Wojciech J. Gajda

In this talk we will introduce the deformation theory of curves with automorphisms using the canonical ideal. We will try to explain how we reduced the problem of lifting curves with automorphisms to a lifting problem of linear representations. Using this theory we will see how this leads us to a new obstruction and a counterexample of the generalized Oort conjecture. The initial Oort conjecture (proven in 2013) asserts that when the group is a cyclic p-group then we can always have a lift from positive characteristic to zero characteristic. The generalized Oort conjecture asking the same question for dihedral groups. Furthermore, we will explain how the method we used can decide whether an action of a metacyclic group can lift or not.