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Euler characteristic of primitive T-hypersurfaces and maximal surfaces

Part of: Polytopes and polyhedra Special varieties

Published online by Cambridge University Press:  23 July 2009

Benoit Bertrand
Benoit Bertrand
Affiliation:
Institut de Mathématiques de Toulouse, IUT de Tarbes, 1 rue Lautréamont, BP 1624, 65016 Tarbes cedex, France ([email protected])
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Abstract

Viro method plays an important role in the study of topology of real algebraic hypersurfaces. The T-primitive hypersurfaces we study here appear as the result of Viro's combinatorial patchworking when one starts with a primitive triangulation. We show that the Euler characteristic of the real part of such a hypersurface of even dimension is equal to the signature of its complex part. We explain how this can be understood in tropical geometry. We use this result to prove the existence of maximal surfaces in some three-dimensional toric varieties, namely those corresponding to Nakajima polytopes.

Keywords

Viro methodcombinatorial patchworkingtropical geometrytopology of real algebraic varietiestropical hypersurfacestoric varieties

MSC classification

Secondary: 14P25: Topology of real algebraic varieties 14M25: Toric varieties, Newton polyhedra 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 9 , Issue 1 , January 2010 , pp. 1 - 27
Copyright
Copyright © Cambridge University Press 2010

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References

1.Bertrand, B., Maximal hypersurfaces and complete intersections in toric varieties, PhD thesis, Université de Genève (2003; available at www.unige.ch/math/folks/bertrand).Google Scholar
2.Bertrand, B., Asymptotically maximal families of hypersurfaces in toric varieties, Geom. Dedicata 118 (2006), 49–70.CrossRefGoogle Scholar
3.Bertrand, B. and Bihan, F., Euler characteristic of real non degenerate tropical complete intersections, preprint arXiv:0710.1222 (2007).Google Scholar
4.Brion, M., Points entiers dans les polytopes convexes, Astérisque, Volume 227, Séminaire Bourbaki, Volume 780 (1994).Google Scholar
5.Brion, M. and Vergne, M., Lattice points in simple polytopes, J. Am. Math. Soc. 10(2) (1997), 371–392.CrossRefGoogle Scholar
6.Dais, D., Über unimodulare, kohärente Triangulierungen von Gitterpolytopen: Beispiele und Anwendungen, Talk given at Grenoble Summer School: Géométrie des Variétés Toriques, 2000.Google Scholar
7.Danilov, V. I., The geometry of toric varieties, Usp. Mat. Nauk 33(2) (1978), 85–134.Google Scholar
8.Danilov, V. and Khovanskii, A., Newton polyhedra and an algorithm for computing Hodge–Deligne numbers, Math. USSR Izv. 29(2) (1987), 279–298.Google Scholar
9.Dimitrios, I., Dais, C.-H. and Ziegler, G. M., All toric local complete intersection singularities admit projective crepant resolutions, Tohoku Math. J. 53(1) (2001), 95–107.Google Scholar
10.Ehrhart, E., Sur un problème de géométrie diophantienne linéaire, I, Polyèdres et réseaux, J. Reine Angew. Math. 226 (1967), 1–29.Google Scholar
11.Ehrhart, E., Un théorème arithmo-géométrique et ses généralisations, L'Ouvert 77 (1994), 33–34.Google Scholar
12.Fulton, W., Introduction to toric varieties (Princeton University Press, 1993).CrossRefGoogle Scholar
13.Gelfand, I., Kapranov, M. and Zelevinsky, A., Discriminants, resultants and multidimensional determinants (Springer, 1994).CrossRefGoogle Scholar
14.Griffiths, P. and Harris, J., Principles of algebraic geometry (Wiley, 1978).Google Scholar
15.Itenberg, I., Topology of real algebraic T-surfaces, Rev. Mat. Complut. 10 (1997), 131–152.CrossRefGoogle Scholar
16.Itenberg, I. and Viro, O., Maximal real algebraic hypersurface of projective space, in preparation (2005).Google Scholar
17.Mikhalkin, G., Real algebraic curves, the moment map and amoebas, Annals Math. (2) 151(1) (2000), 309–326.CrossRefGoogle Scholar
18.Mikhalkin, G., Amoebas of algebraic varieties and tropical geometry, in Different faces of geometry, International Mathematics Series, Volume 3, pp. 257–300 (Kluwer/Plenum, New York, 2004).CrossRefGoogle Scholar
19.Mikhalkin, G., Enumerative tropical algebraic geometry in ℝ2, J. Am. Math. Soc. 18(2) (2005), 313–377.CrossRefGoogle Scholar
20.Rokhlin, V. A., Congruences modulo 16 in Hilbert's sixteenth problem, Funkt. Analiz Prilozhen. 6(4) (1972), 58–64.Google Scholar
21.Sturmfels, B., Viro's theorem for complete intersections, Annali Scuola Norm. Sup. Pisa IV 21(3) (1994), 377–386.Google Scholar
22.Van Lint, J. H. and Wilson, R. M., A course in combinatorics (Cambridge University Press, 1992).Google Scholar
23.Viro, O. J., Construction of M-surfaces, Funkt. Analiz Prilozhen. 13(3) (1979), 71–72.Google Scholar
24.Viro, O. J., Gluing of plane algebraic curves and construction of curves of degree 6 and 7, Lecture Notes in Mathematics, Volume 1060, pp. 187–200 (Springer, 1984).Google Scholar
25.Viro, O. J., Dequantization of real algebraic geometry on logarithmic paper, in European Congress of Mathematics, Volume I, Barcelona, 2000, Progress in Mathematics, Volume 201, pp. 135–146 (Birkhäuser, Basel, 2001).Google Scholar
26.Viro, O. J., Patchworking real algebraic varieties, preprint, Uppsala University (2004).Google Scholar
27.Ziegler, G., Lectures on polytopes (Springer, 1995).Google Scholar