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

The $\mathit{\text{slope}}$ of a fibered $3$-folds $f:X\to B$ is a relative numerical invariant defined by $\lambda (f):=K_f^3\mathrm{/}\mathrm{d}\mathrm{e}\mathrm{g}(f_{\ast }{\omega }_f)$, where $K_f$ is the relative canonical divisor and ${\omega }_f$ 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 $\mathit{\text{minimal}}\mathit{\text{covering}}\mathit{\text{degree}}$ 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 $f$.