Real solutions of a problem in enumerative geometry

TL;DR

This paper investigates the number of real solutions in a specific family of enumerative geometry problems, revealing how solutions vary across configuration space components and providing methods to determine their signs.

Contribution

It introduces a detailed analysis of the real solution count in a 2-parameter family, including solution function calculation and sign determination, with a translation to quiver language.

Findings

Solution function is constant modulo 4 in even cases

Sign of solutions can be explicitly determined

Connected components of configuration space are characterized

Abstract

We study a 2-parameter family of enumerative problems over the reals. Over the complex field, these problems can be solved by Schubert calculus. In the real case the number of solutions can be different on the distinct connected components of the configuration space, resulting in a solution function. The cohomology calculation in the real case only gives the signed sum of the solutions, therefore in general it only gives a lower bound on the range of the solution function. We calculate the solution function for the 2-parameter family and we show that in the even cases the solution function is constant modulo 4. We show how to determine the sign of a solution and describe the connected components of the configuration space. We translate the problem to the language of quivers and also give a geometric interpretation of the sign. Finally, we discuss what aspects might be considered when solving other real enumerative problems.