Non-projective complete log canonical surfaces

Abstract

We construct non-projective complete log canonical algebraic surfaces whose canonical divisors are semi-ample over an algebraically closed field of any characteristic other than the algebraic closure of a finite field. We provide a unified framework to construct such surfaces for any given non-negative Kodaira dimension, namely, zero, one, or two. Furthermore, we show that any complete log canonical algebraic surface with Kodaira dimension minus infinity is automatically projective. This projectivity result confirms that our construction covers all possible values for the Kodaira dimension of non-projective complete log canonical surfaces.