Unitals Embedded in Desarguesian Planes

On polar ovals in cyclic projective planes

“…The next result shows that the set of absolute points 1 2 D q are in general position and is therefore an oval if q is odd. Since this oval is the absolute points of a polarity α, we call such an oval a polar oval with respect to α.…”

“…Suppose π(D q ) is not Desarguesian, and let β be a polarity satisfying the hypothesis. Consider the collineation αβ, where α is the Hall polarity given by (1). By Theorem 4.3, Aut(π(D q )) ⊂ N (G) and so αβ ∈ N (G), where N (G) is the normalizer of G in the automorphism group Aut(π(D q )) of π(D q ).…”

Tangent-Filling Plane Curves Over Finite Fields

“…We conclude that C is still not tangent-filling. [7] REMARK 2.3. Kaji [22] proved that the Gauss map of a smooth plane curve over F q must be purely inseparable.…”

On the equivalence of certain quasi‐Hermitian varieties

“…For $$r=2$$, a quasi‐Hermitian variety of $$\text{PG}(2,{q}^{2})$$ is called a unital or Hermitian arc . Nonclassical unitals have been extensively studied and characterized [6] and many constructions are known; see, for instance, [4]. As far as we know, the only known nonclassical quasi‐Hermitian varieties of $$\text{PG}(r,{q}^{2}),r\ge 3$$ were constructed in [2, 3, 10, 14] and they are not isomorphic among themselves; see [14].…”

“…In [3], quasi‐Hermitian varieties $${{\rm{ {\mathcal M} }}}_{\alpha ,\beta }$$ of $$\text{PG}(r,{q}^{2})$$ with $$r\ge 2$$, depending on a pair of parameters $$\alpha ,\beta $$ from the underlying field $$\text{GF}({q}^{2})$$, were constructed. For $$r=2$$ these varieties are Buekenhout–Metz (BM) unitals, see [5, 6, 11, 12]. As such, for $$r\ge 3$$ we shall call $${{\rm{ {\mathcal M} }}}_{\alpha ,\beta }$$ the BM quasi‐Hermitian varieties of parameters $$\alpha $$ and $$\beta $$ of $$\text{PG}(r,{q}^{2})$$.…”