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

TL;DR

This paper classifies the automorphism groups of numerical Godeaux surfaces with torsion of order 3, 4, or 5, providing explicit stratifications of their moduli spaces and establishing connections to their moduli stacks.

Contribution

It explicitly computes automorphism groups for these surfaces and links the moduli stacks to quotient stacks, especially proving the stack for torsion order 5 is the moduli stack.

Findings

Automorphism groups are classified for torsion orders 3, 4, and 5.

Explicit stratifications of moduli spaces are provided.

The moduli stack for torsion order 5 is identified as a quotient stack.

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.