Isomorphisms between Subiaco {$q$}-clan geometries

Classification of the Hyperovals in PG(2,64)

“…For q 8, Segre [21] showed in 1957 that all hyperovals consist of a conic plus its nucleus. In 1975, Hall [9] showed that there are exactly two classes of hyperovals in PG (2,16), which was later confirmed independently by O'Keefe and Penttila [11] without the use of a computer. In 1994, Penttila and Royle [18] showed that there are exactly six classes of hyperovals in PG(2, 32): five constructions listed in Table 1 and one sporadic construction [12].…”

“…The two main areas of research on hyperovals are construction and classification. Concerning construction, 11 infinite classes are known: the regular hyperoval [2], the translation hyperovals [21], the Segre hyperovals [22], two classes of Glynn hyperovals [8], the Payne hyperovals [14], the Cherowitzo hyperovals [3], three types of Subiaco hyperovals [5,15,16] and the Adelaide hyperovals [4]. Next to these, O'Keefe and Penttila [12] found one sporadic example in PG(2, 32) that has not yet been embedded in an infinite class.…”

Arcs in finite projective spaces

“…Table 1 lists all the known families of hyperovals. The references for these infinite classes of hyperovals are: (1) regular (Bose [9]), (2) translation (Segre [44]), (3) Segre [45], (4) Glynn I and Glynn II [19], (4) Payne [36], (5) Cherowitzo [14], (6) Subiaco (Cherowitzo, Penttila, Pinneri and Royle [16], Payne [37], Payne, Penttila and Pinneri [38]), (7) Adelaide (Cherowitzo, O'Keefe and Penttila [15]).…”

Groups of hyperovals in Desarguesian planes

“…The groups of the Adelaide hyperovals of Cherowitzo-O'Keefe- Penttila (2003) [5] were calculated by Payne-Thas (2005) in [19], the groups of the Subiaco hyperovals of Cherowitzo-Penttila-Pinneri-Royle (1996) [6] were calculated by combined results of O'Keefe- Thas (1996) in [16] and Payne-Penttila-Pinneri (1995) in [18], and the groups of the hyperovals of Payne (1985) [17] were calculated by Thas-Payne-Gevaert (1988) in [23], with all three using the beautiful method of associating a curve of fixed degree with the hyperoval and using Bezout's theorem. (For earlier hyperovals, see, for example [14]).…”