Abstract
We prove that the Hilbert scheme of points on a higher dimensional affine space is non-reduced and has components lying entirely in characteristic p for all primes p. In fact, we show that Vakil’s Murphy’s Law holds up to retraction for this scheme. Our main tool is a generalized version of the Białynicki-Birula decomposition.
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References
American Institute of Mathematics Problem List. Components of Hilbert schemes. http://aimpl.org/hilbertschemes (2010)
Artin, M.: Deformations of Singularities. Notes by C.S. Seshadri and Allen Tannenbaum. Tata Institute of Fundamental Research, Bombay (1976)
Białynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math. 2(98), 480–497 (1973)
Buczyńska, W., Buczyński, J.: Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes. J. Algebraic Geom. 23, 63–90 (2014)
Behrend, K., Bryan, J., Szendrői, B.: Motivic degree zero Donaldson–Thomas invariants. Invent. Math. 192(1), 111–160 (2013)
Bertone, C., Cioffi, F., Roggero, M.: Double-generic initial ideal and Hilbert scheme. Ann. Mat. Pura Appl. (4) 196(1), 19–41 (2017)
Bhatt, B.: p-adic derived de Rham cohomology. arXiv:1204.6560 (2012)
Berthelot, P., Ogus, A.: Notes on Crystalline Cohomology. University of Tokyo Press, Tokyo (1978)
Cartwright, D.A., Erman, D., Velasco, M., Viray, B.: Hilbert schemes of 8 points. Algebra Number Theory 3(7), 763–795 (2009)
Drinfeld, V.: On algebraic spaces with an action of \({G}_m\). arXiv:1308.2604 (2013)
Dimca, A., Szendrői, B.: The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on \(\mathbb{C}^3\). Math. Res. Lett. 16(6), 1037–1055 (2009)
Eisenbud, D.: Commutative Algebra. Graduate Texts in Mathematics. With a View Toward Algebraic Geometry, vol. 150. Springer, New York (1995)
Ekedahl, T.: On non-liftable Calabi–Yau threefolds. arxiv:math/0306435v2 (2004)
Erman, D.: Murphy’s law for Hilbert function strata in the Hilbert scheme of points. Math. Res. Lett. 19(6), 1277–1281 (2012)
Fulton, W., Harris, J.: Representation Theory. Graduate Texts in Mathematics. A First Course, Readings in Mathematics, vol. 129. Springer, New York (1991)
Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math 90, 511–521 (1968)
Hartshorne, R.: Deformation Theory. Graduate Texts in Mathematics, vol. 257. Springer, New York (2010)
Haiman, M., Sturmfels, B.: Multigraded Hilbert schemes. J. Algebraic Geom. 13(4), 725–769 (2004)
Jelisiejew, J.: Elementary components of Hilbert schemes of points. J. Lond. Math. Soc. 100(1), 249–272 (2019)
Jelisiejew, J., Sienkiewicz, Ł.: Białynicki–Birula decomposition for reductive groups. J. Math. Pures Appl. 131, 290–325 (2019)
Landsberg, J.M.: Tensors: Geometry and Applications. Graduate Studies in Mathematics, vol. 128. American Mathematical Society, Providence, RI (2012)
Langer, A.: Lifting zero-dimensional schemes and divided powers. Bull. Lond. Math. Soc. 50(3), 449–461 (2018)
Lee, S.H., Vakil, R.: Mnëv-Sturmfels universality for schemes. In: A Celebration of Algebraic Geometry. Clay Mathematics Proceedings, vol. 18, pp. 457–468. Bulletin of the American Mathematical Society, Providence, RI (2013)
Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1986). (translated from the Japanese by M. Reid)
Martin-Deschamps, M., Perrin, D.: Le schéma de Hilbert des courbes gauches localement Cohen–Macaulay n’est (presque) jamais réduit. Ann. Sci. École Norm. Sup. (4) 29(6), 757–785 (1996)
Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)
Poonen, B.: The moduli space of commutative algebras of finite rank. J. Eur. Math. Soc. (JEMS) 10(3), 817–836 (2008)
Roggero, M., Albert, M.: Private Communication (2016)
Stacks Project. http://math.columbia.edu/algebraic_geometry/stacks-git (2017)
Vakil, R.: Murphy’s law in algebraic geometry: badly-behaved deformation spaces. Invent. Math. 164(3), 569–590 (2006)
Zdanowicz, M.: Liftability of Singularities and Their Frobenius Morphism Modulo \(p^2\). Int. Math. Res. Not. IMRN 14, 4513–4577 (2018)
Acknowledgements
I am very grateful to Piotr Achinger, Jarosław Buczyński and Maciek Zdanowicz for helpful and inspiring conversations and insightful comments of the early versions of this paper. The paper was prepared during the Simons Semester Varieties: Arithmetic and Transformations which is supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.
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J. Jelisiejew Partially supported by NCN Grant 2017/26/D/ST1/00913.
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Jelisiejew, J. Pathologies on the Hilbert scheme of points. Invent. math. 220, 581–610 (2020). https://doi.org/10.1007/s00222-019-00939-5
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DOI: https://doi.org/10.1007/s00222-019-00939-5