Self-intersection of Dualizing Sheaves of Arithmetic Surfaces with Reducible Fibers

Let K be an algebraic number field and OK the ring of integers of K. Let f : X → Spec(OK) be a stable arithmetic surface over OK of genus g ≥ 2. In this short note, we will prove that if f has a reducible geometric fiber, then (ωAr X/OK · ωAr X/OK ) ≥ log 2/6(g − 1). Let K be a number field and OK the ring of integers of K. Let f : X → Spec(OK) be a stable arithmetic surface of genus g ≥ 2. One of interesting problems on arithmetic surfaces is a question whether (ω X/OK · ω Ar X/OK ) > 0. This question is closely related to Bogomolov conjecture, which claims that, for a curve C over a number field and an embedding j : C → Jac(C) into the Jacobian Jac(C), the image j(C) in Jac(C) is discrete in terms of the Néron-Tate height of Jac(C) (cf. [Sz] and [Zh2]). Currently, the positivity of (ω X/OK · ω Ar X/OK ) is known in the following cases: (1) f has a bad reduction. ([Zh1] and [Zh2]) (2) f has a reducible geometric fiber, or Jac(XK) has a complex multiplication. ([Bu]) (3) End(Jac(XK))R is not isomorphic to R, C, or the quaternion division algebra D. ([Zh3]) In this short note, we would like to give an effective lower bound of (ω X/OK · ω Ar X/OK ) under the assumption that f has a reducible geometric fiber, namely, Theorem 1. Let K be an algebraic number field and OK the ring of integers of K. Let f : X → Spec(OK) be a stable arithmetic surface over OK of genus g ≥ 2. Assume that, for P1, . . . , Pn ∈ Spec(OK), geometric fibers XP 1 , . . . , XPn of X at P1, . . . , Pn are reducible. Then, we have ( ω X/OK · ω Ar X/OK ) ≥ n ∑ i=1 log#(OK/Pi) 6(g − 1) . Typeset by AMS-TEX 1