Automorphisms of numerical Godeaux surfaces with torsion of order 3, 4, or 5

Abstract

We compute the automorphisms groups of all numerical Godeaux surfaces, i.e. minimal smooth surfaces of general type with K^2 = 1 and p_g = 0, with torsion of the Picard group of order \nu equals 3, 4, or 5. We present explicit stratifications of the moduli spaces whose strata correspond to different automorphisms groups. Using the automorphisms computation, for each value of \nu we define a quotient stack, and prove that for \nu = 5 this is indeed the moduli stack of numerical Godeaux surfaces with torsion of order 5. Finally, we describe the inertia stacks of the three quotient stacks.