On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence

Casnati, Gianfranco; Jelisiejew, Joachim; Notari, Roberto

doi:10.1216/rmj-2016-46-2-413

Cited by 2 publications

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“…Such a result throws some light on the question of the irreducibility of Hilb d G (P k N ). Along these lines the papers [7,10] and the results proved in Section 3 generalizing the aforementioned decomposition of the apolar polynomial allow us to prove the smoothability of all the k-algebras A with dim k (A) = 11 proving the Main Theorem above.…”

Section: Introduction and Notationmentioning

confidence: 84%

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On the Gorenstein locus of the punctual Hilbert scheme of degree 11

Casnati

Notari

2014

Journal of Pure and Applied Algebra

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Let k be an algebraically closed field of characteristic 0 and let Hilb_d^G(P_k^N) be the open locus of the Hilbert scheme Hilb_d(P_k^N) corresponding to Gorenstein subschemes. We proved in several previous papers that Hilb_d^G(P_k^N) is irreducible for d⩽10 and N⩾1, characterizing its singular locus. In the present paper we prove that also Hilb_{11}^G(P_k^N) is irreducible for each N⩾1. We also give some results about its singular locus

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Section: Introduction and Notationmentioning

confidence: 84%

“…The above proposition allows us to reduce the study of algebras associated to polynomials of the form F = G + x 2 n ∈ P [n] with s > 2 and G ∈ P [n − 1] to S[n − 1]/Ann(G). Indeed Corollary 3.2 of [7] yields…”

Section: Smoothability and Apolar Polynomialmentioning

confidence: 98%

On the Gorenstein locus of the punctual Hilbert scheme of degree 11

Casnati

Notari

2014

Journal of Pure and Applied Algebra

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“…Casnati and Notari [CN16] investigated Gorenstein algebras with H(3) = 1 and classified those with Hilbert function (1,4,4,1,1). The classification results were also used to prove rationality of the Poincarè series of zerodimensional Gorenstein algebras, see [CN09b,CENR13,CJN16].…”

Section: A Brief Historical Surveymentioning

confidence: 99%

Hilbert schemes of points and applications

Jelisiejew¹

2022

Preprint

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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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On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence

Cited by 2 publications

References 22 publications

On the Gorenstein locus of the punctual Hilbert scheme of degree 11

On the Gorenstein locus of the punctual Hilbert scheme of degree 11

Hilbert schemes of points and applications

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