Divisors of expected Jacobian type


  • Josep Àlvarez Montaner
  • Francesc Planas-Vilanova




Divisors whose Jacobian ideal is of linear type have received a lot of attention recently because of its connections with the theory of $D$-modules. In this work we are interested on divisors of expected Jacobian type, that is, divisors whose gradient ideal is of linear type and the relation type of its Jacobian ideal coincides with the reduction number with respect to the gradient ideal plus one. We provide conditions in order to be able to describe precisely the equations of the Rees algebra of the Jacobian ideal. We also relate the relation type of the Jacobian ideal to some $D$-module theoretic invariant given by the degree of the Kashiwara operator.


Aberbach, I. M., Ghezzi, L., and Hà, H. T., Homology multipliers and the relation type of parameter ideals, Pacific J. Math. 226 (2006), no. 1, 1–39. https://doi.org/10.2140/pjm.2006.226.1

Arcadias, R., Minimal resolutions of geometric D-modules, J. Pure Appl. Algebra 214 (2010), no. 8, 1477–1496. https://doi.org/10.1016/j.jpaa.2009.12.001

Blanco, G., and Leykin, A., textsc Kashiwara.m2. A package for Macaulay 2 available at newline https://github.com/Macaulay2/Workshop-2016-Warwick/tree/master/Dmodules.

Bosma, W., Cannon, J., and Playoust, C., The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), no. 3–4, 235–265. https://doi.org/10.1006/jsco.1996.0125

Briançon, J., Granger, M., Maisonobe, Ph., and Miniconi, M., Algorithme de calcul du polynôme de Bernstein: cas non dégénéré, Ann. Inst. Fourier (Grenoble) 39 (1989), no. 3, 553–610.

Bruns, W., and Herzog, J., Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge 1993.

Calderón Moreno, F. J., and Narváez Macarro, L., The module $D f^s$ for locally quasi-homogeneous free divisors, Compositio Math. 134 (2002), no. 1, 59–74. https://doi.org/10.1023/A:1020228824102

Calderón Moreno, F. J., and Narváez Macarro, L., On the logarithmic comparison theorem for integrable logarithmic connections, Proc. Lond. Math. Soc. (3) 98 (2009), no. 3, 585–606. https://doi.org/10.1112/plms/pdn043

Cassou-Noguès, Pi., Étude du comportement du polynôme de Bernstein lors d'une déformation à µ-constant de $x^a+y^b$ avec $(a,b)=1$, Compositio Math. 63 (1987), no. 3, 291–313.

Decker, W., Greuel, G.-M., Pfister, G., and Schönemann, H.: textsc Singular 4-1-2 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2019).

Granger, M., Bernstein-Sato polynomials and functional equations. Algebraic approach to differential equations, 225–291, World Sci. Publ., Hackensack, NJ, 2010. https://doi.org/10.1142/9789814273244_0006

Grayson, D., and Stillman, M., textsc Macaulay 2. newline Available at: http://www.math.uiuc.edu/Macaulay2.

Huneke, C.,The Theory of $d$-Sequences and Powers of Ideals, Adv. in Math. 46 (1982), no. 3, 249–279. https://doi.org/10.1016/0001-8708(82)90045-7

Kashiwara, M., $B$-functions and holonomic systems. Rationality of roots of $B$-functions, Invent. Math. 38 (1976/77), no. 1, 33–53. https://doi.org/10.1007/BF01390168

Kato, M.,The $b$-function of µ-constant deformation of $x^7+y^5$, Bull. College Sci. Univ. Ryukyus 32 (1981), 5–10.

Kato, M., The $b$-function of µ-constant deformation of $x^9+y^4$, Bull. College Sci. Univ. Ryukyus 33 (1982), 5–8.

Muiños, F., and Planas-Vilanova, F., The equations of Rees algebras of equimultiple ideals of deviation one, Proc. Amer. Math. Soc. 141 (2013), no. 4, 1241-1254. https://doi.org/10.1090/S0002-9939-2012-11398-X

Nakamura, Y., On invariants of Reiffen's isolated singularity, Algebraic analysis and the exact WKB analysis for systems of differential equations, 7–13, RIMS Kôkyûroku Bessatsu, B5, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008.

Nakamura, Y., The $b$-function of Reiffen's $(p,4)$ isolated singularity, J. Algebra Appl. 15 (2016), no. 6, 1650115, 11 pp. https://doi.org/10.1142/S0219498816501152

Narváez Macarro, L., Linearity conditions on the Jacobian ideal and logarithmic-meromorphic comparison for free divisors, 245–-269 Contemp. Math. 474, Amer. Math. Soc., Providence, RI, 2008. https://doi.org/10.1090/conm/474/09259

Narváez Macarro, L., A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors, Adv. Math. 281 (2015), 1242-1273. https://doi.org/10.1016/j.aim.2015.06.012

Northcott, D. G., and Rees, D., Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145-158. https://doi.org/10.1017/s0305004100029194

Planas-Vilanova, F., On the module of effective relations of a standard algebra, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 2, 215–-229. https://doi.org/10.1017/S030500419800262X

Ruiz, J. M., The basic theory of power series, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1993. https://doi.org/10.1007/978-3-322-84994-6

Swanson, I., and Huneke, C., Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336 Cambridge University Press, 2006.

Torrelli, T., Logarithmic comparison theorem and $D$-modules: an overview. Singularity theory, 995–1009, World Sci. Publ., Hackensack, NJ, 2007. https://doi.org/10.1142/9789812707499_0040

Walther, U., Survey on the $D$-module $f^s$. With an appendix by Anton Leykin. Math. Sci. Res. Inst. Publ., 67. Vol. I, 391–430, Cambridge Univ. Press, New York, 2015.

Walther, U., The Jacobian module, the Milnor fiber, and the $D$-module generated by $f^s$, Invent. Math. 207 (2017), no. 3, 1239–1287. https://doi.org/10.1007/s00222-016-0684-2

Yano, T., On the theory of $b$-functions, Publ. Res. Inst. Math. Sci. 14 (1978), no. 1, 111–202. https://doi.org/10.2977/prims/1195189282

Yano, T., $b$-functions and exponents of hypersurface isolated singularities, Singularities, Part 2 (Arcata, Calif., 1981), 641–652, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983. https://doi.org/10.24033/bsmf.1455



How to Cite

Àlvarez Montaner, J. ., & Planas-Vilanova, F. . (2021). Divisors of expected Jacobian type. MATHEMATICA SCANDINAVICA, 127(2), 161–184. https://doi.org/10.7146/math.scand.a-126042