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Intrinsic signs and lower bounds in real algebraic geometry

  • Christian Okonek EMAIL logo und Andrei Teleman
Veröffentlicht/Copyright: 8. Juni 2012
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Journal für die reine und angewandte Mathematik (Crelles Journal)
Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2014 Heft 688

Abstract.

A classical result due to Segre states that on a real cubic surface in ℙℝ3 there exist two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines are defined in an intrinsic way, i.e., their definition does not depend on any choices of orientation data. Segre's classification of smooth real cubic surfaces also shows that any such surface contains at least 3 real lines. Starting from these remarks and inspired by the classical problem mentioned above, our article has the following goals: (a) We explain a general principle which leads to lower bounds in real algebraic geometry. (b) We explain the reason for the appearance of intrinsic signs in the classical problem treated by Segre, showing that the same phenomenon occurs in a large class of enumerative problems in real algebraic geometry. (c) We illustrate these principles in the enumerative problem for real lines in real hypersurfaces of degree 2m-3 in ℙℝm.

Received: 2011-12-16
Revised: 2012-04-13
Published Online: 2012-06-08
Published in Print: 2014-03-01

© 2014 by Walter de Gruyter Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Étale duality for constructible sheaves on arithmetic schemes
  3. On the μ-invariant of anticyclotomic p-adic L-functions for CM fields
  4. Turbulence, orbit equivalence, and the classification of nuclear C*-algebras
  5. Fréchet algebraic deformation quantization of the Poincaré disk
  6. Birationally rigid Fano complete intersections. II
  7. Intrinsic signs and lower bounds in real algebraic geometry
https://doi.org/10.1515/crelle-2012-0055

Artikel in diesem Heft

  1. Frontmatter
  2. Étale duality for constructible sheaves on arithmetic schemes
  3. On the μ-invariant of anticyclotomic p-adic L-functions for CM fields
  4. Turbulence, orbit equivalence, and the classification of nuclear C*-algebras
  5. Fréchet algebraic deformation quantization of the Poincaré disk
  6. Birationally rigid Fano complete intersections. II
  7. Intrinsic signs and lower bounds in real algebraic geometry
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