Representation varieties of the fundamental groups of compact non-orientable surfaces

Abstract

Let $\Gamma_g$ be the fundamental group of compact non-orientable surface and K be an algebraically closed field of chracteristic 0. In the paper we describe the structure of representation varieties of the group $\Gamma_g$ into full linear group and into special linear group and their character varieties; namely it is determined a number of their irreducible components, their dimensions and investigated their birational properties, in particular it is proved that all irreducible components of the representation variety into full linear group are Q-rational varieties.