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
Mot clés : dégénérescence de surfaces, singularités, géométrie birationnelle, invariants topologiques
Calabri, Alberto 1; Ciliberto, Ciro 2; Flamini, Flaminio 3; Miranda, Rick 4

1 Università degli Studi di Padova Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Via Trieste, 63 35121 Padova (Italy)
2 Università degli Studi di Roma “Tor Vergata" Dipartimento di Matematica Via della Ricerca Scientifica 00133 Roma (Italy)
3 Università degli Studi di Roma “Tor Vergadata” Dipartimento di Matematica Via della Ricerca Scientifica 00133 Roma (Italy)
4 Colorado State University Department of Mathematics 101 Weber Building Fort Collins, CO 80523–1874 (USA)
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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/

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