Rational Components of Hilbert Schemes

Lella, Paolo; Roggero, Margherita

doi:10.4171/rsmup/126-2

Cited by 33 publications

(62 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First, we observed that we could eliminate a significant number of variables from the equations defining Mf(J ≥m ) as an affine subscheme of A p(m)q(m) , computed using the method developed in [7]; in this way we obtain equations of higher degree than the starting ones, but often more convenient to use (for example, see [7,Appendix]). This feature has already been observed and studied for Gröbner strata in [15]. The bottleneck is that elimination of variables is too time-consuming.…”

Section: Introductionmentioning

confidence: 66%

“…Observe that J ≥4 is not segment w.r.t. any term order (see [7,Appendix]), hence in this case the results of [15] do not apply.…”

Section: Examplesmentioning

confidence: 99%

“…Especially, this criterion allows to determine the minimum integer m 0 such that φ m is an isomorphism for every m ≥ m 0 , and in particular Mf(J ≥m 0 ) is isomorphic to an open subset of Hilb n p(t) (see [4]). Our investigation on marked schemes lies in the framework of the methods and results obtained in the last years by several authors [5,10,15,20,23,24] about families of ideals with a fixed monomial basis for the quotient. Another close framework is the one in [2,19], where the authors study the collection of all monomial ideals J that are initial ideals of a fixed homogeneous ideal I w.r.t.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed in the first part (lines [15][16][17][18][19][20][21][22], the algorithm computes the superminimal reduction H of each monomial x α ∈ B J \ sB J and it imposes the conditions required by Proposition 5.5 (i), in other words the algorithm computes the set D 1 ⊆ A J of Definition 5.2. At the same time, the algorithm constructs the J-marked set G ⊂ K[ C, x].…”

mentioning

confidence: 99%

See 3 more Smart Citations

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

Bertone

Cioffi

Lella

et al. 2013

Journal of Symbolic Computation

Self Cite

View full text Add to dashboard Cite

Let $J\subset S=K[x_0,\ldots,x_n]$ be a monomial strongly stable ideal. The collection $\Mf(J)$ of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S/I$, is called a {\em $J$-marked family}. It can be endowed with a structure of affine scheme, called a {\em $J$-marked scheme}. For special ideals $J$, $J$-marked schemes provide an open cover of the Hilbert scheme $\hilbp$, where $p(t)$ is the Hilbert polynomial of $S/J$. Those ideals more suitable to this aim are the \emph{$m$-truncation} ideals $\underline{J}_{\geq m}$ generated by the monomials of degree $\geq m$ in a saturated strongly stable monomial ideal $\underline{J}$. Exploiting a characterization of the ideals in $\Mf(\underline{J}_{\geq m})$ in terms of a Buchberger-like criterion, we compute the equations defining the $\underline{J}_{\geq m}$-marked scheme by a new reduction relation, called {\em superminimal reduction}, and obtain an embedding of $\Mf(\underline{J}_{\geq m})$ 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 $\phi_m: \Mf(\underline{J}_{\geq m})\hookrightarrow \Mf(\underline{J}_{\geq m+1})$, characterize those $\phi_m$ that are isomorphisms in terms of the monomial basis of $\underline{J}$, especially we characterize the minimum integer $m_0$ such that $\phi_m$ is an isomorphism for every $m\geq m_0$

show abstract

Section: Introductionmentioning

confidence: 66%

“…Observe that J ≥4 is not segment w.r.t. any term order (see [7,Appendix]), hence in this case the results of [15] do not apply.…”

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

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

Bertone

Cioffi

Lella

et al. 2013

Journal of Symbolic Computation

Self Cite

View full text Add to dashboard Cite

show abstract

“…More recently, many papers [2,3,4,5,6,10,11] by Bertone, Cioffi, Marinari, Roggero and the author showed how to use Borel-fixed ideals for a local study of the Hilbert scheme, mainly constructing families of ideals sharing with a fixed Borel-fixed ideals the same basis of the quotient space.…”

Section: Introductionmentioning

confidence: 99%

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

Lella

2012

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

Self Cite

View full text Add to dashboard Cite

Borel-fixed ideals play a key role in the study of Hilbert schemes. Indeed each component and each intersection of components of a Hilbert scheme contains at least one Borelfixed point, i.e. a point corresponding to a subscheme defined by a Borel-fixed ideal. Moreover Borel-fixed ideals have good combinatorial properties, which make them very interesting in an algorithmic perspective. In this paper, we propose 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).

show abstract

The ubiquity of smooth Hilbert schemes

Staal

2020

Math. Z.

View full text Add to dashboard Cite

We investigate the geography of Hilbert schemes that parametrize closed subschemes of projective space with a specified Hilbert polynomial. We classify Hilbert schemes with unique Borel-fixed points via combinatorial expressions for their Hilbert polynomials. We realize the set of all nonempty Hilbert schemes as a probability space and prove that Hilbert schemes are irreducible and nonsingular with probability greater than 0.5. Theorem 1.1. The lexicographic ideal is the unique saturated Borel ideal of codimension c with Hilbert polynomial p if and only if:Borel ideals generalize lexicographic ideals and in characteristic 0 define Borel-fixed points on Hilbert schemes. Some basic general properties of Hilbert schemes have been extracted from these ideals. Rational curves linking Borel-fixed points prove connectedness in [Har66, PS05]. The thesis [Bay82] uses them to give equations for Hilbert schemes and proposes studying their tangent cones. Further, [Ree95] studies their combinatorial properties to give general bounds for radii of Hilbert schemes, and [RS97] proves that lexicographic points are nonsingular. Theorem 1.1 specifies an explicit collection of wellbehaved Hilbert schemes, generalizes the main result of [Got89], and improves our understanding of the geography of Hilbert schemes.This collection of Hilbert schemes is ubiquitous. Our new interpretation of Macaulay's classification identifies an infinite binary tree H c whose vertices are the Hilbert schemes 1 2011-12, by Ontario Graduate Scholarships in 2012-15, and by Gregory G. Smith's NSERC Discovery Grant in 2015-16. THE TREE OF ADMISSIBLE HILBERT POLYNOMIALSWe identify a graph structure on the set of numerical polynomials determining nonempty Hilbert schemes of projective spaces. The pioneering work [Mac27] gives a combinatorial classification of these polynomials. We introduce two binary relations to equip this set with the structure of an infinite binary tree.

show abstract

Rational Components of Hilbert Schemes

Cited by 33 publications

References 14 publications

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

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

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

The ubiquity of smooth Hilbert schemes

Contact Info

Product

Resources

About