Enumeration of Real Conics and Maximal Configurations

TL;DR

This paper proves that for certain enumerative problems involving real conics passing through specified linear spaces in real projective space, all solutions can be real under generic conditions, using tropical geometry methods.

Contribution

It introduces a tropical geometric approach to establish maximality in real enumerative geometry for conics passing through linear constraints.

Findings

All complex conics passing through certain real linear spaces can be realized as real.

The method applies to enumerative problems in any dimension involving conics.

Tropical decompositions are used to prove maximality results.

Abstract

We use floor decompositions of tropical curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in $\RP^n$ is maximal. That is, there exist generic configurations of real linear spaces such that all complex conics passing through these constraints are actually real.