Noether-Type Inequalities for Big Divisors on Algebraic Surfaces
Shi Xu
TL;DR
The paper develops Noether-type lower bounds for the volume of big divisors on smooth projective surfaces by introducing the invariant $\mathfrak{e}(D)$, which depends only on the negative part $N$ in the Zariski decomposition $D=P+N$. Using a detailed analysis of the movable and fixed parts of $|D|$ and a decomposition framework, it proves sharp volume inequalities that separate the pencil and non-pencil cases, and it extends the framework to non-complete surfaces and to foliations via log invariants and pluricanonical systems. The main results yield explicit lower bounds, such as $\mathrm{Vol}(D) \ge \frac{(h^0(D)-1)^2}{h^0(D)-1+\mathfrak{e}}$ for pencil cases and $\mathrm{Vol}(D) \ge h^0(D)-2$ (with refinements depending on the geometry), and analogues for logarithmic and foliated settings, including a bound in terms of the pluricanonical system of a foliation and a relation to the ps-index. These inequalities unify and extend classical Noether-type bounds (e.g., $\mathrm{Vol}(K_X)\ge 2h^0(K_X)-4$) by tying volume to the fixed-part geometry and the fibration structure, with sharp equality scenarios described in precise geometric configurations. The results have implications for the geometry of (non-)complete surfaces and for the pluricanonical geometry of foliations, offering a robust framework for understanding how base loci, fixed components, and fibrations constrain the growth of sections.
Abstract
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.