Algebraic intersection for translation surfaces in the stratum $\mathcal{H}(2)$

TL;DR

This paper investigates the algebraic intersection quantity on genus 2 translation surfaces, providing explicit sequences approaching a conjectured minimal value, thereby advancing understanding of geometric intersection properties.

Contribution

The authors explicitly construct a sequence of genus 2 translation surfaces in the stratum H(2) demonstrating the asymptotic approach to the conjectured minimal KVol value.

Findings

KVol approaches 2 for the constructed sequence

The sequence L(n,n) converges to the conjectured infimum

Provides evidence supporting the conjecture about KVol in H(2)

Abstract

We study the quantity $\mbox{KVol}$ defined as the supremum, over all pairs of closed curves, of their algebraic intersection, divided by the product of their lengths, times the area of the surface. The surfaces we consider live in the stratum $\mathcal{H}(2)$ of translation surfaces of genus $2$, with one conical point. We provide an explicit sequence $L(n,n)$ of surfaces such that $\mbox{KVol}(L(n,n)) \longrightarrow 2$ when $n$ goes to infinity, $2$ being the conjectured infimum for $\mbox{KVol}$ over $\mathcal{H}(2)$.