I will talk about the full exceptional sequences in the perfect derived category D(A) of a graded gentle algebra A, by using the surface model of it. We show that there exists a full exceptional sequence in D(A) if and only if the associated marked surface (S, M) has no punctures and has at least two marked points.
We introduce a braid group action on ordered exceptional dissections of (S, M), and show that it is transitive if the genus of S is zero. Thus in this case the braid group action on full exceptional sequences in D(A) is transitive. We also show that the transitivity is false when the genus is greater than one and the surface has only one boundary component with two marked points on it. So a family of counterexamples is given to a conjecture of Bondal and Polishchuk raised in 1994 on the transitivity of the braid group action on full exceptional sequences in a triangulated category. The talk is based on joint work with Sibylle Schroll arXiv:2205.15830 and ongoing joint work with Fabian Haiden and Sibylle Schroll.