Two examples of surfaces with normal crossing singularities

TL;DR

This paper presents two examples of algebraic surfaces with normal crossing singularities, illustrating different behaviors of their canonical rings and line bundles, and addressing an open problem in algebraic geometry.

Contribution

It provides two explicit examples of surfaces with normal crossing singularities, one showing non-finite generation of the canonical ring and another illustrating a subtle property of the canonical line bundle.

Findings

Canonical ring is not finitely generated in the first example

Canonical line bundle is not ample, but its pullback to normalization is ample

Addresses an open problem in algebraic geometry regarding line bundle ampleness

Abstract

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 [EGA,III.2.6.2] and raised again by Viehweg.