2009/09/11 by Cristina Bertone, Bertone, Cristina, Paolo Lella +3 · 1 citation
Computer Science · Mathematics · #13P10 #14C05 #14Q20 #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.0909.2184
openalex publication_date 2009/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p(t) be an admissible Hilbert polynomial in \PPn of degree d. The Hilbert scheme \hilbnp(t) can be realized as a closed subscheme of a suitable Grassmannian \mathbb G, hence it could be globally defined by homogeneous equations in the Plucker coordinates of \mathbb G and covered by open subsets given by the non-vanishing of a Plucker coordinate, each embedded as a closed subscheme of the affine space AD, D=dim(\mathbb G). However, the number E of Plucker coordinates is so large that effective computations in this setting are practically impossible. In this paper, taking advantage of the symmetries of \hilbnp(t), we exhibit a new open cover, consisting of marked schemes over Borel-fixed ideals, whose number is significantly smaller than E. Exploiting the properties of marked schemes, we prove that these open subsets are defined by equations of degree ≤ d+2 in their natural embedding in \AfD. Furthermore we find new embeddings in affine spaces of far lower dimension than D, and characterize those that are still defined by equations of degree ≤ d+2. The proofs are constructive and use a polynomial reduction process, similar to the one for Grobner bases, but are term order free. In this new setting, we can achieve explicit computations in many non-trivial cases.