Number of singular fibres for non-isotrivial surface fibrations over ℙ^1 in positive characteristics

      Given a certain class of surface fibrations over the rational curve ℙ^1, it is an interesting and classical problem to determine the lower bound for the number of singular fibres of fibrations among this class. In 1981, Beauville first studied such problem over the field of complex numbers ℂ and a lot of results have been then achieved by various people afterwards. On the other hand, such a problem remains widely open in positive characteristics. In this talk, I shall discuss some recent progresses on this problem in positive characteristics. This is a joint work with Gong Chen.