Two Examples of Surfaces with Normal Crossing Singularities

Let S be a surface over C with only normal crossing singularities, abbreviated as nc. That is, each point of S is analytically isomorphic to one of 3 local models: smooth point (x = 0) ⊂ C, double nc (xy = 0) ⊂ C or triple nc (xyz = 0) ⊂ C. The normalization n : S̄ → S is smooth and the preimage of the singular locus D ⊂ S is a nc curve D̄ ⊂ S̄. The dualizing sheaf (or canonical line bundle) ωS is locally free and nωS ∼= ωS̄(D̄). The aim of this note is to give examples of nc surfaces whose canonical line bundle exhibits unexpected behavior.