Structure of gluings along non-reduced base remains unknown

Determine, in the case where (Z,z) is not a reduced point and O_{X,x} → O_{Z,z}, O_{Y,y} → O_{Z,z} are surjective, whether the gluing (X,x) ⊔_{(Z,z)} (Y,y) is smooth or singular and whether it is a complete intersection or Gorenstein, by establishing criteria characterizing these properties.

Background

Prior work described structural properties (Cohen–Macaulayness, complete intersection, etc.) of gluings primarily when the base germ (Z,z) is a reduced point. For non-reduced (Z,z), general structural criteria have not been established in the literature.

This paper develops new classes (weakly large, large, strongly large) and provides criteria and explicit Betti number formulas for these classes, partially addressing the gap. However, the general problem of characterizing when an arbitrary gluing along a non-reduced base is smooth, singular, a complete intersection, or Gorenstein remains open.