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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 order three Brauer classes for cubic surfaces

Andreas-Stephan Elsenhans, Jörg Jahnel (2012)

Open Mathematics

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over ℚ such that Br(S)/Br(ℚ) is a 3-group. Our approach is to associate a Brauer class with every ordered triplet of Galois invariant pairs of Steiner trihedra. We show that all order three Brauer classes may be obtained in this way. To show the effect of the obstruction, we give explicit examples.

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 .

On the S-fundamental group scheme

Adrian Langer (2011)

Annales de l’institut Fourier

We introduce a new fundamental group scheme for varieties defined over an algebraically closed (or just perfect) field of positive characteristic and we use it to study generalization of C. Simpson’s results to positive characteristic. We also study the properties of this group and we prove Lefschetz type theorems.

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