Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-24T10:11:03.693Z Has data issue: false hasContentIssue false
  • English
  • Français

The maximum likelihood degree of a very affine variety

Part of: Local theory Cycles and subschemes Polytopes and polyhedra

Published online by Cambridge University Press:  24 May 2013

June Huh
June Huh*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA email junehuh@umich.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao’s solution to Varchenko’s conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern–Schwartz–MacPherson class. The strengthened version recovers the geometric deletion–restriction formula of Denham et al. for arrangement complements, and generalizes Kouchnirenko’s theorem on the Newton polytope for nondegenerate hypersurfaces.

Keywords

maximum likelihood degreelogarithmic differential formChern–Schwartz–MacPherson class

MSC classification

Secondary: 14B05: Singularities 14C17: Intersection theory, characteristic classes, intersection multiplicities 52B40: Matroids (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 8 , August 2013 , pp. 1245 - 1266
Copyright
© The Author(s) 2013 

References

Aluffi, P., Differential forms with logarithmic poles and Chern–Schwartz–MacPherson classes of singular varieties, C. R. Acad. Sci. Sér. I. Math. 329 (1999), 619–624.Google Scholar
Aluffi, P., Computing characteristic classes of projective schemes, J. Symbolic Comput. 35 (2003), 3–19.CrossRefGoogle Scholar
Aluffi, P., Characteristic classes of singular varieties, in Topics in cohomological studies of algebraic varieties, Trends in Mathematics (Birkhäuser, Basel, 2005), 1–32.Google Scholar
Aluffi, P., Classes de Chern des variétés singulières, revisitées, C. R. Math. Acad. Sci. Paris 342 (2006), 405–410.Google Scholar
Aluffi, P., Limits of Chow groups, and a new construction of Chern–Schwartz–MacPherson classes, Pure Appl. Math. Q. 2 (2006), 915–941.Google Scholar
Aluffi, P., Grothendieck classes and Chern classes of hyperplane arrangements, Int. Math. Res. Not. (2012), doi:10.1093/imrn/rns100.Google Scholar
Catanese, F., Hosten, S., Khetan, A. and Sturmfels, B., The maximum likelihood degree, Amer. J. Math. 128 (2006), 671–697.CrossRefGoogle Scholar
Ciliberto, C., Russo, F. and Simis, A., Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian, Adv. Math. 218 (2008), 1759–1805.Google Scholar
Cohen, D., Denham, G., Falk, M. and Varchenko, A., Critical points and resonance of hyperplane arrangements, Canad. J. Math. 63 (2011), 1038–1057.CrossRefGoogle Scholar
Cox, D., Little, J. and O’Shea, D., Using algebraic geometry, Graduate Texts in Mathematics, vol. 185 (Springer, New York, 1998).Google Scholar
Damon, J., Critical points of affine multiforms on the complements of arrangements, in Singularity theory, London Mathematical Society Lecture Note Series, vol. 263 (Cambridge University Press, Cambridge, 1999), 25–33.Google Scholar
Damon, J., On the number of bounding cycles for nonlinear arrangements, in Arrangements - Tokyo 1998, Advanced Studies in Pure Mathematics, vol. 27 (Kinokuniya, Tokyo, 2000), 51–72.Google Scholar
Dawson, J., A collection of sets related to the Tutte polynomial of a matroid, in Graph theory, Singapore 1983, Lecture Notes in Mathematics, vol. 1073 (Springer, Berlin, 1984), 193–204.Google Scholar
De Concini, C. and Procesi, C., Wonderful models of subspace arrangements, Selecta Math. (N.S.) 1 (1995), 459–494.CrossRefGoogle Scholar
Deligne, P., Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, vol. 163 (Springer, Berlin, 1970).Google Scholar
Denham, G., Garrousian, M. and Schulze, M., A geometric deletion–restriction formula, Adv. Math. 230 (2012), 1979–1994.Google Scholar
Dimca, A. and Papadima, S., Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements, Ann. of Math. (2) 158 (2003), 473–507.Google Scholar
Dolgachev, I., Polar Cremona transformations, Michigan Math. J. 48 (2000), 191–202.Google Scholar
Dolgachev, I., Classical algebraic geometry: a modern view (Cambridge University Press, Cambridge, 2012).Google Scholar
Ein, L. and Shepherd-Barron, N., Some special Cremona transformations, Amer. J. Math. 111 (1989), 783–800.Google Scholar
Einsiedler, M., Kapranov, M. and Lind, D., Non-archimedean amoebas and tropical varieties, J. Reine Angew. Math. 601 (2006), 139–157.Google Scholar
Etingof, P., Kazhdan, D. and Polishchuk, A., When is the Fourier transform of an elementary function elementary?, Selecta Math. (N.S.) 8 (2002), 27–66.Google Scholar
Fassarella, T. and Medeiros, N., On the polar degree of projective hypersurfaces, J. Lond. Math. Soc. (2) 86 (2012), 259–271.CrossRefGoogle Scholar
Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, NJ, 1993).Google Scholar
Fulton, W., Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge [A Series of Modern Surveys in Mathematics], vol. 2, second edition (Springer, Berlin, 1998).Google Scholar
Gordan, P. and Noether, M., Ueber die algebraischen Formen, deren Hesse’sche Determinate identisch verschwindet, Math. Ann. 10 (1876), 547–568.Google Scholar
Goresky, M. and Pardon, W., Chern classes of automorphic vector bundles, Invent. Math. 147 (2002), 561–612.CrossRefGoogle Scholar
Hacking, P., Homology of tropical varieties, Collect. Math. 59 (2008), 263–273.CrossRefGoogle Scholar
Hesse, O., 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 zurück-führen läfsst, die eine Variable weniger enthält, J. Reine Angew. Math. 42 (1851), 117–124.Google Scholar
Hesse, O., Zur Theorie der ganzen homogenen Functionen, J. Reine Angew. Math. 56 (1859), 263–269.Google Scholar
Hoggar, S., Chromatic polynomials and logarithmic concavity, J. Combin. Theory Ser. B 16 (1974), 248–254.Google Scholar
Hoşten, S., Khetan, A. and Sturmfels, B., Solving likelihood equations, Found. Comput. Math. 5 (2005), 389–407.Google Scholar
Hovanskiĭ, A., Newton polyhedra and toroidal varieties, Funktsional. Anal. i Prilozhen. 11 (1977), 56–64.Google Scholar
Huh, J., Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, J. Amer. Math. Soc. 25 (2012), 907–927.CrossRefGoogle Scholar
Huh, J., h-vectors of matroids and logarithmic concavity, Preprint (2012), arXiv:1201.2915.Google Scholar
Jouanolou, J.-P., Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42 (Birkhäuser, Boston, MA, 1983).Google Scholar
Kawamata, Y., On deformations of compactifiable complex manifolds, Math. Ann. 235 (1978), 247–265.Google Scholar
Kouchnirenko, A. G., Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), 1–31.CrossRefGoogle Scholar
Kwieciński, M., Formule du produit pour les classes caractéristiques de Chern–Schwartz–MacPherson et homologie d’intersection, C. R. Acad. Sci. Sér. I. Math. 314 (1992), 625–628.Google Scholar
Lenz, M., The f-vector of a realizable matroid complex is strictly log-concave, Combin. Probab. Comput., to appear.Google Scholar
MacPherson, R., Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432.Google Scholar
Norimatsu, Y., Kodaira vanishing theorem and Chern classes for $\partial $-manifolds, Proc. Japan Acad. Ser. A 54 (1978), 107–108.CrossRefGoogle Scholar
Orlik, P. and Terao, H., Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, vol. 300 (Springer, Berlin, 1992).Google Scholar
Orlik, P. and Terao, H., The number of critical points of a product of powers of linear functions, Invent. Math. 120 (1995), 1–14.Google Scholar
Pachter, L. and Sturmfels, B., Algebraic statistics for computational biology (Cambridge University Press, New York, 2005).Google Scholar
Read, R., An introduction to chromatic polynomials, J. Combin. Theory 4 (1968), 52–71.CrossRefGoogle Scholar
Saito, K., Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 27 (1980), 265–291.Google Scholar
Samuel, P., A propos du théorème des unités, Bull. Sci. Math. (2) 90 (1966), 89–96.Google Scholar
Schneider, R., Convex bodies: the Brunn–Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44 (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
Silvotti, R., On a conjecture of Varchenko, Invent. Math. 126 (1996), 235–248.CrossRefGoogle Scholar
Speyer, D., Tropical geometry, PhD thesis, University of California, Berkeley (2005).Google Scholar
Speyer, D. and Sturmfels, B., The tropical Grassmannian, Adv. Geom. 4 (2004), 389–411.Google Scholar
Tevelev, J., Compactifications of subvarieties of tori, Amer. J. Math. 129 (2007), 1087–1104.Google Scholar
Varchenko, A., Zeta-function of monodromy and Newton’s diagram, Invent. Math. 37 (1976), 253–262.Google Scholar
Varchenko, A., Critical points of the product of powers of linear functions and families of bases of singular vectors, Compositio Math. 97 (1995), 385–401.Google Scholar
Welsh, D., Combinatorial problems in matroid theory, in Combinatorial mathematics and its applications (Oxford, 1969) (Academic Press, London, 1971), 291–306.Google Scholar