Intrinsic signs and lower bounds in real algebraic geometry
By C. Okonek, A. Teleman. 2011
49 citations recorded by the source corpus.
A classical result due to Segre states that on a real cubic surface in ${\mathbb P}^3_\R$ there exists 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: - We explain a general principle which leads to lower bounds in real algebraic geometry, - 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. - We illustrate these principles in the enumerative problem for real lines in real hypersurfaces of degree $2m-3$ in ${\mathbb P}^m_\R$.
Subfields: Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/8a0a6947edc459b3f6a9c6844c6622142e276610
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In Geometry and Topology
Citations in the corpus
- An arithmetic count of the lines on a smooth cubic surface (2017, methodology): “lar, this number is independent of the choice of V. By contrast, a real smooth cubic surface can contain 3, 7, 15, or 27real lines. ItisabeautifulobservationofFinashin–Kharlamov[FK13]andOkonek–Teleman[OT14a] that while the number of real lines on a smooth cubic surface depends on the surface, a cert”
- EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS (2020, unknown): “This builds on results of Finahin–Kharlamov [FK13], J.L. Kass and the second-named author [KW21], M. Levine [Lev19] [Lev20], S. McKean [McK21], Okonek– Teleman [OT14], S. Pauli [Pau20], J. Solomon [Sol06], P.Srinivasan and the second-named author [SW21], and M. Wendt [Wen20].”