Noether-type inequalities for big divisors via control of the negative part

Abstract

Let $X$ be a smooth projective surface over $\mathbb{C}$ and $D$ a big divisor with Zariski decomposition $D=P+N$. We study the relationship between the volume $\mathrm{vol}(D)=P^2$ and the dimension $h^0(D)$. We introduce a numerical invariant $\mathfrak{C}(N)$ depending only on the negative part $N$, which provides a universal baseline control for $\mathrm{vol}(D)$. This allows us to establish Noether-type inequalities relating $\mathrm{vol}(D)$ and $h^0(D)$, where all correction terms are explicitly governed by $\mathfrak{C}(N)$. Our results recover and unify several classical inequalities on surfaces, and apply in particular to adjoint divisors and foliations. We further obtain lower bounds for $\mathrm{vol}(D)$ in terms of the ps-index $ι(D)$, with applications to foliated surfaces.