Galois closures and elementary components of Hilbert schemes of points

• Matthew Satriano ^[1] ; Andrew P. Staal ^[1]
  1. [1] University of Waterloo

     University of Waterloo

     Canadá

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 2, 2024, 35 págs.
• Idioma: inglés
• DOI: 10.1007/s00029-024-00915-9
• Enlaces
  • Texto completo
• Resumen

  • Bhargava and the first-named author of this paper introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz–Mazur. In this paper, we generalize Galois closures and apply them to construct a new infinite family of irreducible components of Hilbert schemes of points. We show that these components are elementary, in the sense that they parametrize algebras supported at a point. Furthermore, we produce secondary families of elementary components obtained from Galois closures by modding out by suitable socle elements.

• Referencias bibliográficas
  • Bhargava, M.: Higher composition laws. III. The parametrization of quartic rings. Ann. Math. (2) 159(3), 1329–1360 (2004).
  • Bhargava, M.: Higher composition laws. IV. The parametrization of quintic rings. Ann. Math. (2) 167(1), 53–94 (2008).
  • Bridgeland, T., King, A., Reid, M.: The McKay correspondence as an equivalence of derived categories. J. Am. Math. Soc. 14(3), 535–554 (2001).
  • Bhargava, M., Satriano, M.: On a notion of “Galois closure” for extensions of rings. J. Eur. Math. Soc. (JEMS) 16(9), 1881–1913 (2014).
  • Cartwright, D.A., Erman, D., Velasco, M., Viray, B.: Hilbert schemes of 8 points. Algebra Number Theory 3(7), 763–795 (2009).
  • Chevalley, S.C.: 2e année: 1958. Anneaux de Chow et applications. Secrétariat mathématique, 11 rue Pierre Curie, Paris (1958).
  • Erman, D., Velasco, M.: A syzygetic approach to the smoothability of zero-dimensional schemes. Adv. Math. 224(3), 1143–1166 (2010).
  • Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math. 90, 511–521 (1968).
  • Grothendieck, A.: Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki,...
  • Haiman, M.: Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14(4), 941–1006 (2001).
  • Hartshorne, R.: Deformation Theory. Graduate Texts in Mathematics, vol. 257. Springer, New York (2010).
  • Ho, W., Satriano, M.: Galois closures of non-commutative rings and an application to Hermitian representations. Int. Math. Res. Not. IMRN...
  • Huibregtse, M.E.: Some elementary components of the Hilbert scheme of points. Rocky Mt. J. Math. 47(4), 1169–1225 (2017).
  • Huibregtse, M.E.: More elementary components of the Hilbert scheme of points (2021). arXiv:2102.00494.
  • Iarrobino, A.: Reducibility of the families of 0-dimensional schemes on a variety. Invent. Math. 15, 72–77 (1972).
  • Iarrobino, A.: The number of generic singularities. Rice Univ. Stud. 59(1), 49–51 (1973).
  • Iarrobino, A.: Compressed algebras: Artin algebras having given socle degrees and maximal length. Trans. Am. Math. Soc. 285(1), 337–378 (1984).
  • Iarrobino, A.: Hilbert scheme of points: overview of last ten years. Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proceedings...
  • Iarrobino, A., Emsalem, J.: Some zero-dimensional generic singularities; finite algebras having small tangent space. Compos. Math. 36(2),...
  • Iarrobino, A., Kanev, V.: Power Sums, Gorenstein algebras, and Determinantal Loci. Lecture Notes in Mathematics, vol. 1721. Springer, Berlin...
  • Ito, Y., Nakajima, H.: McKay correspondence and Hilbert schemes in dimension three. Topology 39(6), 1155–1191 (2000).
  • Jelisiejew, J.: Elementary components of Hilbert schemes of points. J. Lond. Math. Soc. (2) 100(1), 249–272 (2019).
  • Jelisiejew, J.: Pathologies on the Hilbert scheme of points. Invent. Math. 220(2), 581–610 (2020).
  • Katz, N.M., Mazur, B.: Arithmetic moduli of elliptic curves. Annals of Mathematics Studies, vol. 108. Princeton University Press, Princeton...
  • Lichtenbaum, S., Schlessinger, M.: The cotangent complex of a morphism. Trans. Am Math. Soc. 128, 41–70 (1967).
  • Poonen, B.: Isomorphism types of commutative algebras of finite rank over an algebraically closed field. Computational Arithmetic Geometry,...
  • Poonen, B.: The moduli space of commutative algebras of finite rank. J. Eur. Math. Soc. (JEMS) 10(3), 817–836 (2008).
  • Sagan, B.E.: The Symmetric Group. Graduate Texts in Mathematics, vol. 203, 2nd edn. Springer, New York (2001). Representations, combinatorial...
  • Shafarevich, I.R.: Deformations of commutative algebras of class 2. Algebra i Analiz 2(6), 178–196 (1990).
  • Satriano, M., Staal, A.P.: Small elementary components of Hilbert schemes of points. Forum Math. Sigma 11, Paper No. e45, 36 (2023).