Singularities of multiplicative $p$-closed vector fields and global 1-forms of Zariski surfaces

Abstract

We consider quotient surfaces by $p$-closed rational vector fields. First we show that singularities on the quotient surfaces by multiplicative $p$-closed vector fields are tonic singularities. Then we proceed to studying global properties of Zariski surfaces. We see that non-trivial global 1-forms are related to some linear systems with base points. We also give examples of Zariski surfaces admitting non-closed regular 1-forms.