TiME 2019 - F. Catanese - The P

The $P_12$-theorem:

an introduction to the classification of algebraic surfaces via its historical development.

by F. Catanese (Bayreuth U., DE)

Description:

The classification theorem of algebraic surfaces via the 12-th plurigenus $P_{12}$ was achieved by Castelnuovo and Enriques in 1914, and was explained by these authors in an article [CE1] contained in Klein’s Encyklopaedie der mathematischen Wissenschaften.

The program was initiated by Enriques in his first works (see especially [Enr1]) in which he introduced among other things the intersection calculus of curves on an algebraic surface, and the adjunction property of the canonical divisor.

It progressed quickly through the invaluable collaboration with Castelnuovo, who established ([Cas1, Cas2, Cas3]) basic results for the classification (the rationality criterion, $q = P_2 = 0$, the theorem of Castelnuovo and De Franchis, and the criterion for ruledness).

The purpose of the lectures shall be to explain the statement of the classification theorem, to illustrate the crucial steps, to give a flavour of the argument used, from Castelnuovo and Enriques and followers. And to explain what was left open after 1914, and attempted by Enriques in his book `Le superficie algebriche’ [Enr2] (Castelnuovo withdrew from the project since he felt that new methods should be first established, and he was somehow right).

We shall also read (and try to understand) parts of old papers of Castelnuovo and Enriques, comparing with more recent ones.

CONTENTS OF THE LECTURES.

L1. The basic set up. Curves on algebraic surfaces, arithmetic and geometric genus; ramification and canonical divisor. Blow-ups, resolution of rational maps, minimal models. Birational invariants of surfaces.

L2. First important results. Albanese variety, theorem of Castelnuovo-De Franchis. Fibred surfaces: Noether’s theorem, Zeuthen-Segre’s theorem, and Castelnuovo’ criteria for ruledness and for rationality.

L3. The classification theorem. Statement of the $P_{12}$-Theorem through the concept of surfaces isogenous to a product (as in [CL]). The Key theorem (when is $K$ not nef) and the crucial Theorem (surfaces with $q=1, p_g=0$: "abundance" in modern language).

L4. Central methods and ideas. Sketch of the strategy of proof: elliptic fibrations, divisors of elliptic type, modern point of view (Variation of Hodge Structures) and Castelnuovo-Enriques’ argument, as taken up again by Mumford [Mu]. What was done afterwards: Shafarevich, Kodaira, Bombieri-Mumford.

CLASSICAL REFERENCES

Federigo Enriques, Memorie scelte di Geometria. Vol.1-2-3, Zanichelli Editore Bologna (1956-1959-1966).
Federigo Enriques, Ricerche di Geometria sulle superficie algebriche. Memorie Acc. Torino, s. 2, to. XLIV (1893), p. 171-232. (Memorie Scelte vol. 1, pp. 31-106)

Extended and corrected in:

[Enr1] Federigo Enriques, Introduzione alla geometria sopra le superficie algebriche. Memorie della Società Italiana delle Scienze (dei XL), s. 3, to. X, (1896), pp. 1-81 (Memorie Scelte vol. 1, pp. 211-312) .
[CE1] Guido Castelnuovo and Federigo Enriques, Die algebraische Flächen vom Gesichtspunkte der birationalen Transformationen aus. Encyklopädie d. mathematischen Wissenschften, III, 2, Heft 6, pp. 674-768 (Memorie Scelte vol. 3, pp. 197-296).
[Cas1] Guido Castelnuovo, Sur les intégrales de différentielles totales appartenant à une surface irrégulière. Comptes Rendus des S\’eances de l’ Acad\’emie des Sciences de Paris, 140 (janvier-juin 1905), 200-222 (Opere Matematiche vol. II, 411-414).
[Cas2] Guido Castelnuovo, Sulle superficie aventi il genere aritmetico negativo (Estratto da una lettera al prof. M. De Franchis), Rendiconti del Circolo matematico di Palermo, 20 (1905), 55-60 (Opere Matematiche vol. II, 415-422
[Cas3] Guido Castelnuovo, Sugli integrali semplici appartenenti ad una superficie irregolare. Atti della R. Accademia dei Lincei, Classe SFMN. Rendiconti (5), 14 (I-semestre 1905) 545-556, 593-598, 655-663 (Opere Matematiche vol. II, 423-450)
[Enr2] Federigo Enriques, Le Superficie Algebriche. Zanichelli , Bologna (1949) XV + 464 pp.
Guido Castelnuovo, Memorie Scelte. Zanichelli Bologna (1937).
Guido Castelnuovo, Opere Matematiche. Memorie e Note, vol. I-II-III Accademia Nazionale dei Lincei, Roma (2002-2003-2004).

SEMICLASSICAL REFERENCES

Šafarevič, I. R.; Averbuh, B. G.; Vaĭnberg, Ju. R.; Žižčenko, A. B.; Manin, Ju. I.; Moĭšezon, B. G.; Tjurina, G. N.; Tjurin, A. N. Algebraic surfaces. Trudy Mat. Inst. Steklov. 75 1965 1–215.
David Mumford, Lectures on curves on an algebraic surface. With a section by G. M. Bergman. Annals of Mathematics Studies, No. 59 Princeton University Press, Princeton, N.J. 1966 xi+200 pp.
Kunihiko Kodaira, On the structure of complex analytic surfaces. IV. Amer. J. Math. 90 (1968), 1048–1066.
[Mu] David Mumford, Enriques' classification of surfaces in char p. I. 1969 Global Analysis (Papers in Honor of K. Kodaira) pp. 325–339
Enrico Bombieri, Dale Husemoller, Classification and embeddings of surfaces. Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pp. 329–420. Amer. Math. Soc., Providence, R.I., 1975.
Enrico Bombieri, David Mumford, Enriques' classification of surfaces in char. p. II. Complex analysis and algebraic geometry, pp. 23–42. Iwanami Shoten, Tokyo, 1977.
Enrico Bombieri, David Mumford, Enriques' classification of surfaces in char. p. III. Invent. Math. 35 (1976), 197–232.

ONE MODERN REFERENCE

[CL] Fabrizio Catanese and Binru Li, Enriques' classification in characteristic p > 0 : the $P_12$-Theorem. Nagoya Jour. (2018)

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