On the genus of reducible surfaces and degenerations of surfaces

[Sur le genre des surfaces réductibles et dégénérescences des surfaces]

Calabri, Alberto ; Ciliberto, Ciro ; Flamini, Flaminio ; Miranda, Rick

Annales de l'Institut Fourier, Tome 57 (2007), p. 491-516 / Harvested from Numdam

Résumé

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

Abstract

We deal with a reducible projective surface $X$ with so-called Zappatic singularities, which are a generalization of normal crossings. First we compute the $\omega $-genus $p_\omega (X)$ of $X$, i.e. the dimension of the vector space of global sections of the dualizing sheaf $\omega _X$. Then we prove that, when $X$ is smoothable, i.e. when $X$ is the central fibre of a flat family $\pi :\mathcal{X}\rightarrow \Delta $ parametrized by a disc, with smooth general fibre, then the $\omega $-genus of the fibres of $\pi $ is constant.

Détail

Publié le : 2007-01-01

MR 2310949 | Zbl 1125.14018

DOI : https://doi.org/10.5802/aif.2266
Classification:  14J17,  14B07,  14D06,  14D07,  14N20
Mots clés: dégénérescence de surfaces, singularités, géométrie birationnelle, invariants topologiques

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