  In this paper, we first construct varieties of any dimension $n>2$ fibered
over curves with low slopes. These examples violate the conjectural slope
inequality of Barja and Stoppino [BS14b].
  Led by their conjecture, we focus on finding the lowest possible slope when
$n=3$. Based on a characteristic $p > 0$ method, we prove that the sharp lower
bound of the slope of fibered $3$-folds over curves is $4/3$, and it occurs
only when the general fiber is a $(1, 2)$-surface. Otherwise, the sharp lower
bound is $2$. We also obtain a Cornalba-Harris-Xiao type slope inequality for
families of surfaces of general type over curves, and it is sharper than all
previously known results.
  As an application of the slope bound, we deduce a sharp Noether-Severi type
inequality that $K_X^3 \ge 2\chi(X, \omega_X)$ for an irregular minimal
$3$-fold $X$ of general type not having a $(1,2)$-surface Albanese fibration.
It answers a question in [Zha15] and thus completes the full Severi type
inequality for irregular $3$-folds of general type.
