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

Abstract

Over an infinite field K with $\text{char}(K)\ne 2,3,$ 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 ${\text{Hilb}}_{16}^7$ has at least five irreducible components. As a byproduct of our study about ${\text{Hilb}}_{16}^7,$ we also find a new elementary component in ${\text{Hilb}}_{15}^7.$ 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.

Keywords:

• Gorenstein algebra
• Hilbert scheme
• strongly stable ideal
• marked basis

MSC 2010:

• 13H10
• 14C05
• 13P10

Acknowledgments

We thank Joachim Jelisiejew for kindly providing useful knowledge about the equivalence between graded and local Gorenstein Artin K-algebras with Hilbert function of type $(1,n,n,1)$ and about the smoothability of a local Artin K-algebra over any field. We also thank the anonymous referee for very useful comments and suggestions, in particular for highlighting the existence of the two irreducible components of ${\text{Hilb}}_{16}^7$ that can be constructed over the elementary component of ${\text{Hilb}}_8^7.$ We are grateful to Steven Kleiman for several useful discussions on this paper.

Additional information

Funding

The authors are members of GNSAGA (INdAM). The second author is partially supported by Prin 2015 (2015EYPTSB_011—Geometry of Algebraic varieties, Unità locale Tor Vergata, CUP E82F16003030006). Part of the research was conducted during a visit of the first and third author at the Dipartimento di Matematica e Applicazioni of Università di Napoli Federico II, which also financially supported that visit.