Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals. Issue 1 (2nd January 2022)

• Record Type: Journal Article
• Authors: Bertone, Cristina; Cioffi, Francesca; Roggero, Margherita
• Is Part Of: Experimental mathematics. Volume 31:Issue 1(2022)
• Page Start: 120
• Page End: 137
• Publication Date: 2022-01-02
• Subjects: Gorenstein algebra -- Hilbert scheme -- strongly stable ideal -- marked basis; 13H10 -- 14C05 -- 13P10; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2019.1592034
• Languages: English
• ISSNs: 1058-6458

Abstract

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 16 7 has at least five irreducible components. As a byproduct of our study about Hilb 16 7, we also find a new elementary component in Hilb 15 7 . 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.