Divisorial properties and special metrics on hypercomplex twistor spaces

We prove that the general fiber of a compact hypercomplex twistor space with a Kähler fiber has no divisors nor curves. This is first used to prove that, under the same assumption, the trascendental degree of the field of meromoprhic functions is one. The same result allows to prove that these spaces admit no Kähler and not even pluriclosed metrics.