Stratification of moduli spaces of instantons on the Segre product of three lines via 't Hooft bundles

Let $X$ be the Segre product of three projective lines. For a fixed effective divisor $D$ on $X$, we introduce the notions of $D$-'t Hooft, $(D_i,D_j)$-special and $D$-sectional special bundle. The varieties parameterizing these bundles yield a natural stratification of the moduli space of stable instanton bundles with fixed Chern classes. After characterizing the curves associated with these bundles via Serre correspondence, we describe the corresponding Hilbert schemes. Using this description, we analyze the moduli spaces of $h_i$-'t Hooft bundles and the smaller strata of $(h_i,h_j)$-special and $(h_i)$-sectional special bundles. Finally, we provide a detailed study of the low-charge cases.