I'm looking for examples of proper, non-normal schemes over a field $k$, whose global section ring does not coincide with $k$.
Looking in the direction of fiber products of simpler objects is often a "standard" way to construct counter-examples, but in this case most of the properties are stable under base change.
In particular I would be interested in knowing what happens in lower and higher dimensions: is it more likely to find something in dimension 1,2 or higher?