# Singularities and Genus of the k-Ellipse

## TL;DR

This paper investigates the geometric properties of k-ellipses, specifically their singularities and genus, by analyzing their Zariski closure in the complex projective plane, thus resolving an open problem from 2008.

## Contribution

It provides a complete determination of the singularities and genus of k-ellipses, addressing a previously unresolved question in algebraic geometry.

## Key findings

- Identified the singularities of the Zariski closure of k-ellipses.
- Calculated the genus of the Zariski closure of k-ellipses.
- Resolved an open problem from 2008 regarding k-ellipses.

## Abstract

A k-ellipse is a plane curve consisting of all points whose distances from k fixed foci sum to a constant. We determine the singularities and genus of its Zariski closure in the complex projective plane. The paper resolves an open problem stated by Nie, Parrilo and Sturmfels in 2008.