0. The Results. Let f: X → B be a fibration of a compact smooth algebraic surface over a compact Riemann surface B, denote by g ≥ 2 the genus of a generic fiber of f and by q the genus of B. Let s be the number of singular fibers of f and ωX/B be the relative dualizing sheaf. Let C1, · · · , Cn be n mutually disjoint sections of f, and denote by D the divisor �n j=1 Cj. Then the main result we are going to prove in this note is the following Theorem 0.1 Theorem 0.1. If f is not isotrivial and semistable, then (ωX/B + D) 2 < (2g − 2 + n)(2q − 2 + s). In fact our proof shows that, for any n, (ωX/B + D) 2 = (2g − 2 + n)(2q − 2 + s). if and only if f is isotrivial. We can derive several corollaries from this theorem. To state the results, we first introduce some notations. Let k be the function field of B and ¯ k be its algebraic closure. For an algebraic point P ∈ X ( ¯ k), we let CP be the corresponding horizontal curve on X. Let hK(P) = ωX/B · CP [k(P) : k] , d(P) = 2g ( ˜ CP) − 2 [k(P) : k] be respectively the geometric height and the geometric logarithmic discriminant of P. Here ˜ CP is the normalization of CP and [k(P) : k] = F ·CP, where 1 F is a generic fiber of f, is the degree of P. Let bP be the number of ramification points on ˜ CP of the induced map r: ˜ CP → B. Write dP = [k(P) : k]. Then we have Theorem 0.2. If f is semistable and not isotrivial, then