We deal with a reducible projective surface with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the -genus of , i.e. the dimension of the vector space of global sections of the dualizing sheaf . Then we prove that, when is smoothable, i.e. when is the central fibre of a flat family parametrized by a disc, with smooth general fibre, then the -genus of the fibres of is constant.
Nous étudions une surface projective réductible avec des singularités dites Zappatiques, qui sont une généralisation des croisements normaux. Nous calculons d’abord le -genre de , c’est-à-dire la dimension de l’espace vectoriel des sections globales du faisceau dualisant sur . Nous démontrons après que, si est lissifiable, c’est-à-dire si est la fibre centrale d’une famille plate paramétrée par un disque, à fibre générale lisse, alors le -genre des fibres est constant.
Keywords: Degenerations of surfaces, singularities, birational geometry, topological invariants
@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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.2266/}
}
TY - JOUR AU - Calabri, Alberto AU - Ciliberto, Ciro AU - Flamini, Flaminio AU - Miranda, Rick TI - On the genus of reducible surfaces and degenerations of surfaces JO - Annales de l'Institut Fourier PY - 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/ DO - 10.5802/aif.2266 LA - en ID - AIF_2007__57_2_491_0 ER -
%0 Journal Article %A Calabri, Alberto %A Ciliberto, Ciro %A Flamini, Flaminio %A Miranda, Rick %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://aif.centre-mersenne.org/articles/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] On the of degenerations of surfaces and the multiple point formula (to appear on Ann. Math.)
[2] 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 | Zbl
[3] Projective degenerations of surfaces, Gaussian maps, and Fano threefolds, Invent. Math., Volume 114 (1993) no. 3, pp. 641-667 | DOI | MR | Zbl
[4] Pillow degenerations of 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 | Zbl
[5] The braid monodromy of plane algebraic curves and hyperplane arrangements, Comment. Math. Helv., Volume 72 (1997) no. 2, pp. 285-315 | DOI | MR | Zbl
[6] 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] Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993 (The William H. Roever Lectures in Geometry) | MR | Zbl
[8] Families of curves in and Zeuthen’s problem, Mem. Amer. Math. Soc., Volume 130 (1997) no. 617, pp. viii+96 | MR | Zbl
[9] Toroidal embeddings. I, Springer-Verlag, Berlin, 1973 (Lecture Notes in Mathematics, Vol. 339) | MR | Zbl
[10] Toward moduli of singular varieties, Compositio Math., Volume 56 (1985) no. 3, pp. 369-398 | Numdam | MR | Zbl
[11] Stable branch curves and braid monodromies, Algebraic geometry (Chicago, Ill., 1980) (Lecture Notes in Math.), Volume 862, Springer, Berlin, 1981, pp. 107-192 | MR | Zbl
[12] Braid group techniques in complex geometry. III. Projective degeneration of , Classification of algebraic varieties (L’Aquila, 1992) (Contemp. Math.), Volume 162, Amer. Math. Soc., Providence, RI, 1994, pp. 313-332 | MR | Zbl
[13] 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 | Zbl
[14] Vorlesungen über algebraische Geometrie, 1, Teubner, Leipzig, 1921
[15] 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 | Zbl
[16] 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 | Zbl
[17] 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 | DOI | MR | Zbl
Cited by Sources:
