On the genus of reducible surfaces and degenerations of surfaces
Annales de l'Institut Fourier, Volume 57 (2007) no. 2, p. 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 : https://doi.org/10.5802/aif.2266
Classification:  14J17,  14B07,  14D06,  14D07,  14N20
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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {2},
     year = {2007},
     pages = {491-516},
     doi = {10.5802/aif.2266},
     mrnumber = {2310949},
     zbl = {1125.14018},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_2007__57_2_491_0/

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