This note gives two examples of surfaces with normal crossing singularities. In the first example the canonical ring is not finitely generated. In the second, the canonical line bundle is not ample but its pull back to the normalization is ample. The latter answers in the negative a problem left unresolved in III. 2.6.2 of Éléments de géometrie algébrique, 1961, and raised again by Viehweg.
All Science Journal Classification (ASJC) codes
- ampleness criteria
- canonical ring
- singular surface