We study, as hypersurfaces in toric varieties, elliptic Calabi–Yau fourfolds for F-theory compactifications dual to E8 × E8 heterotic strings compactified to four dimensions on elliptic Calabi–Yau threefolds with some choice of vector bundle. We describe how to read off the vector bundle data for the heterotic compactification from the toric data of the fourfold. This map allows us to construct...

Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in toric varieties. We establish th...

There are by now several attempts to make the theory of the field of one element F1 rigorous. In [10] the authors formalize the transition from rings to schemes on a categorial level and apply this machinery to the category of sets to obtain the category of F1-schemes as in [1]. In [3] and [5] the authors extend the definition of rings in order to capture a structure that deserves to be called ...

In an earlier work, we showed that if the Hodge conjecture holds for all complex abelian varieties of CM-type, then the Tate conjecture holds for all abelian varieties over finite fields (Milne 1999b). In this article, we extract from the proof a statement (Theorem 1.1) that sometimes allows one to deduce the Tate conjecture for the powers of a single abelian variety A over a finite field from ...

Increasing collections of West American Salices make possible better interpretations of the variations of expression noted in different species. The present paper adds two varieties to Salix melanopsis Nuttall, that one of the western long-leaf or sandbar willows most closely related to the eastern S. interior Rowlee (S. longifolia Muhl.). Continuing study of the abundant Alaskan collections ma...

The article elucidates the retrospective of breeding flowering plants in Laboratory Ornamental Plants National Botanical Garden (Institute) for about six decades, and also recent results obtained field. mobilization introduction into culture thousands intraspecifi c taxa generated unique collections ornamental plants, which served as a basis obtaining new varieties. More than 300 cultivars have...