Components of Gröbner strata in the Hilbert scheme of points

Lederer, Mathias

doi:10.1112/plms/pdt018

Cited by 4 publications

(3 citation statements)

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“…In fact there is a beautiful theory of Hilbert schemes of points on surfaces, which we do not discuss here, see e.g. [Göt94,Hai01,Hai03,KT01,Led14,Nak99]). In his paper, Fogarty [Fog68,p.…”

Section: A Brief Historical Surveymentioning

confidence: 99%

Hilbert schemes of points and applications

Jelisiejew¹

2022

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Long, long ago 1 , in a galaxy 2 far, far away 3 ...The following is the author's PhD Thesis, defended in September 2017. It has not been edited since (except for the footnotes in the open problems section and for this note). In particular, there is nothing here about applications of Białynicki-Birula decomposition to Hilbert schemes [Jel19, Jel20, SS21], the multigraded Hilbert schemes and border apolarity [BB21, Mań20], fiber-full Hilbert schemes [CRR21, CRR22] and other recent developments. Philosophically speaking, is it reassuring that there are too many new interesting papers on Hilbert schemes of points to summarize here. It is also very interesting that many of the open problems listed below in §1.5 remain actual. Today, the author would add also the more applied open problems (see [JLP22]): a large part of the classical problem of determining the complexity of matrix multiplication can be reduced fruitfully to the study of the Hilbert scheme of points.While the author is not aware of any errors, it is likely they exist here. This work is put on arXiv due/thanks to several people who decided to refer to it and only to grant this piece a quasi-permanent place to rest.1 That is, five years 2 Metaphorically speaking 3 In terms of scientific development ¨

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Section: A Brief Historical Surveymentioning

confidence: 99%

Hilbert schemes of points and applications

Jelisiejew¹

2022

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“…We will now present a natural generating function for the number of standard decompositions of a standard graph G. The analogue of this generating function in the setting of standard sets is discussed in [Led,Section 2.3].…”

Section: Appendix a A Generating Functionmentioning

confidence: 99%

Connect Four and Graph Decomposition

2016

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“…This locus is isomorphic to an affine closed subscheme of an affine space over k with weighted homogeneous defining ideal [RT10], and there is procedures for obtaining a reduced Gröbner basis of a defining ideal [RT10,Kam17]. See also [Rob09,Led11,Led14,BLR13,LR16] for this computable locus of Hilbert schemes.…”

Section: Introductionmentioning

confidence: 99%

Gröbner scheme in the Hilbert scheme and complete intersection monomial ideals

Kambe

2017

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Let k be a commutative ring and S = k[x0, . . . , xn] be a polynomial ring over k with a monomial order. For any monomial ideal J, there exists an affine k-scheme of finite type, called Gröbner scheme, which parameterizes all homogeneous reduced Gröbner bases in S whose initial ideal is J. Here we functorially show that the Gröbner scheme is a locally closed subscheme of the Hilbert scheme if J is a saturated ideal. In the process, we also show that the Gröbner scheme consists of complete intersections if J defines a complete intersection.

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Components of Gröbner strata in the Hilbert scheme of points

Cited by 4 publications

References 18 publications

Hilbert schemes of points and applications

Hilbert schemes of points and applications

Connect Four and Graph Decomposition

Gröbner scheme in the Hilbert scheme and complete intersection monomial ideals

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