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TitleUpgraded methods for the effective computation of marked schemes on a strongly stable ideal
Author(s) Cristina Bertone, Francesca Cioffi, Paolo Lella, margherita Roggero
TypeArticle in Journal
AbstractLet J ⊂ S = K [ x 0 , , x n ] be a monomial strongly stable ideal. The collection M f ( J ) of the homogeneous polynomial ideals I, such that the monomials outside J form a K-vector basis of S / I , is called a J-marked family. It can be endowed with a structure of affine scheme, called a J-marked scheme. For special ideals J, J-marked schemes provide an open cover of the Hilbert scheme H ilb p ( t ) n , where p ( t ) is the Hilbert polynomial of S / J . Those ideals more suitable to this aim are the m-truncation ideals J ̲ ⩾ m generated by the monomials of degree ⩾m in a saturated strongly stable monomial ideal J ̲ . Exploiting a characterization of the ideals in M f ( J ̲ ⩾ m ) in terms of a Buchberger-like criterion, we compute the equations defining the J ̲ ⩾ m -marked scheme by a new reduction relation, called superminimal reduction, and obtain an embedding of M f ( J ̲ ⩾ m ) in an affine space of low dimension. In this setting, explicit computations are achievable in many non-trivial cases. Moreover, for every m, we give a closed embedding ϕ m : M f ( J ̲ ⩾ m ) ↪ M f ( J ̲ ⩾ m + 1 ) , characterize those ϕ m that are isomorphisms in terms of the monomial basis of J ̲ , especially we characterize the minimum integer m 0 such that ϕ m is an isomorphism for every m ⩾ m 0 .
KeywordsHilbert scheme, Strongly stable ideal, Polynomial reduction relation
URL http://www.sciencedirect.com/science/article/pii/S0747717112001241
JournalJournal of Symbolic Computation
Pages263 - 290
Translation No
Refereed No