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On the cohomological strata of families of vector bundles on algebraic surfaces

Edoardo Ballico (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this Note we study certain natural subsets of the cohomological stratification of the moduli spaces of rank 2 vector bundles on an algebraic surface. In the last section we consider the following problem: take a bundle E given by an extension, how can one recognize that E is a certain given bundle? The most interesting case considered here is the case E = T P 3 t since it applies to the study of codimension 1 meromorphic foliations with singularities on P 3 .

On the genus of reducible surfaces and degenerations of surfaces

Alberto Calabri, Ciro Ciliberto, Flaminio Flamini, Rick Miranda (2007)

Annales de l’institut Fourier

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 : 𝒳 Δ parametrized by a disc, with smooth general fibre, then the ω -genus of the fibres of π is constant.

On the moduli b-divisors of lc-trivial fibrations

Osamu Fujino, Yoshinori Gongyo (2014)

Annales de l’institut Fourier

Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro’s result on klt-trivial fibrations.

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