# Upgraded methods for the effective computation of marked schemes on a strongly stable ideal

@article{Bertone2013UpgradedMF, title={Upgraded methods for the effective computation of marked schemes on a strongly stable ideal}, author={Cristina Bertone and Francesca Cioffi and Paolo Lella and Margherita Roggero}, journal={J. Symb. Comput.}, year={2013}, volume={50}, pages={263-290} }

## 24 Citations

A DIVISION ALGORITHM IN AN AFFINE FRAMEWORK FOR FLAT FAMILIES COVERING HILBERT SCHEMES

- Mathematics
- 2012

We study the family of ideals i R = K(x1;:::;xn) whose quotients R=i share the same ane Hilbert polynomial and the same monomial K-vector basis, that we choose to be the sous-escalierN (j) of a…

An overview on marked bases and applications

- Mathematics
- 2018

The Hilbert scheme was introduced by Grothendieck in the 60s. One can simply think of the Hilbert scheme Hilbp(t) as a set containing all the saturated homogeneous ideals I in a certain polynomial…

Quasi-stable ideals and Borel-fixed ideals with a given Hilbert polynomial

- MathematicsApplicable Algebra in Engineering, Communication and Computing
- 2015

The key tool for the proof of both algorithms is the combinatorial structure of a quasi-stable ideal, in particular a special set of generators for the considered ideals, the Pommaret basis.

On the functoriality of marked families

- Mathematics
- 2013

The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme…

Computing Quot schemes via marked bases over quasi-stable modules

- MathematicsJournal of Algebra
- 2020

Marked bases over quasi-stable modules

- Mathematics
- 2015

Let $K$ be a field of any characteristic, $A$ be a Noetherian $K$-algebra and consider the polynomial ring $A[x_0,\dots,x_n]$. The present paper deals with the definition of marked bases for free…

Computable Hilbert Schemes

- MathematicsArXiv
- 2012

This PhD thesis discusses the equations defining the Hilbert scheme as subscheme of a suitable Grassmannian, builds families of ideals that generalize the notion of family of ideals sharing the same initial ideal with respect to a fixed term ordering, and deals with the problem of the connectedness of the Hilbert Scheme of locally Cohen-Macaulay curves in the projective 3-space.

An efficient implementation of the algorithm computing the Borel-fixed points of a Hilbert scheme

- MathematicsISSAC
- 2012

This paper proposes an implementation of the algorithm computing all the saturated Borel-fixed ideals with number of variables and Hilbert polynomial assigned, introduced from a theoretical point of view in the paper "Segment ideals and Hilbert schemes of points", Discrete Mathematics 311 (2011).

## References

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Flat families by strongly stable ideals and a generalization of Gröbner bases

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IDEALS WITH AN ASSIGNED INITIAL IDEAL

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The stratum St(J,� ) (the homogeneous stratum Sth(J,� ) respectively) of a mono- mial ideal J in a polynomial ring R is the family of all (homogeneous) ideals of R whose initial ideal with respect to…

Rational components of Hilbert schemes

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The Gr\"obner stratum of a monomial ideal $\id{j}$ is an affine variety that parametrizes the family of all ideals having $\id{j}$ as initial ideal (with respect to a fixed term ordering). The…

BOREL OPEN COVERING OF HILBERT SCHEMES

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Let p(t) be an admissible Hilbert polynomial in P n of degree d and Gotzmann number r. It is well known that Hilb n(t) can be seen as a closed subscheme of the Grasmannian G(N,s), where N = ` n+r n ´…

The Hilbert schemes of locally Cohen-Macaulay curves in P^3 may after all be connected

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Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in projective 3 space are connected has been hampered by the lack of an answer to a question that was raised by…

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Abstract: We show that the set of the homogeneous saturated ideals with given initial ideal in a fixed term-ordering is locally closed in the Hilbert scheme, and that it is affine if the initial…

The Hilbert schemes of locally Cohen–Macaulay curves in $${\mathbb{P}^3}$$ may after all be connected

- Mathematics
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Progress on the problem whether the Hilbert schemes of locally Cohen–Macaulay curves in $${\mathbb{P}^3}$$ are connected has been hampered by the lack of an answer to a question raised by Robin…

Borel degenerations of arithmetically Cohen-Macaulay curves in ℙ3

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It is conjectured which Borel ideals are on such a component of arithmetically Cohen–Macaulay (ACM) codimension two schemes in �’n, and for a range of BoreL ideals it is proved that they are on the component.