Diagonal quotient surfaces (Q1360917)

Let \(X\) be a smooth projective curve over \(\mathbb{C}\) which admits a finite group \(G\) of automorphisms. Then \(G\times G\) acts on the product surface \(Y=X\times X\), and so for each subgroup \(H\leq G\times G\), the quotient \(Z=H \setminus Y\) is a normal algebraic surface. Here we shall restrict attention to the case that the subgroup \(H\) is the graph of a group automorphism \(\alpha\in \Aut(G)\) of \(G\), i.e. \(H=\Delta_\alpha =\{(g,\alpha (g)):g\in G\}\); we propose to call the resulting quotient surface a (twisted) diagonal quotient surface, and denote it by \(Z_{X,G,\alpha} =\Delta_\alpha \setminus Y\). Such surfaces occur naturally as the (coarse) moduli spaces of certain moduli problems. In this case \(X=X(N)\), the modular curve of level \(N\), and there is a close analogy between these surfaces and the Hilbert modular surfaces studied by Hirzebruch and others [cf. \textit{G. van der Geer}, ``Hilbert modular surfaces'' (1988; Zbl 0634.14022)]. The main objective of this paper is to analyze the geometry of the diagonal quotient surfaces \(Z_{X,G, \alpha}\) and of their desingularizations \(\widetilde Z_\alpha\) by calculating their numerical invariants such as their Betti and Chern numbers, and to determine their place in the Enriques-Kodaira classification table.