## Tohoku Mathematical Journal 2006 March SECOND SERIES VOL. 58, NO. 1

 Tohoku Math. J. 58 (2006), 33-69

Title CATANESE-CILIBERTO SURFACES OF FIBER GENUS THREE WITH UNIQUE SINGULAR FIBER

Author Hirotaka Ishida

(Received December 17, 2003, revised April 27, 2005)
Abstract. In this paper, we study a minimal surface of general type with $p_g=q=1, K_S^2=3$ which we call a Catanese-Ciliberto surface. The Albanese map of this surface gives a fibration of curves over an elliptic curve. For an arbitrary elliptic curve $E$, we obtain the Catanese-Ciliberto surface which satisfies $\mathop{\rm Alb}(S)\cong E$, has no $(-2)$-curves and has a unique singular fiber. Furthermore, we show that the number of the isomorphism classes satisfying these conditions is four if $E$ has no automorphism of complex multiplication type.

2000 Mathematics Subject Classification. Primary 14D05; Secondary 14J29, 14D06.

Key words and phrases. Surface of gneral type, fibration of curves, elliptic curve.