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# Documents  Floris, Enrica | enregistrements trouvés : 2

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## Invariance of plurigenera for foliations on surfaces Floris, Enrica | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry

Let $X$ be a smooth algebraic surface. A foliation $F$ on $X$ is, roughly speaking, a subline bundle $T_F$ of the tangent bundle of $X$. The dual of $T_F$ is called the canonical bundle of the foliation $K_F$. In the last few years birational methods have been successfully used in order to study foliations. More precisely, geometric properties of the foliation are translated into properties of the canonical bundle of the foliation. One of the most important invariants describing the properties of a line bundle $L$ is its Kodaira dimension $\kappa(L)$, which measures the growth of the global sections of $L$ and its tensor powers. The Kodaira dimension of a foliation $F$ is defined as the Kodaira dimension of its canonical bundle $\kappa(K_F)$. In their fundamental works, Brunella and McQuillan give a classfication of foliations on surfaces on the model of Enriques-Kodaira classification of surfaces. The next step is the study of the behaviour of families of foliations. Brunella proves that, for a family of foliations $(X_t, F_t)$ of dimension one on surfaces, satisfying certain hypotheses of regularity, the Kodaira dimension of the foliation does not depend on $t$. By analogy with Siu's Invariance of Plurigenera, it is natural to ask whether for a family of foliations $(X_t, F_t)$ the dimensions of global sections of the canonical bundle and its powers depend on $t$. In this talk we will discuss to which extent an Invariance of Plurigenera for foliations is true and under which hypotheses on the family of foliations it holds. Let $X$ be a smooth algebraic surface. A foliation $F$ on $X$ is, roughly speaking, a subline bundle $T_F$ of the tangent bundle of $X$. The dual of $T_F$ is called the canonical bundle of the foliation $K_F$. In the last few years birational methods have been successfully used in order to study foliations. More precisely, geometric properties of the foliation are translated into properties of the canonical bundle of the foliation. One of the ...

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## On the B-Semiampleness Conjecture Floris, Enrica | CIRM H

Multi angle

Research talks

An lc-trivial fibration $f : (X, B) \to Y$ is a fibration such that the log-canonical divisor of the pair $(X, B)$ is trivial along the fibres of $f$. As in the case of the canonical bundle formula for elliptic fibrations, the log-canonical divisor can be written as the sum of the pullback of three divisors: the canonical divisor of $Y$; a divisor, called discriminant, which contains informations on the singular fibres; a divisor, called moduli part, that contains informations on the variation in moduli of the fibres. The moduli part is conjectured to be semiample. Ambro proved the conjecture when the base $Y$ is a curve. In this talk we will explain how to prove that the restriction of the moduli part to a hypersurface is semiample assuming the conjecture in lower dimension. This is a joint work with Vladimir Lazić. An lc-trivial fibration $f : (X, B) \to Y$ is a fibration such that the log-canonical divisor of the pair $(X, B)$ is trivial along the fibres of $f$. As in the case of the canonical bundle formula for elliptic fibrations, the log-canonical divisor can be written as the sum of the pullback of three divisors: the canonical divisor of $Y$; a divisor, called discriminant, which contains informations on the singular fibres; a divisor, called moduli ...

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