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On jet schemes of pfaffian ideals

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Abstract

Jet schemes and arc spaces received quite a lot of attention by researchers after their introduction, due to J. Nash, and established their importance as an object of study in M. Kontsevich’s motivic integration theory. Several results point out that jet schemes carry a rich amount of geometrical information about the original object they stem from, whereas, from an algebraic point of view, little is know about them. In this paper we study some algebraic properties of jet schemes ideals of pfaffian varieties and we determine under which conditions the corresponding jet scheme varieties are irreducible.

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Notes

  1. For instance, the computations we performed on the quotient ring of \(I^{6,3}_2\) with [14] break up after a few minutes, whereas with [3] it exhausts the available 250 Gb RAM memory in a few hours time.

References

  1. Abeasis, S., Del Fra, A.: Young diagrams and ideals of pfaffians. Adv. Math. 35, 158–178 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  2. Avramov, L.: A class of factorial domains. Serdica 5, 378–379 (1979)

    MathSciNet  MATH  Google Scholar 

  3. Abbott, J., Bigatti, A.M., Robbiano, L.: CoCoA: a system for doing Computations in commutative algebra. http://cocoa.dima.unige.it

  4. Bruns, W., Herzog, J.: On the computation of \(a\)-invariants. Manuscr. Math. 77, 201–213 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. De Negri, E.: Some results on Hilbert series and \(a\)-invariant of pfaffian ideals. Math. J. Toyama Univ. 24, 93–106 (2001)

    MathSciNet  MATH  Google Scholar 

  6. De Negri, E., Gorla, E.: Invariants of ideals generated by pfaffians. In: Commutative Algebra and Its Connections to Geometry. Contemporary Mathematics, vol. 555, pp. 47–62. Amer. Math. Soc., Providence (2011)

  7. De Negri, E., Gorla, E.: G-biliaison of ladder pfaffian varieties. J. Algebra 321(9), 2637–2649 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. De Negri, E., Sbarra, E.: Gröbner bases of ideals cogenerated by pfaffians. J. Pure Appl. Algebra 215(5), 812–821 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. De Concini, C., Procesi, C.: A characteristic free approach to invariant theory. Adv. Math. 21, 330–354 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  10. Docampo, R.: Arcs on determinantal varieties. Trans. Am. Math. Soc. 365, 2241–2269 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ghorpade, S., Jonov, B., Sethuraman, B.A.: Hilbert series of certain jet schemes of determinantal varieties. Pac. J. Math. 272(1), 147–175 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ghorpade, S., Krattenthaler, C.: The Hilbert series of Pfaffian rings. In: Algebra, Arithmetic and Geometry with Applications (West Lafayette, IN, 2000), pp. 337–356. Springer, Berlin (2004)

  13. Goward, R.A., Smith, K.: The jet scheme of a monomial scheme. Commun. Algebra 34, 1591–1598 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grayson, D.R., Stillman, M.E., Michael, E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2

  15. Herzog, J., Trung, N.V.: Gröbner bases and multiplicity of determinantal and pfaffian ideals. Adv. Math. 96, 1–37 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jonov, B.: Initial complex associated to a jet scheme of a determinantal variety. JPAA 215, 806–811 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Józefiak, T., Pragacz, P.: Ideals generated by pfaffians. J. Algebra 61, 189–198 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kleppe, H., Laksov, D.: The algebraic structure and deformation of pfaffian schemes. J. Algebra 64, 167–189 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  19. Košir, T., Sethuraman, B.A.: Determinantal varieties over truncated polynomial rings. J. Pure Appl. Algebra 195, 75–95 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kurano, K.: Relations on pfaffians I: plethysm formulas. J. Math. Kyoto Univ. 31(1), 713–731 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Leyton-Álvarez, M.: Deforming spaces of \(m\)-jets of isolated hypersurfaces singularities. J. Algebra 508, 81–97 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Marinov, V.: Perfection of ideals generated by pfaffians of alternating matrices. Serdica 9, 31–42 (1983) (pp. 122–131)

  23. Mustaţă, M.: Jet schemes of locally complete intersection canonical singularities. Invent. math. 145, 397–424 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Neubauer, M.G., Sethuraman, B.A.: Commuting pairs in the centralizers of 2-regular matrices. J. Algebra 214, 174–181 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pogudin, G.: Products of ideals and jet schemes. J. Algebra 502, 61–78 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sethuraman, B.A., Šivic, K.: Jet schemes of the commuting matrix pairs scheme. Proc. Am. Math. Soc. 137(12), 3953–3967 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yuen, C.: Jet schemes of determinantal varieties. In: Algebra, Geometry and Their Interactions, Volume 448 of Contemporary Mathematics, pp. 261–270. Amer. Math. Soc., Providence (2007)

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Acknowledgements

Our computations have been performed on a dedicated server, which has been acquired thanks to the support of the University of Pisa, within the call “Bando per il cofinanziamento dell’acquisto di medio/grandi attrezzature scientifiche 2016”. The authors were partially supported by INdAM-GNSAGA.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146, Genoa, Italy

    Emanuela De Negri

  2. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Enrico Sbarra

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  1. Emanuela De Negri
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  2. Enrico Sbarra
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Correspondence to Emanuela De Negri.

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De Negri, E., Sbarra, E. On jet schemes of pfaffian ideals. Collect. Math. 70, 479–491 (2019). https://doi.org/10.1007/s13348-019-00242-9

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