On Huh’s conjectures for the polar degree
Authors:
Dirk Siersma, Joseph Steenbrink and Mihai Tibăr
Journal:
J. Algebraic Geom. 30 (2021), 189-203
DOI:
https://doi.org/10.1090/jag/741
Published electronically:
December 16, 2019
MathSciNet review:
4233181
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Abstract |
References |
Additional Information
Abstract: We prove a precise version of a general conjecture on the polar degree stated by June Huh. We confirm Huh’s conjectural list of all projective hypersurfaces with isolated singularities and polar degree equal to 2.
References
- V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi. MR 966191
- J. W. Bruce, An upper bound for the number of singularities on a projective hypersurface, Bull. London Math. Soc. 13 (1981), no. 1, 47–50. MR 599641, DOI https://doi.org/10.1112/blms/13.1.47
- J. W. Bruce and C. T. C. Wall, On the classification of cubic surfaces, J. London Math. Soc. (2) 19 (1979), no. 2, 245–256. MR 533323, DOI https://doi.org/10.1112/jlms/s2-19.2.245
- Alexandru Dimca, On polar Cremona transformations, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 9 (2001), no. 1, 47–53. To Mirela Ştefănescu, at her 60’s. MR 1946153
- Alexandru Dimca and Stefan Papadima, Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments, Ann. of Math. (2) 158 (2003), no. 2, 473–507. MR 2018927, DOI https://doi.org/10.4007/annals.2003.158.473
- Igor V. Dolgachev, Polar Cremona transformations, Michigan Math. J. 48 (2000), 191–202. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786486, DOI https://doi.org/10.1307/mmj/1030132714
- Thiago Fassarella and Nivaldo Medeiros, On the polar degree of projective hypersurfaces, J. Lond. Math. Soc. (2) 86 (2012), no. 1, 259–271. MR 2959304, DOI https://doi.org/10.1112/jlms/jds005
- P. Gordan and M. Nöther, Ueber die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet, Math. Ann. 10 (1876), no. 4, 547–568 (German). MR 1509898, DOI https://doi.org/10.1007/BF01442264
- O. Hesse, Über die Bedingung, unter welcher eine homogene ganze Function von $n$ unabhängigen Variabeln durch lineäre Substitutionen von $n$ andern unabhängigen Variabeln auf eine homogene Function sich zurückführen läßt, die eine Variable weniger enthält, J. Reine Angew. Math. 42 (1851), 117–124 (German). MR 1578739, DOI https://doi.org/10.1515/crll.1851.42.117
- Otto Hesse, Zur Theorie der ganzen homogenen Functionen, J. Reine Angew. Math. 56 (1859), 263–269 (German). MR 1579101, DOI https://doi.org/10.1515/crll.1859.56.263
- June Huh, Milnor numbers of projective hypersurfaces with isolated singularities, Duke Math. J. 163 (2014), no. 8, 1525–1548. MR 3210967, DOI https://doi.org/10.1215/00127094-2713700
- Dirk Siersma, Classification and deformation of singularities, University of Amsterdam, Amsterdam, 1974. With Dutch and English summaries; Doctoral dissertation, University of Amsterdam. MR 0350775
- J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525–563. MR 0485870
- Joseph Steenbrink, Intersection form for quasi-homogeneous singularities, Compositio Math. 34 (1977), no. 2, 211–223. MR 453735
- J. H. M. Steenbrink, Semicontinuity of the singularity spectrum, Invent. Math. 79 (1985), no. 3, 557–565. MR 782235, DOI https://doi.org/10.1007/BF01388523
- Mihai Tibăr, Polynomials and vanishing cycles, Cambridge Tracts in Mathematics, vol. 170, Cambridge University Press, Cambridge, 2007. MR 2360234
- A. N. Varchenko, Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface, Dokl. Akad. Nauk SSSR 270 (1983), no. 6, 1294–1297 (Russian). MR 712934
References
- V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Monodromy and asymptotics of integrals, translated from the Russian by Hugh Porteous, translation revised by the authors and James Montaldi, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. MR 966191
- J. W. Bruce, An upper bound for the number of singularities on a projective hypersurface, Bull. London Math. Soc. 13 (1981), no. 1, 47–50. MR 599641, DOI https://doi.org/10.1112/blms/13.1.47
- J. W. Bruce and C. T. C. Wall, On the classification of cubic surfaces, J. London Math. Soc. (2) 19 (1979), no. 2, 245–256. MR 533323, DOI https://doi.org/10.1112/jlms/s2-19.2.245
- Alexandru Dimca, On polar Cremona transformations, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 9 (2001), no. 1, 47–53. MR 1946153
- Alexandru Dimca and Stefan Papadima, Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments, Ann. of Math. (2) 158 (2003), no. 2, 473–507. MR 2018927, DOI https://doi.org/10.4007/annals.2003.158.473
- Igor V. Dolgachev, Polar Cremona transformations, Michigan Math. J. 48 (2000), 191–202. MR 1786486, DOI https://doi.org/10.1307/mmj/1030132714
- Thiago Fassarella and Nivaldo Medeiros, On the polar degree of projective hypersurfaces, J. Lond. Math. Soc. (2) 86 (2012), no. 1, 259–271. MR 2959304, DOI https://doi.org/10.1112/jlms/jds005
- P. Gordan and M. Nöther, Ueber die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet, Math. Ann. 10 (1876), no. 4, 547–568 (German). MR 1509898, DOI https://doi.org/10.1007/BF01442264
- O. Hesse, Über die Bedingung, unter welcher eine homogene ganze Function von $n$ unabhängigen Variabeln durch lineäre Substitutionen von $n$ andern unabhängigen Variabeln auf eine homogene Function sich zurückführen läßt, die eine Variable weniger enthält, J. Reine Angew. Math. 42 (1851), 117–124 (German). MR 1578739, DOI https://doi.org/10.1515/crll.1851.42.117
- Otto Hesse, Zur Theorie der ganzen homogenen Functionen, J. Reine Angew. Math. 56 (1859), 263–269 (German). MR 1579101, DOI https://doi.org/10.1515/crll.1859.56.263
- June Huh, Milnor numbers of projective hypersurfaces with isolated singularities, Duke Math. J. 163 (2014), no. 8, 1525–1548. MR 3210967, DOI https://doi.org/10.1215/00127094-2713700
- Dirk Siersma, Classification and deformation of singularities, with Dutch and English summaries, Doctoral dissertation, University of Amsterdam, University of Amsterdam, Amsterdam, 1974. MR 0350775
- J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525–563. MR 0485870
- Joseph Steenbrink, Intersection form for quasi-homogeneous singularities, Compositio Math. 34 (1977), no. 2, 211–223. MR 453735
- J. H. M. Steenbrink, Semicontinuity of the singularity spectrum, Invent. Math. 79 (1985), no. 3, 557–565. MR 782235, DOI https://doi.org/10.1007/BF01388523
- Mihai Tibăr, Polynomials and vanishing cycles, Cambridge Tracts in Mathematics, vol. 170, Cambridge University Press, Cambridge, 2007. MR 2360234
- A. N. Varchenko, Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface, Dokl. Akad. Nauk SSSR 270 (1983), no. 6, 1294–1297 (Russian). MR 712934
Additional Information
Dirk Siersma
Affiliation:
Institute of Mathematics, Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The Netherlands
MR Author ID:
161855
Email:
D.Siersma@uu.nl
Joseph Steenbrink
Affiliation:
IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands
MR Author ID:
166630
Email:
j.steenbrink@math.ru.nl
Mihai Tibăr
Affiliation:
University Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France
Email:
mihai-marius.tibar@univ-lille.fr
Received by editor(s):
November 21, 2018
Received by editor(s) in revised form:
April 4, 2019
Published electronically:
December 16, 2019
Additional Notes:
The first and third authors express their gratitude to the Mathematisches Forschungsinstitut Obewolfach for supporting this research project through the Research in Pairs program 2017 and acknowledge the support of the Labex CEMPI (ANR-11-LABX-0007-01)
Article copyright:
© Copyright 2019
University Press, Inc.
