Number of singular fibres of surface fibrations over \(\mathbb{P} 1\) (Q2226654)

For \(f\) a nonisotrivial surface fibration of fibre genus \(g>0\) over an algebraically closed field \(k\) of finite characteristic \(p\) the authors study the optimum lower bound \(L\) for the number of singular fibres of \(f\) with respect to \(p\). They find that for \(p\leq 2g-1\), \(L=1\); for \(p=2g+1\) and \(f\) hyperelliptic, \(L=2\); for \(p=2\) or \(3\) and \(g=1\), \(L=2\).