Higher order selfdual toric varieties

The notion of higher order dual varieties of a projective variety, introduced in \cite{P83}, is a natural generalization of the classical notion of projective duality. In this paper we present geometric and combinatorial characterizations of those equivariant projective toric embeddings that satisfy higher order selfduality. We also give several examples and general constructions. In particular, we highlight the relation with Cayley-Bacharach questions and with Cayley configurations.Comment: 21 page

Inflectional loci of scrolls

Let $X\subset \mathbb P^N$ be a scroll over a smooth curve $C$ and let \L=\mathcal O_{\mathbb P^N}(1)|_X denote the hyperplane bundle. The special geometry of $X$ implies that some sheaves related to the principal part bundles of \L are locally free. The inflectional loci of $X$ can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas imply that the only uninflected scrolls are the balanced rational normal scrolls.Comment: 9 pages, improved version. Accepted in Mathematische Zeitschrif

Classifying smooth lattice polytopes via toric fibrations

We define Q-normal lattice polytopes. Natural examples of such polytopes are Cayley sums of strictly combinatorially equivalent lattice polytopes, which correspond to particularly nice toric fibrations, namely toric projective bundles. In a recent paper Batyrev and Nill have suggested that there should be a bound, N(d), such that every lattice polytope of degree d and dimension at least N(d) decomposes as a Cayley sum. We give a sharp answer to this question for smooth Q-normal polytopes. We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a Cayley sum of strictly combinatorially equivalent polytopes if n is greater than or equal to 2d+1. The proof relies on the study of the nef value morphism associated to the corresponding toric embedding.Comment: Revised version, minor changes. To appear in Advances in Mat

The effects of a sport drink containing carbohydrates and electrolytes with or without caffeine on 20km cycling time trial performance in men and women

Caffeine is a commonly used substance by athletes and has become more widespread since it\u27s legalization by the World Anti Doping Association (WADA) and the International Olympic Committee (IOC). Extensive research has been done on the effect of caffeine on exercise performance, however, most studies were done at low intensity, and none compared differences in performance between men and women. Thus, 18 university aged (9 men and 9 women) individuals were recruited to perform three 20km cycling time trials including a familiarization trial (FAM), a carbohydrate plus placebo condition (CHO+P) and carbohydrate plus caffeine (5mg/kg) condition (CHO+C). Time to complete the 20km distance was significantly decreased in the CHO+C trial compared to the CHO+P trial in both men and women. This was concomitant with an increase in post-exercise blood glucose levels in the CHO+C trial but not in the CHO+P trial. However, ratings of perceived exertion (RER), respiratory exchange ratio (RER), tympanic temperature and a novel measure of arousal (long term excitement, LTE) were not different between trials. It is concluded that caffeine can improve performance when combined carbohydrate over carbohydrate alone. Moreover, these performance improvements are not different between sexes. Of the subjective (RPE), physiological (HR, RER) and excitement (LTE) variables that were measured, it is unclear why caffeine had this effect. Nonetheless, these findings are intriguing, especially for those athletes looking to gain an advantage in activity that requires intense work over the span of 20-35min, as well as WADA in their decisions to examine the use of caffeine during competition. Caffeine is currently a legal ergogenic aid and these data add to the significant body of literature indicating its powerful ergogenic properties in many activities