## Rocky Mountain Journal of Mathematics

### Some elementary components of the Hilbert scheme of points

Mark E. Huibregtse

#### Abstract

Let $K$ be an algebraically closed field of characteristic~$0$, and let $H^{\mu }_{\mathbb {A}^n_{K }}$ denote the Hilbert scheme of $\mu$ points of $\mathbb {A}^n_{K}$. An \textit {elementary component} $E$ of $H^{\mu }_{\mathbb {A}^n_{K }}$ is an irreducible component such that every $K$-point $[I]\in E$ represents a length-$\mu$ closed subscheme $Spec (K [x_1,\ldots ,x_n]/I)\subseteq \mathbb {A}^n_{K}$ that is supported at one point. Iarrobino and Emsalem gave the first explicit examples (with $\mu > 1$) of elementary components \cite {Iarrob-Emsalem}; in their examples, the ideals $I$ were homogeneous (up to a change of coordinates corresponding to a translation of $\mathbb {A}^n_{K}$). We generalize their construction to obtain new examples of elementary components.

#### Article information

Source
Rocky Mountain J. Math., Volume 47, Number 4 (2017), 1169-1225.

Dates
First available in Project Euclid: 6 August 2017

https://projecteuclid.org/euclid.rmjm/1501984946

Digital Object Identifier
doi:10.1216/RMJ-2017-47-4-1169

Mathematical Reviews number (MathSciNet)
MR3689951

Zentralblatt MATH identifier
06790011

Subjects
Primary: 14C05: Parametrization (Chow and Hilbert schemes)

#### Citation

Huibregtse, Mark E. Some elementary components of the Hilbert scheme of points. Rocky Mountain J. Math. 47 (2017), no. 4, 1169--1225. doi:10.1216/RMJ-2017-47-4-1169. https://projecteuclid.org/euclid.rmjm/1501984946

#### References

• G. Borges dos Santos, A. Henni and M. Jardim, Commuting matrices and the Hilbert scheme of points on affine spaces, preprint, arXiv:1304.3028v3 [math.AG], 2013.
• D. Cartwright, D. Erman, M. Velasco and B. Viray, Hilbert schemes of $8$ points, Alg. Num. Theory 3 (2009), 763–795.
• G. Casnati, J. Jelisiejew and R. Notari, Irreducibility of the Gorenstein loci of Hilbert schemes via ray families, Alg. Num. Theory 9 (2015), 1525–1570.
• D. Erman and M. Velasco, A syzygetic approach to the smoothability of zero-dimensional schemes, Adv. Math. 224 (2010), 1143–1166.
• T.S. Gustavsen, D. Laksov and R. Skjelnes, An elementary, explicit, proof of the existence of Hilbert schemes of points, J. Pure Appl. Alg. 210 (2007), 705–720.
• R. Hartshorne, Deformation theory, Grad. Texts Math. 257, Springer-Verlag, New York, 2010.
• M. Huibregtse, An elementary construction of the multigraded Hilbert scheme of points, Pacific J. Math. 223 (2006), 269–315.
• A. Iarrobino, Reducibility of the families of $0$-dimensional schemes on a variety, Invent. Math. 15 (1972), 72–77.
• ––––, The number of generic singularities, Volume I: Geometry of singularities, Rice Univ. Stud. 59 (1973), 49–51.
• ––––, Compressed algebras and components of the punctual Hilbert scheme, Lect. Notes Math. 1124, Springer-Verlag, 1985.
• A. Iarrobino and J. Emsalem, Some zero-dimensional generic singularities: Finite algebras having small tangent space, Compositio Math. 36 (1978), 145–188.
• M. Kreuzer and M. Kriegl, Gröbner bases for syzygy modules of border bases, J. Alg. Appl. 13 (2014).
• M. Kreuzer and L. Robbiano, Computational commutative algebra 2, Springer, Berlin, 2005.
• ––––, Deformations of border bases, Collect. Math. 59 (2008),275–297.
• ––––, The geometry of border bases, J. Pure Appl. Alg. 215 (2011), 2005–2018.
• I.R. Shafarevich, Deformations of commutative algebras of class $2$, Alg. Anal. 2 (1990), 178–196 (in Russian), Leningrad Math. J. 2 (1991), 1335–1351 (in English).
• Wolfram Research, Inc., Mathematica, Version 9, Champaign, IL, 2012.