## Tohoku Mathematical Journal

### Catanese-Ciliberto surfaces of fiber genus three with unique singular fiber

Hirotaka Ishida

#### 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 $\Alb(S)\isom E$, 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 58, Number 1 (2006), 33-69.

Dates
First available in Project Euclid: 18 April 2006

https://projecteuclid.org/euclid.tmj/1145390205

Digital Object Identifier
doi:10.2748/tmj/1145390205

Mathematical Reviews number (MathSciNet)
MR2221791

Zentralblatt MATH identifier
1112.14012

#### Citation

Ishida, Hirotaka. Catanese-Ciliberto surfaces of fiber genus three with unique singular fiber. Tohoku Math. J. (2) 58 (2006), no. 1, 33--69. doi:10.2748/tmj/1145390205. https://projecteuclid.org/euclid.tmj/1145390205

#### References

• M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414--452.
• F. Catanese and C. Ciliberto, Surfaces with $p_g=q=1$, Problems inthe theory of surfaces and their classification (Cortona, 1988), 49--79, Sympos. Math., XXXII, Academic Press, London, 1991.
• F. Catanese and C. Ciliberto, Symmetric products of ellipticcurves and surfaces of general type with $p_g=q=1$, J. Algebraic Geom. 2 (1993), 389--411.
• R. Hartshorne, Algebraic Geometry, Graduate Texts Math. 52, Springer-Verlag, New York-Heidelberg, 1977.
• E. Horikawa, On algebraic surfaces with pencils of curves of genus 2, Complex analysis and algebraic geometry, 79--90, Iwanami Shoten, Tokyo, 1977.
• E. Horikawa, Algebraic surfaces of general type with small $c_1^2$, I, Ann. of Math. (2) 104 (1976), 357--388; Algebraic surfaces of general type with small $c_1^2$, II, Invent. Math. 37 (1976), 121--155; Algebraic surfaces of general type with small $c_1^2$, III, Invent. Math. 47 (1978/79), 209--248; Algebraic surfaces of general type with small $c_1^2$, IV, Invent. Math. 50 (1979), 103--128; Algebraic surfaces of general type with small $c_1^2$, V, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 745--755.
• M. Namba, Geometry of projective algebraic curves, Monogr. Textbooks Pure Appl. Math. 88, Dekker, New York, 1984.
• T. Oda, Vector bundles on an elliptic curve, Nagoya Math. J. 43 (1971), 41--72.
• T. Takahashi, Certain algebraic surfaces of general type withirregularity one and their canonical mappings, Tohoku Math. J. (2) 50 (1998), 261--290.