Noether Line

In mathematics, the Noether inequality, named after Max Noether, is a property of compact minimal complex surfaces that restricts the topological type of the underlying topological 4-manifold. It holds more generally for minimal projective surfaces of general type over an algebraically closed field.

Formulation of the inequality

Let ''X'' be a smooth minimal projective surface of general type defined over an algebraically closed field (or a smooth minimal compact complex surface of general type) with canonical divisor ''K'' = −''c''₁(''X''), and let ''p''_g = ''h''⁰(''K'') be the dimension of the space of holomorphic two forms, then
:$p_g \le \frac c_1(X)^2 + 2.$

For complex surfaces, an alternative formulation expresses this inequality in terms of topological invariants of the underlying real oriented four manifold. Since a surface of general type is a Kähler surface, the dimension of the maximal positive subspace in intersection form on the second cohomology is given by ''b''₊ = 1 + 2''p''_g. Moreover, by the Hirzebruch signature theorem ''c''₁² (''X'') = 2''e'' + 3''σ'', where ''e'' = ''c''₂(''X'') is the topological Euler characteristic and ''σ'' = ''b''₊ − ''b''₋ is the signature of the intersection form. Therefore, the Noether inequality can also be expressed as
:$b_+ \le 2 e + 3 \sigma + 5$

or equivalently using ''e'' = 2 – 2 ''b''₁ + ''b''₊ + ''b''₋
:$b_- + 4 b_1 \le 4b_+ + 9.$

Combining the Noether inequality with the Noether formula 12χ=''c''₁²+''c''₂ gives
:$5 c_1(X)^2 - c_2(X) + 36 \ge 12q$
where ''q'' is the irregularity of a surface, which leads to
a slightly weaker inequality, which is also often called the Noether inequality:
:$5 c_1(X)^2 - c_2(X) + 36 \ge 0 \quad (c_1^2(X)\text)$
:$5 c_1(X)^2 - c_2(X) + 30 \ge 0 \quad (c_1^2(X)\text).$

Surfaces where equality holds (i.e. on the Noether line) are called Horikawa surfaces.

Proof sketch

It follows from the minimal general type condition that ''K''² > 0. We may thus assume that ''p''_g > 1, since the inequality is otherwise automatic. In particular, we may assume there is an effective divisor ''D'' representing ''K''. We then have an exact sequence
:$0 \to H^0(\mathcal_X) \to H^0(K) \to H^0( K, _D) \to H^1(\mathcal_X) \to$

so $p_g - 1 \le h^0(K, _D).$

Assume that ''D'' is smooth. By the adjunction formula ''D'' has a canonical linebundle $\mathcal_D(2K)$, therefore $K, _D$ is a special divisor and the Clifford inequality applies, which gives
:$h^0(K, _D) - 1 \le \frac \deg_D(K) = \frac K^2.$

In general, essentially the same argument applies using a more general version of the Clifford inequality for local complete intersections with a dualising line bundle and 1-dimensional sections in the trivial line bundle. These conditions are satisfied for the curve ''D'' by the adjunction formula and the fact that ''D'' is numerically connected.

References

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*{{Citation , doi=10.1007/BF02106598 , last1=Noether , first1 = Max, title=Zur Theorie der eindeutigen Entsprechungen algebraischer Gebilde, journal=Math. Ann., volume=8 , issue=4, year=1875, pages=495–533

Inequalities
Algebraic surfaces