TY - JOUR

T1 - On varieties of almost minimal degree I

T2 - Secant loci of rational normal scrolls

AU - Brodmann, M.

AU - Park, E.

N1 - Funding Information:
The first named author thanks the KIAS in Seoul and the KAIST in Daejeon for their hospitality and financial support offered during the preparation of this paper. The second named author was supported by the Korea Research Foundation Grant by the Korean Government (1KRF-352-2006-2-C00002). This paper was started when the second named author was conducting Post Doctoral Research at the Institute of Mathematics in the University of Zurich. He thanks them for their hospitality. The authors also thank the referee for his/her careful study of the manuscript and the improvements he/she suggested.

PY - 2010/11

Y1 - 2010/11

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.

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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 -