Intrinsic signs and lower bounds in real algebraic geometry

TL;DR

This paper explores intrinsic signs and lower bounds in real algebraic geometry, illustrating how classical problems like Segre's classification of real lines on cubic surfaces reveal fundamental principles applicable to enumerative problems in higher dimensions.

Contribution

It introduces a general principle for deriving lower bounds in real algebraic geometry and explains the role of intrinsic signs in classical and modern enumerative problems.

Findings

Intrinsic signs explain the classification of real lines on cubic surfaces.

The same phenomenon of signs appears in enumerative problems for hypersurfaces.

The principles are illustrated in real hypersurfaces of degree 2m-3 in projective space.

Abstract

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 ${\mathbb P}^m_\R$.