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Abstract  The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points $${\mathcal{H}}$$ such that every line ℓ meets $${\mathcal{H}}$$ in half of the points of ℓ. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q 2) were those of the elliptic quadric $${\mathsf{Q}^-(5,q)}$$ , q odd. We show in this paper that there exists a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 52), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3· A 7-hemisystem of $${\mathsf{Q}^-(5,5)}$$ , first constructed by Cossidente and Penttila. [ABSTRACT FROM AUTHOR]

Abstract  The concept of a tight set of points of a generalised quadrangle was introduced by S. E. Payne in 1987, and that of an m-ovoid of a generalised quadrangle was introduced by J. A. Thas in 1989, and we unify these two concepts by defining intriguing sets of points. We prove that every intriguing set of points in a generalised quadrangle is an m-ovoid or a tight set, and we state an intersection result concerning these objects. In the classical generalised quadrangles, we construct new m-ovoids and tight sets. In particular, we construct m-ovoids of W(3,q), q odd, for all even m; we construct (q+1)/2-ovoids of W(3,q) for q odd; and we give a lower bound on m for m-ovoids of H(4,q 2). [ABSTRACT FROM AUTHOR]

Abstract Cultivated Solanum tuberosum L. subsp. andigena is well known as a rich source of valuable traits for potato breeding, especially for resistance to diseases and pests. The golden potato cyst nematode, Globodera rostochiensis Woll., is considered to be one of todays most serious hindrances to potato production in Europe and North America. Thus, the breeding of new cultivars that have resistance to PCN is of great importance. The USPG (USA) and VIR (Russian) potato genebanks, as well as others, maintain many samples of primitive cultivated and wild potato species originating from Latin America. Many of these samples are assumed to be genetically duplicated because the material in both genebanks came from the same original source. A joint investigation of new genotypes of subsp. andigena forms resistant to potato cyst nematode (PCN) was carried out on samples of subsp. andigena at VIR with reputed duplicate samples at USPG. After careful screening, 14 samples which possessed resistance to PCN were identified. A high level of this resistance was transmitted to sexual progeny at a high frequency for all of the selections. Eleven of the accessions found to be resistant have reputed duplicates in USPG that were not previously known to be resistant. Thus, this research not only broadens the choice of parents available for resistance breeding, but identifies model materials for future research to test the parity of PCN resistance among reputed duplicate samples in the two genebanks. [ABSTRACT FROM AUTHOR]

We construct an infinite family of symplectic spreads in spaces of odd rank and characteristic. [ABSTRACT FROM AUTHOR]