Details:
Title  A Borel open cover of the Hilbert scheme  Author(s)  Cristina Bertone, Paolo Lella, margherita Roggero  Type  Article in Journal  Abstract  Let 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 nonvanishing 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 Borelfixed 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 nontrivial cases.  Keywords  Hilbert scheme, Borelfixed ideal, Marked scheme  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717113000023 
Language  English  Journal  Journal of Symbolic Computation  Volume  53  Number  0  Pages  119  135  Year  2013  Edition  0  Translation 
No  Refereed 
No 
