Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs


Author: June Huh
Journal: J. Amer. Math. Soc. 25 (2012), 907-927
MSC (2010): Primary 14B05, 05B35
DOI: https://doi.org/10.1090/S0894-0347-2012-00731-0
Published electronically: February 8, 2012
MathSciNet review: 2904577
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The chromatic polynomial $ \chi _G(q)$ of a graph $ G$ counts the number of proper colorings of $ G$. We give an affirmative answer to the conjecture of Read and Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic polynomial form a log-concave sequence. The proof is obtained by identifying $ \chi _G(q)$ with a sequence of numerical invariants of a projective hypersurface analogous to the Milnor number of a local analytic hypersurface. As a by-product of our approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor number with the Newton polytope; we also characterize homology classes of $ \mathbb{P}^n \times \mathbb{P}^m$ corresponding to subvarieties and answer a question posed by Trung-Verma.


References [Enhancements On Off] (What's this?)

  • 1. M. Aigner, Whitney numbers, Combinatorial Geometries, 139-160, Encyclopedia Math. Appl., 29, Cambridge Univ. Press, Cambridge, 1987. MR 0921072
  • 2. P. Aluffi, Computing characteristic classes of projective schemes, J. Symbolic Comput., 35 (2003), no. 1, 3-19. MR 1956868 (2004b:14007)
  • 3. F. Brenti, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, Jerusalem Combinatorics '93, Contemp. Math. 178, Amer. Math. Soc., Providence, RI, 1994, pp. 71-89. MR 1310575 (95j:05026)
  • 4. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993. MR 1251956 (95h:13020)
  • 5. D. Cox, J. Little, and D. O'Shea, Using Algebraic Geometry, Graduate Texts in Mathematics, 185, Springer, New York, 1998. MR 2122859 (2005i:13037)
  • 6. A. Dimca, Singularities and Topology of Hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 1194180 (94b:32058)
  • 7. A. Dimca and S. Papadima, Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements, Ann. of Math. (2) 158 (2003), no. 2, 473-507. MR 2018927 (2005a:32028)
  • 8. W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993. MR 1234037 (94g:14028)
  • 9. T. Gaffney, Integral closure of modules and Whitney equisingularity, Invent. Math. 107 (1992), no. 2, 301-322. MR 1144426 (93d:32055)
  • 10. T. Gaffney, Multiplicities and equisingularity of ICIS germs, Invent. Math. 123 (1996), no. 2, 209-220. MR 1374196 (97b:32051)
  • 11. M. Gromov, Convex sets and Kähler manifolds, Advances in Differential Geometry and Topology, World Sci. Publ., Teaneck, NJ, 1990, pp. 1-38. MR 1095529 (92d:52018)
  • 12. J. Harris, Algebraic Geometry. A First Course, Corrected reprint of the 1992 original., Graduate Texts in Mathematics, 133, Springer-Verlag, New York, 1995. MR 1416564 (97e:14001)
  • 13. R. Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc., 80 (1974), 1017-1032. MR 0384816 (52:5688)
  • 14. A. P. Heron, Matroid polynomials, Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), Inst. of Math. and its Appl., Southend-on-Sea, 1972, pp. 164-202. MR 0340058 (49:4814)
  • 15. C. Huneke and I. Swanson, Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series, 336, Cambridge University Press, Cambridge, 2006. MR 2266432 (2008m:13013)
  • 16. J.-P. Jouanolou, Théorèmes de Bertini et Applications, Progress in Mathematics, 42, Birkhäuser Boston, Inc., Boston, MA, 1983. MR 0725671 (86b:13007)
  • 17. K. Kaveh and A. G. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Preprint: arXiv:0904.3350v2.
  • 18. A. G. Khovanskii, Algebra and mixed volumes, Appendix 3 in: Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Translated from the Russian by A. B. Sosinskii, Grundlehren der Mathematischen Wissenschaften, 285, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1988. MR 0936419 (89b:52020)
  • 19. A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1-31. MR 0419433 (54:7454)
  • 20. J. P. S. Kung, The geometric approach to matroid theory, Gian-Carlo Rota on Combinatorics, 604-622, Contemp. Mathematicians, Birkhäuser Boston, Boston, MA, 1995. MR 1392975
  • 21. R. Lazarsfeld, Positivity in Algebraic Geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 48, Springer-Verlag, Berlin, 2004. MR 2095471 (2005k:14001a)
  • 22. R. Lazarsfeld and M. Mustaţă, Convex bodies associated to linear series, Ann. Sci. École Norm. Supér. (4) 42 (2009), no. 5, 783-835. MR 2571958 (2011e:14012)
  • 23. R. D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423-432. MR 0361141 (50:13587)
  • 24. D. Marker, Model Theory. An Introduction, Graduate Texts in Mathematics, 217. Springer-Verlag, New York, 2002. MR 1924282 (2003e:03060)
  • 25. D. G. Northcott and D. Rees, Reduction of ideals in local rings, Proc. Cambridge Phil. Soc. 50 (1954), 145-158. MR 0059889 (15:596a)
  • 26. A. Okounkov, A Brunn-Minkowski inequality for multiplicities, Invent. Math. 125 (1996), no. 3, 405-411. MR 1400312 (99a:58074)
  • 27. J. Oxley, Matroid Theory, Oxford Science Publications, Oxford University Press, New York, 2011. MR 2849819
  • 28. P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167-189. MR 0558866 (81e:32015)
  • 29. P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften, 300, Springer-Verlag, Berlin, 1992. MR 1217488 (94e:52014)
  • 30. R. Randell, Morse theory, Milnor fibers and minimality of hyperplane arrangements, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2737-2743. MR 1900880 (2003e:32048)
  • 31. R. C. Read, An introduction to chromatic polynomials, J. Combinatorial Theory 4 (1968), 52-71. MR 0224505 (37:104)
  • 32. D. Rees and R.Y. Sharp, On a theorem of B. Teissier on multiplicities of ideals in local rings, J. London Math. Soc. (2) 18 (1978), no. 3, 449-463. MR 0518229 (80e:13009)
  • 33. G.-C. Rota, Combinatorial theory, old and new, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3, Gauthier-Villars, Paris, 1971, pp. 229-233. MR 0505646 (58:21703)
  • 34. P. Samuel, La notion de multiplicité en algèbre et géométrie algébrique, J. Math. Pures Appl. (9) 30 (1951), 159-274. MR 0048103 (13:980c)
  • 35. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993. MR 1216521 (94d:52007)
  • 36. I. R. Shafarevich, Basic Algebraic Geometry. 1. Varieties in Projective Space, Second edition, Springer-Verlag, Berlin, 1994. MR 1328833 (95m:14001)
  • 37. G. C. Shephard, Inequalities between mixed volumes of convex sets, Mathematika, 7 (1960), 125-138. MR 0146736 (26:4256)
  • 38. R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph Theory and Its Applications: East and West (Jinan 1986), Ann. New York Acad. Sci., 576, 1989, pp. 500-535. MR 1110850 (92e:05124)
  • 39. R. P. Stanley, Foundations I and the development of algebraic combinatorics, Gian-Carlo Rota on Combinatorics, 105-107, Contemp. Mathematicians, Birkhäuser Boston, Boston, MA, 1995. MR 1392967
  • 40. R. P. Stanley, Positivity problems and conjectures in algebraic combinatorics, Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295-319. MR 1754784 (2001f:05001)
  • 41. R. P. Stanley, An introduction to hyperplane arrangements, Geometric Combinatorics, IAS/Park City Math. Ser., 13, Amer. Math. Soc., Providence, RI, 2007, pp. 389-496. MR 2383131
  • 42. B. Teissier, Cycles évanescents, sections planes et conditions de Whitney, Singularités à Cargèse (Rencontre Singularités Géom. Anal., Inst. Etudes Sci., Cargèse, 1972), Astérisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973, pp. 285-362. MR 0374482 (51:10682)
  • 43. B. Teissier, Appendix: Sur une inégalité à la Minkowski pour les multiplicités, in: D. Eisenbud and H. Levine, An algebraic formula for the degree of a $ C^\infty $ map germ, Ann. of Math. (2) 106 (1977), no. 1, 38-44. MR 0467800 (57:7651)
  • 44. B. Teissier, Du théorème de l'index de Hodge aux inégalités isopérimétriques, C. R. Acad. Sci. Paris Ser. A-B 288 (1979), no. 4, 287-289. MR 0524795 (80k:14014)
  • 45. B. Teissier, Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney, Algebraic Geometry (La Rábida, 1981), Lecture Notes in Math., 961, Springer, Berlin, 1982, pp. 314-491. MR 708342 (85i:32019)
  • 46. N. V. Trung, Positivity of mixed multiplicities, Math. Ann. 319 (2001), no. 1, 33-63. MR 1812818 (2001m:13042)
  • 47. N. V. Trung and J. K. Verma, Mixed multiplicities of ideals versus mixed volumes of polytopes, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4711-4727. MR 2320648 (2008e:13029)
  • 48. D. Welsh, Matroid Theory, London Mathematical Society Monographs, 8, Academic Press, London-New York, 1976. MR 0427112 (55:148)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 14B05, 05B35

Retrieve articles in all journals with MSC (2010): 14B05, 05B35


Additional Information

June Huh
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: huh14@illinois.edu, junehuh@umich.edu

DOI: https://doi.org/10.1090/S0894-0347-2012-00731-0
Keywords: Chern-Schwartz-MacPherson class, characteristic polynomial, chromatic polynomial, Milnor number, Okounkov body
Received by editor(s): July 10, 2011
Received by editor(s) in revised form: January 17, 2012
Published electronically: February 8, 2012
Additional Notes: The author acknowledges support from National Science Foundation grant DMS 0838434 “EMSW21-MCTP: Research Experience for Graduate Students”.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society