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We study the quantity $\mbox{KVol}$ defined as the supremum, over all pairsof closed curves, of their algebraic intersection, divided by the product oftheir lengths, times the area of the surface. The surfaces we consider live inthe stratum $\mathcal{H}(2)$ of translation surfaces of genus $2$, with oneconical 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, $2$ beingthe conjectured infimum for $\mbox{KVol}$ over $\mathcal{H}(2)$.

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