Hyperconics (a.k.a. Regular Hyperovals)

Definition

A hyperconic is a conic together with its nucleus. In the literature this hyperoval is also known as a regular hyperoval or as a complete conic. Since conics exist over any field, hyperconics exist in all Desarguesian planes of even order.

o-polynomials

The o-polynomials that correspond to hyperconics are:
  1. f(t) = t2
  2. f(t) = tq/2 = t1/2
  3. f(t) = tq-2 = t-1
(Note that the form given in 4 is slightly different from that in [OKPe91a]. This corrects a minor error in that paper, since the form given there does not give a hyperoval when s = 1.)

Rational Function form of o-polynomials

The polynomial form of the o-polynomial is frequently too cumbersome and its length is field dependent. When a rational function can be used to describe the o-polynomial, the form is easier to work with and is field independent. We have, for the non-monomial hyperconics, the rational forms:

Automorphism Group

For q > 4, the full automorphism group of a hyperconic has order (q+1)q(q-1)h and is isomorphic to . (For q = 2 and 4 see below.) [Hi79]. The group fixes the nucleus of the conic and acts triply transitively on the points of the conic. The secant, tangent and exterior lines of the conic each lie in a single orbit under the action of the group.

Ovals contained in this Hyperoval

From the definition of a hyperconic it is clear that the removal of some point must leave a conic. However, if q > 4, removal of a different point will leave an oval having more than five points in common with this conic. Such an oval can not be a conic (since there is a unique conic through five points, no three collinear). These ovals are called pointed conics. A hyperconic therefore contains one conic and q pointed conics when q > 4.

Special Properties in Small Planes

PG(2,2): A hyperoval in the Fano plane consists of only four points. By the fundamental theorem of projective planes, there is a unique map sending these four points to the standard four points. Thus, all hyperovals are projectively equivalent to the standard one. The o-polynomial of this standard hyperconic is f(t) = t2. (Technically this is not correct, it should be f(t) = t, but in this field t = t2 and for aesthetic reasons we prefer the quadratic version.) The automorphism group here is isomorphic to S4, and so is transitive on the points of the hyperconic. Consequently, the four ovals contained in this hyperconic are projectively equivalent, and are conics. There are 7 hyperconics and 28 conics in this plane.

PG(2,4): A hyperoval in this plane consists of six points. The pointed conics contained in such a hyperoval consist of five points, and so, are conics. Since all conics are projectively equivalent, we see that the automorphism group of the hyperoval must be transitive on the points of the hyperoval. (Besides the hyperconics in PG(2,2) and PG(2,4) the only other hyperoval with a transitive automorphism group is the Lunelli-Sce hyperoval in PG(2,16) [Ko78]). In fact, the group is isomorphic to S6. There are 168 hyperconics in this plane. There is only one o-polynomial, f(t) = t2, over this field.

Given a hyperconic, there are 56 hyperconics which meet the given one in an even number of points. These 56 hyperconics have several interesting properties.

  1. They form an orbit of PSL(3,4).
  2. Any two distinct hyperconics in this set intersect in 0 or 2 points.
  3. If two of these hyperconics have 2 points in common, then there are exactly two other hyperconics that are disjoint from both of the original pair.
  4. With the hyperconics as points, and blocks the sets of 11 hyperconics equal to or disjoint from a given hyperconic, these hyperconics form the biplane on 56 points (i.e. a symmetric (56, 11, 2) 2-design).