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For any relatively minimal hyperelliptic fibration f with slope four, there exists the inequality with respect to the relative Euler-Poincare characteristic χ(f) of f and the genus g (f) of a fiber of f. This inequality restricts the extent of pairs (g(f), x(f)) for relatively minimal hyperelliptic fibrations f with slope four which exist. Hence, for given suitable integers g and z, we consider the existence of a relatively minimal hyperelliptic fibration f with g(f) = g, x(f) = z and slope four. The main purpose in this paper, for any positive integer g, is to prove that there exists a relatively minimal hyperelliptic fibration f with g(f) = g, χ(f) ≥ z(g) and slope four, where z(X) is a certain polynomial of degree two.

The existence of hyperelliptic fibrations with slope four and high relative Euler-Poincaré characteristic