Definition
Noether inequality (also written as Noether’s inequality) is a numerical bound in the theory of complex algebraic surfaces. For a smooth, minimal projective surface $X$ of general type, the inequality states
$$ K_X^{2} ;\ge; 2,p_g(X) ;-; 4, $$
where $K_X$ denotes the canonical divisor of $X$ and $p_g(X)=h^{0}(X,\mathcal{O}_X(K_X))$ is the geometric genus of $X$.
Overview
The inequality provides a lower bound for the self‑intersection number $K_X^{2}$ in terms of the geometric genus. It is a fundamental tool in the Enriques–Kodaira classification of algebraic surfaces, particularly in the study of surfaces of general type. Equality in the inequality characterises surfaces that lie on the so‑called Noether line in the $(p_g, K^2)$-plane; such surfaces have special geometric properties (e.g., the canonical map is birational onto its image).
The statement originally applied to complex surfaces, but an analogous inequality holds over algebraically closed fields of characteristic zero, and variants have been proved in positive characteristic under additional hypotheses.
Etymology / Origin
The inequality is named after the German mathematician Max Noether (1844–1921), who contributed significantly to the birational geometry of algebraic surfaces in the late 19th century. The result first appeared in Noether’s work on the canonical series of surfaces and was later incorporated into the modern framework of surface classification.
Characteristics
| Aspect | Description |
|---|---|
| Scope | Applies to smooth, minimal, projective surfaces of general type. Minimality means that the surface contains no $(-1)$-curves (exceptional curves of the first kind). |
| Invariants involved | - $K_X^{2}$: self‑intersection of the canonical divisor (a measure of the “size” of the canonical class). - $p_g(X)$: geometric genus, the dimension of the space of holomorphic 2‑forms. |
| Typical proof ideas | Uses Riemann‑Roch theorem for surfaces, properties of the canonical map, and the fact that for surfaces of general type the canonical system is big. Classical proofs involve analysis of the image of the canonical map and its degree. |
| Equality cases | Surfaces attaining $K_X^{2}=2p_g-4$ are said to lie on the Noether line. Examples include certain Horikawa surfaces (double covers of $\mathbb{P}^2$ branched over a curve of degree $2p_g+2$). |
| Generalizations / refinements | - Noether–Castelnuovo inequality for irregular surfaces ($q>0$). - Severi inequality relating $K_X^{2}$ to $\chi(\mathcal{O}_X)$ for surfaces with maximal Albanese dimension. - Bogomolov–Miyaoka–Yau inequality gives an upper bound: $K_X^{2} \le 9\chi(\mathcal{O}_X)$. |
| Limitations | The inequality does not hold for non‑minimal surfaces (a blow‑up adds a $(-1)$-curve, decreasing $K^2$). It also fails for surfaces that are not of general type (e.g., ruled or rational surfaces). |
Related Topics
- Enriques–Kodaira classification – the overall scheme for classifying complex surfaces.
- Canonical divisor – the divisor class associated with holomorphic 2‑forms.
- Geometric genus ($p_g$) – dimension of the space of global holomorphic 2‑forms.
- Irregularity ($q$) – dimension of $H^1(X,\mathcal{O}_X)$.
- Noether line – the line $K^2 = 2p_g - 4$ in the $(p_g,K^2)$-plane.
- Horikawa surfaces – examples of surfaces attaining equality in Noether’s inequality.
- Bogomolov–Miyaoka–Yau inequality – an upper bound counterpart for $K^2$.
- Severi inequality – another lower bound involving the Albanese map.
References
- Barth, W., Hulek, K., Peters, C., & Van de Ven, A. (2004). Compact Complex Surfaces (2nd ed.). Springer.
- Badescu, L. (2001). Algebraic Surfaces. Springer.
- Horikawa, E. (1976). “Algebraic surfaces of general type with small $c_1^2$.” Ann. of Math., 104, 357–387.
These sources contain the formal statement, proof sketches, and examples of surfaces satisfying or attaining the Noether inequality.