A volume maximizing canonical surface in 3-space

Abstract

Answering a question posed by Enriques, we construct a minimal smooth algebraic surface S of general type over the complex numbers with K 2 = 45 and pg = 4, and with birational canonical map. The canonical system |K S| has a fixed part and the degree of the canonical image is 19. The surface we construct is rigid, S is indeed a ball quotient. It is obtained as an Abelian covering of the plane branched over an arrangement of lines already considered by Hirzebruch, and it is the first such example which is regular (q = 0).