Stratification of $\mathcal{H}$

In between, we expect a series of ﬂips, dictated by the structure of the preimage of Δ under the quotient map π : D → F . More precisely, let H := π−1(supp Δ); then H is a union of hyperplane sections of D, and hence is stratiﬁed by closed subsets, where a stratum is determined by the number of independent sheets (“independent sheets” means that their deﬁning equations have linearly independent differentials) of H containing the general point of the stratum. The stratiﬁcation of H induces a stratiﬁcation of supp Δ, where the strata of supp Δ are indexed by the “number of sheets” (in D, not in F = Γ \D). Roughly speaking, Looijenga predicts that a stratum of supp Δ corresponding to k (at least) sheets meeting (in D) is ﬂipped to a dimension k − 1 locus on the GIT side.