Quotients of abelian and hyperelliptic surfaces by rational vector fields (Q913878)

Let S be a non-singular complete algebraic surface defined over an algebraically closed field k of characteristic \(p>0.\) A rational vector field D on S is called a p-closed vector field if there exists a rational function f on S such that \(D^ p=fD\). The quotient surface \(S^ D\) defined by D is normal. Let \(\tilde S^ D\) denote the minimal desingularization of \(S^ D\). The authors investigate the case where S is an abelian surface of a hyperelliptic surface. The conclusion is as follows: (1) \((D)\equiv 0\) (numerical equivalence) \(\Leftrightarrow \kappa (\tilde S^ D)=0\Leftrightarrow \tilde S^ D\) is an abelian surface or a hyperelliptic surface according to the type of S. (2) D has only divisorial singularities and \((D)\not\equiv 0\Leftrightarrow \kappa (\tilde S^ D)=1.\) (3) D has non-divisorial singularities \(\Leftrightarrow \kappa (\tilde S^ D)=2\).