Infinite Transitivity on Affine Varieties

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.

Fermat's Equation in Matrices

The Fermat equation is solved in integral two by two matrices of determinant one as well as in finite order integral three by three matrices.

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

We construct holomorphic families of proper holomorphic embeddings of \mathbb {C}^{k} into \mathbb {C}^{n} (0\textless k\textless n-1), so that for any two different parameters in the family, no holomorphic automorphism of \mathbb {C}^{n} can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of \mathbb {C}^{n}, we derive the existence of families of holomorphic \mathbb {C}^{*}-actions on \mathbb {C}^{n} (n\ge5) so that different actions in the family are not conjugate. This result is surprising in view of the long-standing holomorphic linearization problem, which, in particular, asked whether there would be more than one conjugacy class of \mathbb {C}^{*}-actions on \mathbb {C}^{n} (with prescribed linear part at a fixed point).

Diophantine approximation on projective varieties I: algebraic distance and metric Bézout theorem

"Vegeu el resum a l'inici del document del fitxer adjunt."

The Brauer group and the second Abel-Jacobi map for 0-cycles on algebraic varieties

This paper focuses on the connection between the Brauer group and the 0-cycles of an algebraic variety. We give an alternative construction of the second l-adic Abel-Jacobi map for such cycles, linked to the algebraic geometry of Severi-Brauer varieties on X. This allows us then to relate this Abel-Jacobi map to the standard pairing between 0-cycles and Brauer groups (see [M], [L]), completing results from [M] in this direction. Second, for surfaces, it allows us to present this map according to the more geometrical approach devised by M. Green in the framework of (arithmetic) mixed Hodge structures (see [G]). Needless to say, this paper owes much to the work of U. Jannsen and, especially, to his recently published older letter [J4] to B. Gross.

Theta-duality on Prym varieties and a Torelli Theorem

Let $\pi : \widetilde C \to C$ be an unramified double covering of irreducible smooth curves and let $P$ be the attached Prym variety. We prove the scheme-theoretic theta-dual equalities in the Prym variety $T(\widetilde C)=V^2$ and $T(V^2)=\widetilde C$, where $V^2$ is the Brill-Noether locus of $P$ associated to $\pi$ considered by Welters. As an application we prove a Torelli theorem analogous to the fact that the symmetric product $D^{(g)}$ of a curve $D$ of genus $g$ determines the curve.

Slow-fast systems on algebraic varieties bordering piecewise-smooth dynamical systems

This article extends results contained in Buzzi et al. (2006) [4], Llibre et al. (2007, 2008) [12,13] concerning the dynamics of non-smooth systems. In those papers a piecewise C-k discontinuous vector field Z on R-n is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F :U -> R a polynomial function defined on the open subset U subset of R-n. The set F-1 (0) divides U into subdomains U-1, U-2,...,U-k, with border F-1(0). These subdomains provide a Whitney stratification on U. We consider Z(i) :U-i -> R-n smooth vector fields and we get Z = (Z(1),...., Z(k)) a discontinuous vector field with discontinuities in F-1(0). Our approach combines several techniques such as epsilon-regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an epsilon-regularization of Z (see Sotomayor and Teixeira, 1996 [18]; Llibre and Teixeira, 1997 [15]). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16]), in systems with hysteresis (Seidman, 2006 [17]) and in mechanical systems with impacts (di Bernardo et al., 2008 [5]). (C) 2011 Elsevier Masson SAS. All rights reserved.

Impact damage on selected varieties of apples, pears and other fruits.

During the past years, different laboratory impact response studies have been carrj.ec out in following fruits: apples (2 varieties), pears (4 varieties), Asian or Nashi pears (4 varieties), melons (2 varieties), peaches (2 varieties) and avocados. The methodology of the tests is described, as well as the results and observations obtained in each group of tests. Impact response is compared to bruising susceptibility, bruise characteristics (appearance and structural features) and varietal and ripeness differences.

A dissertation on the varieties of mankind : being an address read before the Cuvierian and Physiological societies of Wesleyan University, August 2, 1843 /

Mode of access: Internet.

The graphic arts; a treatise on the varieties of drawing, painting, and engraving, in comparison with each other and with nature,

First published in 1882.

On Affine Connections in a Riemannian Manifold with a Circulant Metric and two Circulant Affinor Structures

Ива Р. Докузова, Димитър Р. Разпопов - В настоящата статия е разгледан клас V оттримерни риманови многообразия M с метрика g и два афинорни тензора q и S. Дефинирана е и друга метрика ¯g в M. Локалните координати на всички тези тензори са циркулантни матрици. Намерени са: 1) зависимост между тензора на кривина R породен от g и тензора на кривина ¯R породен от ¯g; 2) тъждество за тензора на кривина R в случая, когато тензорът на кривина ¯R се анулира; 3) зависимост между секционната кривина на прозволна двумерна q-площадка {x, qx} и скаларната кривина на M.

The algebraic density property for affine toric varieties

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén–Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X∖Y≠TX∖Y≠T. For toric surfaces we are able to classify those which possess a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella's famous classification of complete algebraic vector fields in the affine plane.

Spodoptera frugiperda (J. E. Smith) (Lepidoptera: Noctuidae) in cotton: vertical distribution of egg masses, effects of adult density and plant age on oviposition behavior

O trabalho teve por objetivo determinar, em casa-de-vegetação, a densidade adequada de adultos, a idade ideal das plantas e a distribuição vertical de ovos nas diferentes partes da planta, visando à realização de estudos de resistência e à melhoria das práticas de manejo de Spodoptera frugiperda em algodoeiro. Os experimentos de oviposição de S. frugiperda em relação à densidade de adultos, relação entre plantas de algodoeiro e oviposição de S. frugiperda e não preferência para oviposição de S. frugiperda em variedades de algodoeiro foram realizados com plantas da variedade de algodoeiro BRS Ita 90. A não preferência (antixenose) para oviposição foi acompanhada nas variedades FiberMax 966, FiberMax 977, DeltaOpal, DeltaPenta, Acala 90, Coodetec 408, Coodetec 409, Coodetec 410, BRS Cedro, BRS Ipê, BRS Aroeira, IPR 96, IPR 20, BRS Araçá, IAC 24 e BRS Ita 90. Concluiu-se que S. frugiperda prefere ovipositar em plantas com cerca de 60 dias de idade, na superfície inferior das folhas situadas no terço superior dos testes de plantas sob condições de estufa. Uma densidade de, pelo menos, três pares de adultos S. frugiperda por planta foi suficiente para realizar testes de não preferência para oviposição em casa de vegetação. As variedades Coodetec 408, BRS Aroeira, BRS Araçá, BRS Ita 90 e DeltaPenta apresentaram resistência do tipo não preferência para oviposição de S. frugiperda. Em regiões com altas infestações de S. frugiperda, seria prudente para o cultivo utilizar a variedade de algodão BRS Ita 90.