The moduli space of polygons with area center

Abstract

A point $p$ is said to be an area center of a polygon if all of the triangles composed of $p$ and its edges have one and the same area. We construct a moduli space $AC_n$ of such $n$-gons and study its geometry and arithmetic. For every $n\geq 5$, the moduli space is proved to be a rational complete intersection subvariety in $\mathbb{A}^n$. With the help of some subvarieties of low degree in $AC_n$, we also find a unified method of construction of good-looking polygons with area center.