We give a detailed proof of Hirzebruch's remarkableresult that the symmetric Hilbert modular surface oflevel for

is PSL2 (lF7)equivariantly isomorphic to the complexprojective plane. We identify the curves F1 , F2, F4, F28 explicitlyas plane curves defined by invariants of degrees4,12,18,21 for a three-dimensional representation ofPSL2 (lF7), and we explain theirgeometry. For example, F1 is the Klein curve,F12 isthe Steinerian of the Klein curve and F18 isessentially the Cay lean of the Klein curve. Thecurves F12and FI8 are birationally equivalent to the Hessianof the Klein curve, which was shown to be defined bya cocompact arithmetic group by Fricke; Hirzebruch'stheory gives another uniformization using subgroupsof SL2( ℤ). We compute the group ofinvariant line bundles on the Hessian and offer theHessian as a challenge to extending Doglachev'srecent work on the invariant vector bundles onmodular curves to the case of triangle groups﹛p, q, r﹜ in whichp, q, r are notpairwise relatively prime. The curve FI4 maps to the21-point orbit in ℙ2. Using our explicitidentification of F1 , F2,F4,F14,F28 we are able to complete Hirzebruch'sidentification of the nonsymmetric Hilbert modularsurface.

Hirzebruch [1977] proved the remarkable result that thecomplex projective plane ℙ2 is a minimalmodel of the symmetric Hilbert modular surface oflevel for the extended Hilbert modular group of.Furthermore, the identification is equivariant fornatural actions of PSL2( 𝔽7 )on the two surfaces. This result also enabled him toidentify the (nonsymmetric) Hilbert modular surfaceassociated to this group: namely, it is obtainedfrom ℙ2 by a certain sequence of blowingsup and then passing to a two-sheeted coveringbranched along a certain curve on the resultingsurface.