On pseudo-horizontal surfaces and $\mathbb{Z}_2$-Thurston norms 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 $\mathbb{Z}_2$-homology classes. Using such pseudo-horizontal surfaces, we can determine the $\mathbb{Z}_2$-Thurston norm for every $\mathbb{Z}_2$-homology classes in small Seifert manifolds. We find several families of examples that in the same $\mathbb{Z}_2$-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 $\mathbb{Z}_2$-taut.