The \textit{ slope} of a fibered $3$-folds $f:X \to B$ is a relative numerical invariant defined by $\lambda(f) := K_{ f}^{ 3}/\mathrm{ deg}(f_{ \ast}\omega_{ f})$, 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 \textit{ 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 $f$.
Categories: math.AG