The bogomolov conjecture for totally degenerate Abelian varieties

We prove the Bogomolov conjecture for a totally degenerate abelian variety A over a function field. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place.

Tropical varieties for non-archimedean analytic spaces

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

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Strong exceptional sequences of vector bundles on certain Fano varieties

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Evaluation of susceptibility of pear and plum varieties and rootstocks to Ca. P. pyri and Ca. P. prunorum using Real-Time PCR

Real-time PCR was used to quantify phytoplasma concentration in fifty inoculated trees from five Prunus rootstocks and in forty-eight symptomatic pear and Japanese plum trees from orchards. Seasonal fluctuation of Ca. P. prunorum in different Prunus rootstocks, over three years, showed that the highest percentage detected by nested-PCR was in the ‘Garnem’ rootstock on nearly all sampling dates. Intra-varietal differences were also observed. Phytoplasma titer could be estimated by real time PCR in some trees of the rootstocks ‘Garnem’, ‘Barrier’, ‘GF-677’ and ‘Marianna’, and ranged from 4.7x105 to 3.18x109 phytoplasmas per gram of tissue. Quantification by real-time PCR was not possible in the ‘Cadaman’ trees analyzed, probably due to a lower phytoplasma titer in this variety. Samples from infected trees from commercial plots had different phytoplasma concentration and detection percentage depending on the variety, both being lower in ‘Fortune’ and ‘606’ Japanese plum and in ‘Blanquilla’ pear trees.

Homologia de Lawson i descens

La meva recerca en els tres anys de gaudiment de beca s'ha centrat en l'estudi de teories semitopològiques definides a través d'espais de cicles algebraics, introduïts per Friedlander i Lawson. Hem estudiat propietats de descens d'aquestes teories i hem construït una successió espectral que calcula explícitament la cohomologia mòrfica d'una varietat tòrica. D'altra banda, estem treballant en l'estudi de propietats d'invariància homotòpica per la cohomologia mòrfica, així com en l'estructura algebraica dels grups d'homologia de Lawson.

The varieties of regional change

Many metropolitan areas have experienced extreme boom-bust cycles over the past century. Some places, like Detroit, grew enormously as industrial powerhouses and then declined, while other older cities, like Boston, seem quite resilient. Education does a reasonable job of explaining urban resilience. In this paper, we present a simple model where education increases the level of entrepreneurship. In this model, human capital spillovers occur at the city level because skilled workers produce more product varieties and thereby increase labor demand. We decompose empirically the causes of the connection between skills and urban success and find that skills are associated with growth in productivity or entrepreneurship, not with growth in quality of life, at least outside of the West. We also find that skills seem to have depressed housing supply growth in the West, but not in other regions, which supports the view that educated residents in that region have fought for tougher land-use controls. We also present evidence that skills have had a disproportionately large impact on unemployment during the current recession.

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.

Arithmetical problems in number fields, abelian varieties and modular forms

Number theory, a fascinating area in mathematics and one of the oldest, has experienced spectacular progress in recent years. The development of a deep theoretical background and the implementation of algorithms have led to new and interesting interrelations with mathematics in general which have paved the way for the emergence of major theorems in the area. This report summarizes the contribution to number theory made by the members of the Seminari de Teoria de Nombres (UB-UAB-UPC) in Barcelona. These results are presented in connection with the state of certain arithmetical problems, and so this monograph seeks to provide readers with a glimpse of some specific lines of current mathematical research.

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus $2$: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.

Post-market environmental monitoring of Bt maize in Spain: Non-target effects of varieties derived from the event MON810 on predatory fauna

The Spanish Government has established post-market environmental monitoring (PMEM) as mandatory for genetically modified (GM) crop varieties cultivated in Spain. In order to comply with this regulation, effects of Bt maize varieties derived from the event MON810 on the predatory fauna were monitored for two years in northeast and central Spain. The study was carried out with a randomized block design in maize fields of 3-4 ha on which the abundance of plant-dwelling predators and the activity-density of soil-dwelling predators in Bt vs. non-Bt near-isogenic varieties were compared. To this end, the plots were sampled by visual inspection of a certain number of plants and pitfall traps 6 or 7 times throughout two seasons. No significant differences in predator densities on plants were found between Bt and non-Bt varieties. In the pitfall traps, significant differences between the two types of maize were found only in Staphylinidae, in which trap catches in non-Bt maize were higher than in Bt maize in central Spain. Based on the statistical power of the assays, surrogate arthropods for PMEM purposes are proposed; Orius spp. and Araneae for visual sampling and Carabidae, Araneae, and Staphylinidae for pitfall trapping. The other predator groups recorded in the study, Nabis sp. and Coccinellidae in visual sampling and Dermaptera in pitfall trapping, gave very poor power results. To help to establish a standardized protocol for PMEM of genetically modified crops, the effect-detecting capacity with a power of 0.8 of each predator group is given.

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.

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus $2$: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.

$E_{1}$-Formality of complex algebraic varieties

Let $X$ be a smooth complex algebraic variety. Morgan showed that the rational homotopy type of $X$ is a formal consequence of the differential graded algebra defined by the first term $E_{1}(X,W)$ of its weight spectral sequence. In the present work, we generalize this result to arbitrary nilpotent complex algebraic varieties (possibly singular and/or non-compact) and to algebraic morphisms between them. In particular, our results generalize the formality theorem of Deligne, Griffiths, Morgan and Sullivan for morphisms of compact Kähler varieties, filling a gap in Morgan"s theory concerning functoriality over the rationals. As an application, we study the Hopf invariant of certain algebraic morphisms using intersection theory.