• DOI:10.2748/TMJ/1145390205
• Corpus ID: 121949976

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

@article{Ishida2006CataneseCilibertoSO,
  title={Catanese-Ciliberto surfaces of fiber genus three with unique singular fiber},
  author={Hirotaka Ishida},
  journal={Tohoku Mathematical Journal},
  year={2006},
  volume={58},
  pages={33-69}
}

• H. Ishida
• Published 30 March 2006
• Mathematics
• Tohoku Mathematical Journal

In this paper, we study a minimal surface of general type with pg = q = 1, K2 S = 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) ∼= 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. 0. Introduction. Let S be a minimal algebraic surface of general type over C. A proper surjective morphism f : S → C from an algebraic surface S to a non-singular algebraic curve C is called a fibration of curves of genus g if fibers of f are connected and the genus of the generic fiber is g . It is important to study the structures of the fibrations for surfaces of general type. For instance, Horikawa studied surfaces with fibrations of curves of genus two [5, 6]. We set pg (S) = dimH 2(S,OS) and q(S) = dimH 1(S,OS). Let K2 S be the self intersection number of the canonical divisor KS of S. In this paper, we are interested in the case pg (S) = q(S) = 1 and K2 S = 3. If q(S) = 1, then the Albanese map a : S → Alb(S) gives a fibration of curves over the elliptic curve E = Alb(S). Let g be the genus of a general fiber of a. Catanese and Ciliberto studied this surface in [2, 3] and showed that the genus g is two or three. DEFINITION. Let S be a minimal algebraic surface of general type over C. S is called a Catanese-Ciliberto surface if S satisfies pg = q = 1 and K2 S = 3. We also denote by KS the invertible sheaf associated to the divisor KS . In the case g = 3, Catanese and Ciliberto showed that the direct image V = a∗KS/E of the relative canonical sheaf KS/E = KS ⊗OS (a∗Ω1 E)∨ ∼= KS is an indecomposable vector bundle of rank three and degree one over the elliptic curve E. Therefore, there exists a point P ∈ E such that detV ∼= OE(P ). Let p : PE(a∗KS) → E be the P2-bundle associated with a∗KS and ω : S → PE(a∗KS) the relative canonical map. They obtained the following theorem. THEOREM 0.1 (Catanese-Ciliberto [3, Theorem 3.1]). Let S, a,E, g, P, p and ω be as above, and H the tautological divisor of PE(a∗KS), i.e., it satisfies p∗OPE(a∗KS)(H) ∼= a∗KS . If g = 3, then we have: 2000 Mathematics Subject Classification. Primary 14D05; Secondary 14J29, 14D06.