Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T04:28:31.916Z Has data issue: false hasContentIssue false
  • English
  • Français

Sums of three squares and Noether–Lefschetz loci

Part of: Families, fibrations Forms and linear algebraic groups Special varieties

Published online by Cambridge University Press:  03 April 2018

Olivier Benoist
Olivier Benoist*
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000 Strasbourg, France email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in $\mathbb{P}^{3}$ whose function field has level 2 is dense in the set of those that have no real points.

Keywords

sums of squaresHodge theoryreal algebraic geometryvariations of Hodge structures

MSC classification

Primary: 11E25: Sums of squares and representations by other particular quadratic forms
Secondary: 14P99: None of the above, but in this section 14D07: Variation of Hodge structures 14M12: Determinantal varieties
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 5 , May 2018 , pp. 1048 - 1065
Copyright
© The Author 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Atiyah, M. F., Complex analytic connections in fibre bundles , Trans. Amer. Math. Soc. 85 (1957), 181–207.CrossRefGoogle Scholar
Audin, M., The topology of torus actions on symplectic manifolds, Progress in Mathematics, vol. 93 (Birkhäuser, Basel, 1991).CrossRefGoogle Scholar
Benoist, O., On Hilbert’s 17th problem in low degree , Algebra Number Theory 11 (2017), 929–959.CrossRefGoogle Scholar
Benoist, O. and Wittenberg, O., On the integral Hodge conjecture for real varieties, II, Preprint (2018), arXiv:1801.00873.Google Scholar
Berg, C., Christensen, J. P. R. and Jensen, C. U., A remark on the multidimensional moment problem , Math. Ann. 243 (1979), 163–169.CrossRefGoogle Scholar
Biswas, I., Huisman, J. and Hurtubise, J., The moduli space of stable vector bundles over a real algebraic curve , Math. Ann. 347 (2010), 201–233.CrossRefGoogle Scholar
Blekherman, G., Nonnegative polynomials and sums of squares , in Semidefinite optimization and convex algebraic geometry, MOS-SIAM Series on Optimization, vol. 13 (SIAM, Philadelphia, PA, 2013), 159–202.Google Scholar
Blekherman, G., Hauenstein, J., Ottem, J. C., Ranestad, K. and Sturmfels, B., Algebraic boundaries of Hilbert’s SOS cones , Compos. Math. 148 (2012), 1717–1735.CrossRefGoogle Scholar
Bochnak, J., Coste, M. and Roy, M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36, translation of 1987 French original, revised by the authors (Springer, Berlin, 1998).CrossRefGoogle Scholar
Bruns, W. and Vetter, U., Determinantal rings, Lecture Notes in Mathematics, vol. 1327 (Springer, Berlin, 1988).CrossRefGoogle Scholar
Bruzzo, U., Grassi, A. and Lopez, A., Existence and density of general components of the Noether–Lefschetz locus on normal threefolds, Preprint (2017), arXiv:1706.02081.Google Scholar
Cassels, J. W. S., Ellison, W. J. and Pfister, A., On sums of squares and on elliptic curves over function fields , J. Number Theory 3 (1971), 125–149.CrossRefGoogle Scholar
Ciliberto, C., Harris, J. and Miranda, R., General components of the Noether–Lefschetz locus and their density in the space of all surfaces , Math. Ann. 282 (1988), 667–680.CrossRefGoogle Scholar
Ciliberto, C. and Lopez, A., On the existence of components of the Noether–Lefschetz locus with given codimension , Manuscripta Math. 73 (1991), 341–357.CrossRefGoogle Scholar
Colliot-Thélène, J.-L., The Noether–Lefschetz theorem and sums of 4 squares in the rational function field R(x, y) , Compos. Math. 86 (1993), 235–243.Google Scholar
Dimca, A., Monodromy and Betti numbers of weighted complete intersections , Topology 24 (1985), 369–374.CrossRefGoogle Scholar
Eisenbud, D., The geometry of syzygies, Graduate Texts in Mathematics, vol. 229 (Springer, New York, ny, 2005).Google Scholar
Gille, P. and Szamuely, T., Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101 (Cambridge University Press, Cambridge, 2006).CrossRefGoogle Scholar
Harris, J. and Tu, L. W., On symmetric and skew-symmetric determinantal varieties , Topology 23 (1984), 71–84.CrossRefGoogle Scholar
Hilbert, D., Ueber die Darstellung definiter Formen als Summe von Formenquadraten , Math. Ann. 32 (1888), 342–350.CrossRefGoogle Scholar
Hilbert, D., Über ternäre definite Formen , Acta Math. 17 (1893), 169–197.CrossRefGoogle Scholar
Huybrechts, D., Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, vol. 158 (Cambridge University Press, Cambridge, 2016).CrossRefGoogle Scholar
Jouanolou, J.-P., Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42 (Birkhäuser, Boston, MA, 1983).Google Scholar
Kim, S.-O., Noether–Lefschetz locus for surfaces , Trans. Amer. Math. Soc. 324 (1991), 369–384.CrossRefGoogle Scholar
Kleiman, S. L. and Altman, A. B., Bertini theorems for hypersurface sections containing a subscheme , Comm. Algebra 7 (1979), 775–790.CrossRefGoogle Scholar
Krasnov, V. A., Characteristic classes of vector bundles on a real algebraic variety , Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), 716–746.Google Scholar
Lam, T. Y., The algebraic theory of quadratic forms, Mathematics Lecture Note Series, revised second printing (Benjamin Cummings, Reading, MA, 1980).Google Scholar
Landau, E., Über die Darstellung definiter binärer Formen durch Quadrate , Math. Ann. 57 (1903), 53–64.CrossRefGoogle Scholar
Leep, D. B. and Starr, C. L., Polynomials in ℝ[x, y] that are sums of squares in ℝ(x, y) , Proc. Amer. Math. Soc. 129 (2001), 3133–3141.CrossRefGoogle Scholar
Manetti, M., Degenerate double covers of the projective plane , in New trends in algebraic geometry, Warwick, 1996, London Mathematical Society Lecture Note Series, vol. 264 (Cambridge University Press, Cambridge, 1999), 255–281.CrossRefGoogle Scholar
Mangolte, F., Variétés algébriques réelles, Cours Spécialisés, vol. 24 (Société Mathématique de France, Paris, 2017).Google Scholar
Pfister, A., Zur Darstellung von - 1 als Summe von Quadraten in einem Körper , J. Lond. Math. Soc. 40 (1965), 159–165.CrossRefGoogle Scholar
Pfister, A., Zur Darstellung definiter Funktionen als Summe von Quadraten , Invent. Math. 4 (1967), 229–237.CrossRefGoogle Scholar
Silhol, R., Real algebraic surfaces, Lecture Notes in Mathematics, vol. 1392 (Springer, Berlin, 1989).CrossRefGoogle Scholar
van Hamel, J., Divisors on real algebraic varieties without real points , Manuscripta Math. 98 (1999), 409–424.CrossRefGoogle Scholar
van Hamel, J., Algebraic cycles and topology of real algebraic varieties, CWI Tract, vol. 129 (Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 2000).Google Scholar
Voisin, C., Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés, vol. 10 (Société Mathématique de France, Paris, 2002).Google Scholar
Voisin, C., On integral Hodge classes on uniruled or Calabi-Yau threefolds , in Moduli spaces and arithmetic geometry, Advanced Studies in Pure Mathematics, vol. 45 (Mathematical Society Japan, Tokyo, 2006), 43–73.CrossRefGoogle Scholar