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Inégalités numériques pour les surfaces de type général. Appendice: ”L’inégalité $$p_ g\geq 2q-4$$ pour les surfaces de type général” par A. Beauville. (French) Zbl 0543.14026
The aim of this article is to improve Noether’s classical inequality: $$K^ 2\geq 2p_ g-4,$$ for minimal complex algebraic surfaces of general type, by using the irregularity of the surface. We show the following inequalities:
$$K^ 2\geq 2p_ g+(q-4),$$ with equality only for the product of two curves of genus 2; $$K^ 2\geq 2p_ g,$$ if the surface is irregular. They are derived from the following inequalities: $$K^ 2\geq 3p_ g-2q+9,$$ if the canonical system is composed with a pencil; $$K^ 2\geq 3p_ g+q-7$$, if the canonical map is birational [F. Jongmans, Mém. Soc. Sci. Liège, IV. Ser.7, 367-468 (1947; Zbl 0036.377)]; $$K^ 2\geq 2p_ g+4(q-4)$$, if the canonical map is of degree 2, with equality only for the product of two curves; one of them of genus 2, the other one of genus at least 2; $$K^ 2\geq 3p_ g-4,$$ if the canonical map is of degree 3 and the surface is irregular.
Finally, an appendix written by A. Beauville is devoted to the proof of the inequality $$p_ g\geq 2q-4.$$
In a later addendum [Bull. Soc. Math. Fr. 111, 301-302 (1983)] a proof, due to Xiao Gang, of the following result is given: $$K^ 2\geq 2p_ g$$, if the surface is irregular and the canonical system is composed with a pencil.

##### MSC:
 14J25 Special surfaces 14N05 Projective techniques in algebraic geometry
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##### References:
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