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

We study a volume related quantity $\mathrm{KVol}$ on 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 $\mathrm{KVol}(L(n, n)) \rightarrow 2$ when n goes to infinity, $2$ being the conjectured infimum for $\mathrm{KVol}$ over ${\mathcal{H}(2)}$.