Count irreducible components by codimension

Determine, for the varieties \(\mathcal{Y}_j\) and their union \(\bigcup_{j=0}^{n-2}\mathcal{Y}_j\) arising from intersections of compatible \(A_1\)-hypersurface singularities with parameters \(q=(q_1,\ldots,q_n)\), the number of irreducible components of each codimension \(c\), and identify precisely those codimensions for which the counts are nonzero.

Background

For an intersection of compatible $A_1$-hypersurface singularities, the paper defines a sequence of loci $\mathcal{Y}_0\subset\cdots\subset\mathcal{Y}_{n-2}$, obtained from coordinate subspaces intersected with the singularity. The quantity $\mathcal{A}(n;q;j)$ counts the irreducible components of $\mathcal{Y}_j$, while $\mathcal{A}(n;q)$ counts all components across the resolution procedure.

The authors determine the total number of components and observe that the codimension of a component of $\mathcal{Y}_j$ is $\sum_{a\in\{a_1,\ldots,a_{n-j}\}}2q_a+j$, which can vary for fixed $j$ according to the selected values of $q_a$. They leave unresolved the refined enumeration by codimension, both for each individual $\mathcal{Y}_j$ and for the union of all resolution centers.