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TitleA Borel open cover of the Hilbert scheme
Author(s) Cristina Bertone, Paolo Lella, margherita Roggero
TypeArticle in Journal
AbstractLet p ( t ) be an admissible Hilbert polynomial in P^n of degree d. The Hilbert scheme Hilb_p ( t )^n can be realized as a closed subscheme of a suitable Grassmannian G , hence it could be globally defined by homogeneous equations in the Plücker coordinates of G and covered by open subsets given by the non-vanishing of a Plücker coordinate, each embedded as a closed subscheme of the affine space A^D , D = dim ( G ) . However, the number E of Plücker coordinates is so large that effective computations in this setting are practically impossible. In this paper, taking advantage of the symmetries of Hilb_p ( t )^n , 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 A^D . 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 Gröbner bases, but are term order free. In this new setting, we can achieve explicit computations in many non-trivial cases.
KeywordsHilbert scheme, Borel-fixed ideal, Marked scheme
URL http://www.sciencedirect.com/science/article/pii/S0747717113000023
JournalJournal of Symbolic Computation
Pages119 - 135
Translation No
Refereed No