Gröbner scheme in the Hilbert scheme and complete intersection monomial ideals

2017/09/03 by Yuta Kambe, Kambe, Yuta
Computer Science · Mathematics · #13F20 #13P10 #14C05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1709.00701

openalex publication_date 2017/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k be a commutative ring and S=k[x₀, …, x_n] be a polynomial ring over k with a monomial order. For any monomial ideal J, there exists an affine k-scheme of finite type, called Gröbner scheme, which parameterizes all homogeneous reduced Gröbner bases in S whose initial ideal is J. Here we functorially show that the Gröbner scheme is a locally closed subscheme of the Hilbert scheme if J is a saturated ideal. In the process, we also show that the Gröbner scheme consists of complete intersections if J defines a complete intersection.

Citations

• Ideals with an assigned initial ideal

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