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.

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).

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.

Selection and hydroponic growth of bread wheat cultivars for bioregenerative life support systems

As part of the ESA-funded MELiSSA program, the suitability, the growth and the development of four bread wheat cultivars were investigated in hydroponic culture with the aim to incorporate such a cultivation system in an Environmental Control and Life Support System (ECLSS). Wheat plants can fulfill three major functions in space: (a) fixation of CO2 and production of O2, (b) production of grains for human nutrition and (c) production of cleaned water after condensation of the water vapor released from the plants by transpiration. Four spring wheat cultivars (Aletsch, Fiorina, Greina and CH Rubli) were grown hydroponically and compared with respect to growth and grain maturation properties. The height of the plants, the culture duration from germination to harvest, the quantity of water used, the number of fertile and non-fertile tillers as well as the quantity and quality of the grains harvested were considered. Mature grains could be harvested after around 160 days depending on the varieties. It became evident that the nutrient supply is crucial in this context and strongly affects leaf senescence and grain maturation. After a first experiment, the culture conditions were improved for the second experiment (stepwise decrease of EC after flowering, pH adjusted twice a week, less plants per m2) leading to a more favorable harvest (higher grain yield and harvest index). Considerably less green tillers without mature grains were present at harvest time in experiment 2 than in experiment 1. The harvest index for dry matter (including roots) ranged from 0.13 to 0.35 in experiment 1 and from 0.23 to 0.41 in experiment 2 with modified culture conditions. The thousand-grain weight for the four varieties ranged from 30.4 to 36.7 g in experiment 1 and from 33.2 to 39.1 g in experiment 2, while market samples were in the range of 39.4–46.9 g. Calcium levels in grains of the hydroponically grown wheat were similar to those from field-grown wheat, while potassium, magnesium, phosphorus, iron, zinc, copper, manganese and nickel levels tended to be higher in the grains of experimental plants. It remains a challenge for future experiments to further adapt the nutrient supply in order to improve senescence of vegetative plant parts, harvest index and the composition of bread wheat grains.

Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka–Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930s, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview chapter we present three classes of properties: (1) density property, (2) flexibility, and (3) Oka–Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

On the Closure of the Tame Automorphism Group of Affine Three-Space

We provide explicit families of tame automorphisms of the complex affine three-space which degenerate to wild automorphisms. This shows that the tame subgroup of the group of polynomial automorphisms of C3 is not closed, when the latter is seen as an infinite-dimensional algebraic group.

Participatory on-farm appraisal of improved pearl millet varieties in Eritrea

Pearl millet is Eritrea’s second most important cereal crop. Embedded in Eritrea’s millet improvement programme, this study summarizes the results of an on-farm appraisal of millet varieties at 3 study sites in western Eritrea. The participating communities were asked to rank improved and traditional varieties. Men generally preferred the improved varieties for their higher yields, while women preferred the traditional landraces owing to taste and food quality.

Flexible varieties and automorphism groups

Given an irreducible affine algebraic variety X of dimension n≥2 , we let SAut(X) denote the special automorphism group of X , that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus X reg , then it is infinitely transitive on X reg . In turn, the transitivity is equivalent to the flexibility of X . The latter means that for every smooth point x∈X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X) . We also provide various modifications and applications.

Effect of combined drought and heat stress on heat shock proteins in wheat varieties

Introduction: The re sponse of crop plants ex posed on drought or heat shock is related to de crease in the synthesis of normal proteins, accompanied by increased translation of heat shock proteins (HSPs). Though drought and heat stress have been studied individually, little is known about their combined effect on plants. Methods: The wheat (Triticum aestivum L.) varieties (Katya-tolerant, Sadovo or Mladka-susceptible) were potted in soil. Eight-day-old plants were ex posed to with drawing water for seven days. Heat shock was realized in growth chamber at 40 °C for 6h. A combination of drought and heat shock was per formed by subjecting drought-stressed plants to heat shock treatment. Expression of HSPs in the first leaf of wheat varieties was analyzed by SDS electrophoresis and immunoblotting. Polyclonal antibodies against HSP20, HSP60, HSP110 and mononclonal antibodies against HSP70 were used to distinguish the mentioned HSPs. Results: The leaf relative water content (RWC), which indicated the level of plant dehydration decreased significantly (34 %) under drought stressed conditions The electrolyte leakage of ions (EL), representing the level of the cell membrane stability in creased mark edly (68 %), especially under combination of drought and heat. Maximum EL was ob served in drought susceptible varieties Sadovo and Mladka. Drought and heat shock combination in the wheat plants resulted in the induction of specific HSPs. Conclusions: Our results demonstrate that the response of the wheat plants to a combination of drought and heat stress is different from the response of plants to each of these stresses applied separately. Induction of synergetic effect on HSP expression in case of combination between drought and heat was discussed in the case of two contrasting wheat varieties.

Uniqueness of the Embedding Continuous Convolution Semigroup of a Gaussian Probability Measure on the Affine Group and an Application in Mathematical Finance

Let {μ(i)t}t≥0 ( i=1,2 ) be continuous convolution semigroups (c.c.s.) of probability measures on Aff(1) (the affine group on the real line). Suppose that μ(1)1=μ(2)1 . Assume furthermore that {μ(1)t}t≥0 is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then μ(1)t=μ(2)t for all t≥0 . We end up with a possible application in mathematical finance.

No adverse effect of genetically modified antifungal wheat on decomposition dynamics and the soil fauna community - a field study

The cultivation of genetically modified (GM) plants has raised several environmental concerns. One of these concerns regards non-target soil fauna organisms, which play an important role in the decomposition of organic matter and hence are largely exposed to GM plant residues. Soil fauna may be directly affected by transgene products or indirectly by pleiotropic effects such as a modified plant metabolism. Thus, ecosystem services and functioning might be affected negatively. In a litterbag experiment in the field we analysed the decomposition process and the soil fauna community involved. Therefore, we used four experimental GM wheat varieties, two with a race-specific antifungal resistance against powdery mildew (Pm3b) and two with an unspecific antifungal resistance based on the expression of chitinase and glucanase. We compared them with two non-GM isolines and six conventional cereal varieties. To elucidate the mechanisms that cause differences in plant decomposition, structural plant components (i.e. C:N ratio, lignin, cellulose, hemicellulose) were examined and soil properties, temperature and precipitation were monitored. The most frequent taxa extracted from decaying plant material were mites (Cryptostigmata, Gamasina and Uropodina), springtails (Isotomidae), annelids (Enchytraeidae) and Diptera (Cecidomyiidae larvae). Despite a single significant transgenic/month interaction for Cecidomyiidae larvae, which is probably random, we detected no impact of the GM wheat on the soil fauna community. However, soil fauna differences among conventional cereal varieties were more pronounced than between GM and non-GM wheat. While leaf residue decomposition in GM and non-GM wheat was similar, differences among conventional cereals were evident. Furthermore, sampling date and location were found to greatly influence soil fauna community and decomposition processes. The results give no indication of ecologically relevant adverse effects of antifungal GM wheat on the composition and the activity of the soil fauna community.