On pseudo-horizontal surfaces in small Seifert $3$-manifolds.

We give a criterion for the existence for pseudo-horizontal surfaces in small Seifert fibered manifolds. We calculate the genuses for such surfaces and detect their mod 2 homology classes. We find several families of examples that in the same mod 2 homology class the genus of a pseudo-horizontal surface is less than the genus of the pseudo-vertical surface. Hence the pseudo-vertical surfaces is not geometric incompressible.