Abstract
The notion of sub-objects and their disjoint union is introduced for a dimensional dual arc. This naturally motivates a problem to decompose a dimensional dual hyperoval (DHO for short) into the disjoint union of some subdual arcs, including a subDHO, because such an expression is useful to calculate its universal cover, as suggested by an elementary observation about its ambient space. Under mild restrictions, a criterion is obtained for a DHO of rank over two element field to be extended to a DHO of rank so that is a disjoint union of and some subDHO of rank . Under the choice of a complement to , such as well as are uniquely determined by , if they exist. Several known families of DHOs are examined whether they can be extended to DHOs in the above form, but no example is found unless they are bilinear. If a subDHO is bilinear over a specified complement, the criterion is satisfied, and thus there exists a unique pair of DHOs satisfying the above conditions. This makes clear the meaning of the construction called “extension" by Dempwolff and Edel for bilinear DHOs.
Citation
Satoshi Yoshiara. "Disjoint unions of dimensional dual hyperovals." Innov. Incidence Geom. 14 43 - 76, 2015. https://doi.org/10.2140/iig.2015.14.43
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