On the genus of reducible surfaces and degenerations of surfaces
Annales de l'Institut Fourier, Volume 57 (2007) no. 2, pp. 491-516.

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.

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.

DOI: 10.5802/aif.2266
Classification: 14J17,  14B07,  14D06,  14D07,  14N20
Keywords: Degenerations of surfaces, singularities, birational geometry, topological invariants
     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},
     zbl = {1125.14018},
     mrnumber = {2310949},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2266/}
TI  - On the genus of reducible surfaces  and degenerations of surfaces
JO  - Annales de l'Institut Fourier
PY  - 2007
DA  - 2007///
SP  - 491
EP  - 516
VL  - 57
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2266/
UR  - https://zbmath.org/?q=an%3A1125.14018
UR  - https://www.ams.org/mathscinet-getitem?mr=2310949
UR  - https://doi.org/10.5802/aif.2266
DO  - 10.5802/aif.2266
LA  - en
ID  - AIF_2007__57_2_491_0
ER  - 
%0 Journal Article
%T On the genus of reducible surfaces  and degenerations of surfaces
%J Annales de l'Institut Fourier
%D 2007
%P 491-516
%V 57
%N 2
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2266
%R 10.5802/aif.2266
%G en
%F AIF_2007__57_2_491_0
Calabri, Alberto; Ciliberto, Ciro; Flamini, Flaminio; Miranda, Rick. On the genus of reducible surfaces  and degenerations of surfaces. Annales de l'Institut Fourier, Volume 57 (2007) no. 2, pp. 491-516. doi : 10.5802/aif.2266. https://aif.centre-mersenne.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., Tome 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.) Tome 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., Tome 72 (1997) no. 2, pp. 285-315 | Article | MR: 1470093 | Zbl: 0959.52018

[6] The birational geometry of degenerations (Friedman, Robert; Morrison, David R., eds.), Progress in Mathematics, Tome 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, Tome 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., Tome 130 (1997) no. 617, viii+96 pages | 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., Tome 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.) Tome 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.) Tome 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.) Tome 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 101-119 | MR: 756848 | Zbl: 0576.32034

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

[15] Teicher, M. Hirzebruch surfaces: degenerations, related braid monodromy, Galois covers, Algebraic geometry: Hirzebruch 70 (Warsaw, 1998) (Contemp. Math.) Tome 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, Tome 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), Tome 30 (1949), pp. 123-146 | Article | MR: 36545 | Zbl: 0041.48006

Cited by Sources: