On syzygies of Veronese embedding of arbitrary projective varieties

Euisung Park

doi:10.1016/j.jalgebra.2009.03.022

On syzygies of Veronese embedding of arbitrary projective varieties

Euisung Park

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

For a very ample line bundle L on a projective scheme X, let φ_Lℓ : X {right arrow, hooked} P H⁰ (X, L^ℓ), ℓ ≥ 1, be the embedding defined by the complete linear series | L^ℓ |. In this paper we study the problem how the Castelnuovo-Mumford regularity of φ_L (X) effects on the defining equations of φ_Lℓ (X) and the syzygies among them. We show that if φ_L (X) ⊂ P H⁰ (X, L) is m-regular, then (X, L^ℓ) satisfies property N_{2 ℓ - m + 1} for frac(m, 2) ≤ ℓ ≤ m - 2, and (X, L^ℓ) satisfies property N_ℓ for all ℓ ≥ m - 1.

Original language	English
Pages (from-to)	108-121
Number of pages	14
Journal	Journal of Algebra
Volume	322
Issue number	1
DOIs	https://doi.org/10.1016/j.jalgebra.2009.03.022
Publication status	Published - 2009 Jul 1

Keywords

Algebraic geometry
Minimal free resolution
Projective variety

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2009.03.022

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abstract = "For a very ample line bundle L on a projective scheme X, let φLℓ : X {right arrow, hooked} P H0 (X, Lℓ), ℓ ≥ 1, be the embedding defined by the complete linear series | Lℓ |. In this paper we study the problem how the Castelnuovo-Mumford regularity of φL (X) effects on the defining equations of φLℓ (X) and the syzygies among them. We show that if φL (X) ⊂ P H0 (X, L) is m-regular, then (X, Lℓ) satisfies property N2 ℓ - m + 1 for frac(m, 2) ≤ ℓ ≤ m - 2, and (X, Lℓ) satisfies property Nℓ for all ℓ ≥ m - 1.",

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N2 - For a very ample line bundle L on a projective scheme X, let φLℓ : X {right arrow, hooked} P H0 (X, Lℓ), ℓ ≥ 1, be the embedding defined by the complete linear series | Lℓ |. In this paper we study the problem how the Castelnuovo-Mumford regularity of φL (X) effects on the defining equations of φLℓ (X) and the syzygies among them. We show that if φL (X) ⊂ P H0 (X, L) is m-regular, then (X, Lℓ) satisfies property N2 ℓ - m + 1 for frac(m, 2) ≤ ℓ ≤ m - 2, and (X, Lℓ) satisfies property Nℓ for all ℓ ≥ m - 1.

AB - For a very ample line bundle L on a projective scheme X, let φLℓ : X {right arrow, hooked} P H0 (X, Lℓ), ℓ ≥ 1, be the embedding defined by the complete linear series | Lℓ |. In this paper we study the problem how the Castelnuovo-Mumford regularity of φL (X) effects on the defining equations of φLℓ (X) and the syzygies among them. We show that if φL (X) ⊂ P H0 (X, L) is m-regular, then (X, Lℓ) satisfies property N2 ℓ - m + 1 for frac(m, 2) ≤ ℓ ≤ m - 2, and (X, Lℓ) satisfies property Nℓ for all ℓ ≥ m - 1.

KW - Algebraic geometry

KW - Minimal free resolution

KW - Projective variety

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On syzygies of Veronese embedding of arbitrary projective varieties

Abstract

Keywords

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