Cohomology of GIT quotients and an application to moduli of curves

In several papers, F. Kirwan developed a machinery to compute the intersection cohomology of GIT quotients of projective varieties. In this talk, I will first explain this procedure, based on stratifications, a partial desingularisation of the quotient and decomposition theorem, in a general setting. Then, I will focus on an application to the moduli space of complete intersections of a smooth quadric and a cubic surface in the projective space, i.e. non-hyperelliptic genus 4 curves.