It was a real pleasure for me to participate in the symposium in honor of Shiing-shen Chern. In my lecture I intended to give a survey on Hilbert modular surfaces. But actually I discussed examples of such Hilbert modular surfaces for which specific information is available on their structure as algebraic surfaces. The paper presented here is an extended version of the talk.
Algebraic surfaces are often investigated by means of their pluricanonical maps (see for example Bombieri [1973]). The properties of the canonical map itself (given by the secctions of the canonical bundle or in the case of Hilbert modular surfaces by the cusp forms of weight 2) are relatively complicated (compare Beauville [1979]). In a certain range, namely for minimal surfaces of general type with
\[ 2p_g - 4 \leq K^2 \leq 2p_g - 2 ,\]
Horikawa’s results are available [1975, 1976a, 1976b, 1978, 1979]. To apply them, one has to prove that the surface being studied is minimal. For Hilbert modular surfaces this is a difficult problem, which was attacked first by van der Geer and Van de Ven [1977]. Van der Geer has obtained many results on the structure of special Hilbert modular surfaces [1978, 1979] including some of the surfaces studied here.
The rough classification of Hilbert modular surfaces according to rational, \( K3 \), elliptic, and general type was considered in [Hirzebruch and Van de Ven 1974; Hirzebruch and Zagier 1977; Hirzebruch 1978]. The present paper tries to show that in some cases a finer classification of the surfaces of general type can be obtained.
