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In this paper we study a surface which has many intriguing and puzzling aspects: on one hand it is related to the Fano surface of lines of a cubic threefold, and on the other hand it is related to a ball quotient occurring in the realm of hypergeometric functions, as studied by Deligne and Mostow. It is moreover connected to a surface constructed by Hirzebruch in his works for constructing surfaces with Chern ratio equal to 3 by arrangements of lines on the plane. Furthermore, we obtain somedoi:10.1090/s0002-9939-2011-10847-5 fatcat:ceytsnlgijc4fh4vm52jvzsnxy