On the genus of reducible surfaces and degenerations of surfaces

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

Annales de l'Institut Fourier, Tome 57 (2007) no. 2, pp. 491-516.

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 $p_{\omega }\left(X\right)$ 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 $\pi :𝒳\to \Delta$ paramétrée par un disque, à fibre générale lisse, alors le $\omega$-genre des fibres est constant.

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 $p_{\omega }\left(X\right)$ 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 $\pi :𝒳\to \Delta$ parametrized by a disc, with smooth general fibre, then the $\omega$-genus of the fibres of $\pi$ is constant.

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