On the Gonality type invariants and the slope of a fibered $3$-fold

The slope of a fibered $3$-folds f:X → B is a relative numerical invariant defined by λ(f) := Kf³/deg(f_ωf), where Kf is the relative canonical divisor and ωf is the relative dualizing sheaf. Establishing slope inequalities is a fundamental problem in the geography of fibered spaces. In this paper, we introduce a new invariant called the minimal covering degree as a gonality-type invariant and study a lower bound of the slope increasing with the covering gonality and the minimal covering degree of the general fiber of f.