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A Saito criterion for holonomic divisors

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Abstract

We show that a holonomic divisor is free if and only if applying all logarithmic derivations to a generic function with isolated critical point yields a complete intersection Artin algebra.

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References

  1. Abe, T., Horiguchi, T., Masuda, M., Murai, S., Sato, T.: Hessenberg varieties and hyperplane arrangements, arXiv:1611.00269, (2016)

  2. Abe, T., Maeno, T., Murai, S., Numata, Y.: Solomon—Terao algebra of hyperplane arrangements, arXiv:1802.04056, (2018)

  3. Bruns, W., Herzog, J.: Cohen–Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  4. Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 4-1-1—a computer algebra system for polynomial computations, http://www.singular.uni-kl.de, (2018)

  5. Flenner, H.: Die Sätze von Bertini für lokale Ringe. Math. Ann. 229(2), 97–111 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. Grauert, H., Remmert, R.: Analytische Stellenalgebren, Springer-Verlag, Berlin, Unter Mitarbeit von O. Riemenschneider, Die Grundlehren der mathematischen Wissenschaften, Band 176 (1971)

  7. Grothendieck, A.: Techniques de construction en géométrie analytique.VI. Étude locale des morphismes: germes d’espaces analytiques, platitude, morphismes simples, Séminaire Henri Cartan, 13ième année, 1960/61 (Henri Cartan, ed.), Secrétariat mathématique, Paris, pp. 1–13 (1962)

  8. Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, Inst. Hautes Études Sci. Publ. Math. 32, 361 (1967)

  9. Houzel, C.: Géométrie analytique locale, II. Théorie des morphismes finis, Séminaire Henri Cartan, 13ième année, 1960/61 (Henri Cartan, ed.), Secrétariat mathématique, Paris, pp. 1–22 (1962)

  10. Houzel, C.: Géométrie analytique locale, III. Séminaire Henri Cartan, 13ième année, 1960/61 (Henri Cartan, ed.), Secrétariat mathématique, Paris, pp. 1–25 (1962)

  11. Kunz, E.: Kähler Differentials, Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig (1986)

    Book  Google Scholar 

  12. Saito, K.: Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27(2), 265–291 (1980)

    MathSciNet  MATH  Google Scholar 

  13. Scheja, G.: Differentialmoduln lokaler analytischer Algebren, Schriftenreihe Math. Inst. Univ. Fribourg, no. 2, Univ. Fribourg, Switzerland, (1969/70)

  14. Scheja, G., Storch, U.: Differentielle Eigenschaften der Lokalisierungen analytischer Algebren. Math. Ann. 197, 137–170 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Mathias Schulze.

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Epure, R., Schulze, M. A Saito criterion for holonomic divisors. manuscripta math. 160, 1–8 (2019). https://doi.org/10.1007/s00229-018-1065-5

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  • DOI: https://doi.org/10.1007/s00229-018-1065-5

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