Lower bound for KVol on the minimal stratum of translation surfaces

In this paper we are interested in algebraic intersection of closed curves of a given length on translation surfaces. We study the quantity KVol, defined in Cheboui et al. (Bull Soc Math France 149(4):613–640, 2021) and studied in Cheboui et al. (2021), Cheboui et al. (C R Math Acad Sci Paris 359:65–70, 2021), Boulanger et al. (Ann Henri Lebesgue, 2024), and Boulanger (Algebraic intersection, lengths and Veech surfaces, 2023. arXiv:2309.17165), and we construct families of translation surfaces in each connected component of the minimal stratum \(\mathcal {H}(2g-2)\) $\mathcal{H}(2g-2)$ of the moduli space of translation surfaces of genus \(g \ge 2\) $g\ge 2$ such that KVol is arbitrarily close to the genus of the surface, which is conjectured to be the infimum of KVol on \(\mathcal {H}(2g-2)\) $\mathcal{H}(2g-2)$ .