Over an infinite field K with char(K)≠2,3, we investigate smoothable Gorenstein K-points in a punctual Hilbert scheme and obtain the following results: (i) every K-point defined by local Gorenstein K-algebras with Hilbert function (1,7,7,1) is smoothable (this is the only case non treated in the range considered by Iarrobino and Kanev in 1999; (ii) the Hilbert scheme Hilb^7_16 has at least five irreducible components. As a byproduct of our study about Hilb^7_16, we also find a new elementary component in Hilb^7_15. We face the problem from a new point of view, that is based on properties of double-generic initial ideals and of marked schemes. The properties of marked schemes give us a simple method to compute the Zariski tangent space to a Hilbert scheme at a given K-point, which is very useful in this context. We also test our tools to find the already known result that K-points defined by local Gorenstein K-algebras with Hilbert function (1,5,5,1) are smoothable. The problem that we consider is strictly related to the study of the irreducibility of the Gorenstein locus in a Hilbert scheme and, more generally, of the irreducibility of a Hilbert scheme, which is a very open question.

Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals / Bertone, Cristina; Cioffi, Francesca; Roggero, Margherita. - In: EXPERIMENTAL MATHEMATICS. - ISSN 1058-6458. - 31:1(2022), pp. 120-137. [10.1080/10586458.2019.1592034]

Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals

Francesca Cioffi;
2022

Abstract

Over an infinite field K with char(K)≠2,3, we investigate smoothable Gorenstein K-points in a punctual Hilbert scheme and obtain the following results: (i) every K-point defined by local Gorenstein K-algebras with Hilbert function (1,7,7,1) is smoothable (this is the only case non treated in the range considered by Iarrobino and Kanev in 1999; (ii) the Hilbert scheme Hilb^7_16 has at least five irreducible components. As a byproduct of our study about Hilb^7_16, we also find a new elementary component in Hilb^7_15. We face the problem from a new point of view, that is based on properties of double-generic initial ideals and of marked schemes. The properties of marked schemes give us a simple method to compute the Zariski tangent space to a Hilbert scheme at a given K-point, which is very useful in this context. We also test our tools to find the already known result that K-points defined by local Gorenstein K-algebras with Hilbert function (1,5,5,1) are smoothable. The problem that we consider is strictly related to the study of the irreducibility of the Gorenstein locus in a Hilbert scheme and, more generally, of the irreducibility of a Hilbert scheme, which is a very open question.
2022
Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals / Bertone, Cristina; Cioffi, Francesca; Roggero, Margherita. - In: EXPERIMENTAL MATHEMATICS. - ISSN 1058-6458. - 31:1(2022), pp. 120-137. [10.1080/10586458.2019.1592034]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11588/751171
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