Algebraic intersection in regular polygons

TL;DR

This paper investigates the algebraic intersection function on moduli spaces of translation surfaces derived from regular polygons, establishing sharp bounds for specific Teichmüller discs related to right-angled triangles.

Contribution

It provides explicit bounds for the KVol function on Teichmüller discs of Veech surfaces from regular polygons, identifying the extremal surface.

Findings

Established sharp bounds for KVol on Teichmüller discs.

Identified the unique surface attaining the lower bound.

Connected geometric properties of polygons to intersection theory.

Abstract

We study the function $$\mbox{KVol} : (X,\omega)\mapsto \mbox{Vol} (X,\omega) \sup_{\alpha,\beta} \frac{\mbox{Int} (\alpha,\beta)}{l_g (\alpha) l_g (\beta)}$$ defined on the moduli spaces of translation surfaces. More precisely, let $\mathcal T_n$ be the Teichm\"uller discs of the original Veech surface $(X_n,\omega_n)$ arising from right-angled triangle with angles $(\pi/2,\pi/n,(n-2)\pi/2n)$ by the unfolding construction for $n\geq 5$. For $n \equiv 1 \mod 2$ and any $(X,\omega)\in \mathcal T_n$, we establish the (sharp) bounds $$ \frac{n}{2} \cot \frac{\pi}{n} \leq \mbox{KVol}(X,\omega) \leq \frac{n}{2} \cot \frac{\pi}{n} \cdot \frac1{\sin \frac{2\pi}{n}}.$$ The lower bound is uniquely realized at $(X_n,\omega_n)$.