M-curves and symmetric products

TL;DR

This paper proves that certain symmetric products of real algebraic curves called M-curves are M-varieties for specific values of n, extending understanding of their geometric properties over real numbers.

Contribution

It establishes that the n-th symmetric product of a real M-curve is an M-variety for n=2, 3, and n≥2g−1, providing new insights into their structure.

Findings

Symmetric products for n=2, 3 are M-varieties.

Symmetric products for n≥2g−1 are M-varieties.

Extends known properties of M-curves to their symmetric products.

Abstract

Let $(X , \sigma)$ be a geometrically irreducible smooth projective M-curve of genus $g$ defined over the field of real numbers. We prove that the $n$-th symmetric product of $(X , \sigma)$ is an M-variety for $n=2 ,3$ and $n\geq 2g -1$.