classified by Fujita (1985) [3, Theorem 2.1(a)] by providing a complete list of pairs (X̃,p), where X̃ ⊂ P{doubel-struck}Kr+1 is a variety of minimal degree, p ε P{doubel-struck}Kr+1\X̃ and Xp=X ⊂ P{doubel-struck}Kr.

AB - To provide a geometrical description of the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let X̃ ⊂ P{doubel-struck}Kr+1 be a variety of minimal degree and of codimension at least 2, and consider Xp=πp(X̃)⊂PKr where p ε P{doubel-struck}Kr+1\X̃. By Brodmann and Schenzel (2007) [1], it turns out that the cohomological and local properties of Xp are governed by the secant locus σp(X̃) of X̃ with respect to p.Along these lines, the present paper is devoted to giving a geometric description of the secant stratification of X̃, that is of the decomposition of P{doubel-struck}Kr+1 via the types of secant loci. We show that there are at most six possibilities for the secant locus σ p(X̃), and we precisely describe each stratum of the secant stratification of X̃, each of which turns out to be a quasi-projective variety.As an application, we obtain a different geometrical description of non-normal del Pezzo varieties X ⊂ P{doubel-struck}Kr, first classified by Fujita (1985) [3, Theorem 2.1(a)] by providing a complete list of pairs (X̃,p), where X̃ ⊂ P{doubel-struck}Kr+1 is a variety of minimal degree, p ε P{doubel-struck}Kr+1\X̃ and Xp=X ⊂ P{doubel-struck}Kr.

UR - http://www.scopus.com/inward/record.url?scp=77952882434&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952882434&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2010.02.009

DO - 10.1016/j.jpaa.2010.02.009

M3 - Article

AN - SCOPUS:77952882434

VL - 214

SP - 2033

EP - 2043

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 11

ER -