Noether-Type Inequalities for Big Divisors on Algebraic Surfaces

Let $D$ be a big integral divisor on a smooth projective surface $X$. In this paper, we study Noether-type inequalities for $D$. The key ingredient is the introduction of a numerical invariant $\mathfrak{e}(D)$, which depends only on the negative part $N$ of $D$. In particular, $\mathfrak{e}(D)=0$ if $D$ is nef. As an application, we establish an inequality between the volume and the pluri-sectional index of a canonical foliated surface of general type.