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On a question of Demailly-Peternell-Schneider

Meng Chen, Qi Zhang (2013)

Journal of the European Mathematical Society

We give an affirmative answer to an open question posed by Demailly-Peternell-Schneider in 2001 and recently by Peternell. Let f : X Y be a surjective morphism from a log canonical pair ( X , D ) onto a -Gorenstein variety Y . If - ( K X + D ) is nef, we show that K Y is pseudo-effective.

On covering and quasi-unsplit families of curves

Laurent Bonavero, Cinzia Casagrande, Stéphane Druel (2007)

Journal of the European Mathematical Society

Given a covering family V of effective 1-cycles on a complex projective variety X , we find conditions allowing one to construct a geometric quotient q : X Y , with q regular on the whole of X , such that every fiber of q is an equivalence class for the equivalence relation naturally defined by V . Among other results, we show that on a normal and -factorial projective variety X with canonical singularities and dim X 4 , every covering and quasi-unsplit family V of rational curves generates a geometric extremal...

On log canonical divisors that are log quasi-numerically positive

Shigetaka Fukuda (2004)

Open Mathematics

Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.

On some numerical properties of Fano varieties

Cinzia Casagrande (2004)

Bollettino dell'Unione Matematica Italiana

This is the text of a talk given at the XVII Convegno dell’Unione Matematica Italiana held at Milano, September 8-13, 2003. I would like to thank Angelo Lopez and Ciro Ciliberto for the kind invitation to the conference. I survey some numerical conjectures and theorems concerning relations between the index, the pseudo-index and the Picard number of a Fano variety. The results I refer to are contained in the paper [3], wrote in collaboration with Bonavero, Debarre and Druel.

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.

On the number of compatibly Frobenius split subvarieties, prime F -ideals, and log canonical centers

Karl Schwede, Kevin Tucker (2010)

Annales de l’institut Fourier

Let X be a projective Frobenius split variety with a fixed Frobenius splitting θ . In this paper we give a sharp uniform bound on the number of subvarieties of X which are compatibly Frobenius split with θ . Similarly, we give a bound on the number of prime F -ideals of an F -finite F -pure local ring. Finally, we also give a bound on the number of log canonical centers of a log canonical pair. This final variant extends a special case of a result of Helmke.

On the Picard number of divisors in Fano manifolds

Cinzia Casagrande (2012)

Annales scientifiques de l'École Normale Supérieure

Let  X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in  X . We consider the image 𝒩 1 ( D , X ) of  𝒩 1 ( D ) in  𝒩 1 ( X ) under the natural push-forward of 1 -cycles. We show that ρ X - ρ D codim 𝒩 1 ( D , X ) 8 . Moreover if codim 𝒩 1 ( D , X ) 3 , then either X S × T where S is a Del Pezzo surface, or codim 𝒩 1 ( D , X ) = 3 and X has a fibration in Del Pezzo surfaces onto a Fano manifold T such that ρ X - ρ T = 4 .

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