On the genus of reducible surfaces and degenerations of surfaces
[Sur le genre des surfaces réductibles et dégénérescences des surfaces]
Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 491-516.

Nous étudions une surface projective réductible X avec des singularités dites Zappatiques, qui sont une généralisation des croisements normaux. Nous calculons d’abord le ω-genre p ω (X) de X, c’est-à-dire la dimension de l’espace vectoriel des sections globales du faisceau dualisant ω X sur X. Nous démontrons après que, si X est lissifiable, c’est-à-dire si X est la fibre centrale d’une famille plate π:𝒳Δ paramétrée par un disque, à fibre générale lisse, alors le ω-genre des fibres est constant.

We deal with a reducible projective surface X with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the ω-genus p ω (X) of X, i.e. the dimension of the vector space of global sections of the dualizing sheaf ω X . Then we prove that, when X is smoothable, i.e. when X is the central fibre of a flat family π:𝒳Δ parametrized by a disc, with smooth general fibre, then the ω-genus of the fibres of π is constant.

DOI : https://doi.org/10.5802/aif.2266
Classification : 14J17,  14B07,  14D06,  14D07,  14N20
Mots clés : dégénérescence de surfaces, singularités, géométrie birationnelle, invariants topologiques
@article{AIF_2007__57_2_491_0,
     author = {Calabri, Alberto and Ciliberto, Ciro and Flamini, Flaminio and Miranda, Rick},
     title = {On the genus of reducible surfaces  and degenerations of surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {491--516},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {2},
     year = {2007},
     doi = {10.5802/aif.2266},
     mrnumber = {2310949},
     zbl = {1125.14018},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2266/}
}
Calabri, Alberto; Ciliberto, Ciro; Flamini, Flaminio; Miranda, Rick. On the genus of reducible surfaces  and degenerations of surfaces. Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 491-516. doi : 10.5802/aif.2266. http://archive.numdam.org/articles/10.5802/aif.2266/

[1] Calabri, Alberto; Ciliberto, Ciro; Flamini, Flaminio; Miranda, Rick On the K 2 of degenerations of surfaces and the multiple point formula (to appear on Ann. Math.)

[2] Calabri, Alberto; Ciliberto, Ciro; Flamini, Flaminio; Miranda, Rick On the geometric genus of reducible surfaces and degenerations of surfaces to unions of planes, The Fano Conference, Univ. Torino, Turin, 2004, pp. 277-312 | MR 2112579 | Zbl 1071.14057

[3] Ciliberto, Ciro; Lopez, Angelo; Miranda, Rick Projective degenerations of K3 surfaces, Gaussian maps, and Fano threefolds, Invent. Math., Volume 114 (1993) no. 3, pp. 641-667 | Article | MR 1244915 | Zbl 0807.14028

[4] Ciliberto, Ciro; Miranda, Rick; Teicher, Mina Pillow degenerations of K3 surfaces, Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001) (NATO Sci. Ser. II Math. Phys. Chem.), Volume 36, Kluwer Acad. Publ., Dordrecht, 2001, pp. 53-63 | MR 1866890 | Zbl 1006.14014

[5] Cohen, Daniel C.; Suciu, Alexander I. The braid monodromy of plane algebraic curves and hyperplane arrangements, Comment. Math. Helv., Volume 72 (1997) no. 2, pp. 285-315 | Article | MR 1470093 | Zbl 0959.52018

[6] Friedman, Robert; Morrison, David R. The birational geometry of degenerations, Progress in Mathematics, 29, Birkhäuser Boston, Mass., 1983 (Based on papers presented at the Summer Algebraic Geometry Seminar held at Harvard University, Cambridge, Mass. June 11–July 29, 1981)

[7] Fulton, William Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993 (The William H. Roever Lectures in Geometry) | MR 1234037 | Zbl 0813.14039

[8] Hartshorne, Robin Families of curves in 3 and Zeuthen’s problem, Mem. Amer. Math. Soc., Volume 130 (1997) no. 617, pp. viii+96 | MR 1401493 | Zbl 0894.14001

[9] Kempf, G.; Knudsen, Finn Faye; Mumford, D.; Saint-Donat, B. Toroidal embeddings. I, Springer-Verlag, Berlin, 1973 (Lecture Notes in Mathematics, Vol. 339) | MR 335518 | Zbl 0271.14017

[10] Kollár, János Toward moduli of singular varieties, Compositio Math., Volume 56 (1985) no. 3, pp. 369-398 | Numdam | MR 814554 | Zbl 0666.14003

[11] Moishezon, B. G. Stable branch curves and braid monodromies, Algebraic geometry (Chicago, Ill., 1980) (Lecture Notes in Math.), Volume 862, Springer, Berlin, 1981, pp. 107-192 | MR 644819 | Zbl 0476.14005

[12] Moishezon, Boris; Teicher, Mina Braid group techniques in complex geometry. III. Projective degeneration of V 3 , Classification of algebraic varieties (L’Aquila, 1992) (Contemp. Math.), Volume 162, Amer. Math. Soc., Providence, RI, 1994, pp. 313-332 | MR 1272706 | Zbl 0815.14023

[13] Morrison, David R. The Clemens-Schmid exact sequence and applications, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) (Ann. of Math. Stud.), Volume 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 101-119 | MR 756848 | Zbl 0576.32034

[14] Severi, Francesco Vorlesungen über algebraische Geometrie, 1, Teubner, Leipzig, 1921

[15] Teicher, M. Hirzebruch surfaces: degenerations, related braid monodromy, Galois covers, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998) (Contemp. Math.), Volume 241, Amer. Math. Soc., Providence, RI, 1999, pp. 305-325 | MR 1720873 | Zbl 0993.14017

[16] Zappa, Guido Su alcuni contributi alla conosceuza della struttura topologica delle superficie algebriche, dati dal metodo dello spezzamento in sistemi di piani, Pont. Acad. Sci. Acta, Volume 7 (1943), pp. 4-8 | MR 26362 | Zbl 0061.34908

[17] Zappa, Guido Alla ricerca di nuovi significati topologici dei generi geometrico e aritmetico di una superficie algebrica, Ann. Mat. Pura Appl. (4), Volume 30 (1949), pp. 123-146 | Article | MR 36545 | Zbl 0041.48006