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Ring Geometry: Partial Ovoids

Partial Ovoids:

A section of an ovoid is a partial ovoid. P is a finite classical polar space. A proper partial ovoid is not an ovoid, instead it is a maximal. A maximal is a partial ovoid that is not contained within a larger partial ovoid. In a case of rank 2 a maximal becomes a partial spread in the dual space. The following lemmas are used to show that a partial ovoid defined by W(3,q) also defines and arc K defined by PG(2,q). [1] These concepts can be applied later to partial spreads.

Lemma 1:

Given O is a partial ovoid defined by W(3,q). The order of O is greater than q2-q+1 for any point P that does not belong to O. The set K is defined by {P} unified with {Pᗮ intersecting with O}. K is an arc in the plane Pᗮ.


 * Assume l is a line and part of the ambient projective space defined by PG(3,q). The line meets O in C ≥ 2 points.
 * l is not a generator of W(3,q).
 * If there exists a line meeting O in at least three points the order of O is less than or equal to q2-q+1.
 * Thus every line in PG(3,q) meets O in no more than 2 points which means that the set K is an arc in the projective plane Pᗮ.

[1] [7]

Explanation:

Think of a line in space that meets the partial ovoid O in 2 or more points. The line does not belong to the generators of W(3,q). If the line exists than the order of O is less than or equal to q2-q+1. Since every line defined by PG(3,q) cannot meet the partial ovoid in more than 2 points the arc K exists in the projective plane that is perpendicular to O.

Lemma 2:

Given O is a partial ovoid of W(3,q) and q is odd. Show that the order of O is less than or equal to q2-q+1.


 * Assume the order of O is greater than q2-q+1.
 * Consider a point P that does not belong to O.
 * By lemma 1, K is an arc in the projective plane Pᗮ.
 * Q is odd means that the order of K is less than or equal to q+1.
 * Since the order of K is less than or equal to q+1 and K is an arc in the projective plane Pᗮ, the order of the intersection of the projective plane and O is less than or equal to q.

Explanation:

From lemma 1 we know there is a line in the projective space that meets the partial ovoid O. We also know that the arc K exists perpendicular to O. Lemma 2 states that the order of O is less than q2-q+1 because the order of K is less than or equal to q+1. The order of the intersection between the arc and the ovoid is less than or equal to q.


 * Assume g is a generator of W(3,q) that meets O in a unique point.
 * The order of the intersection of the unique point and O is equal to 1.
 * Any plane not equal to the plane created by the unique point on g meets O in at most q-1 points that are not the unique point.
 * Thus the order of O is less than or equal to 1+q(q-1) which when distributed out is equal to q2-q+1.

When proving the upper bound of the size of the maximal created by W(3,q) for q is even, use extendability of arcs in PG(2,q).

[1]

Explanation:

There is a second generator in W(3,q) that meets the ovoid at a unique point. Since the intersection is unique the order of the intersection is one. Any plane that is not defined by the unique point does not meet the ovoid in the same number of points. This means the order of O is equal to q2-q+1.

Lemma 3:

Given O is a maximal defined by W(3,q), q is even, and has a size of q2+1-ẟ. Show that when ẟ is less than q, O can be extended.


 * Assume the order of O is equal to q2+1-ẟ and ẟ is a value between but not equal to o and q.
 * Since O is a maximal there is a generator g defined by W(3,q) that does not meet O.
 * Due to the previous statement all planes through g meet O in at most q points.
 * If all planes through g meet O in at most q-2 points the order of O is less than or equal to (q-2)(q-1) which when distributed equals q2-q-2.
 * Thus there is a plane on g with q-1 or q points of O.

P is defined as a perpendicular plane. P is within g. Which means the arc K in the plane is equal to q or q+1. Both cases lead to K being extended to a hyperoval that contains a point T belonging to g in the set {P}.


 * Assume a second generator l that does not equal g defined by W(3,q) that is on T.
 * Assume l meets O at a point P. The plane P does not contain q which intersects the plane in a line through T.
 * T contains a point R on the arc K.
 * R does not belong to O.
 * Thus the order of the plane intersecting with O is equal to q. Which means no generator of W(3,q) on T meets O.
 * O can be extended with T when the order of the plane intersecting with O is equal to q-1. Which means at most one generator defined by W(3,q) on T can meet O.
 * The sum of the order of the intersection of the plane and O is equal to q-1+1+ẟ is greater than q2-q+1

Since ẟ is greater than q a contradiction ensues. This means at least one of the planes contains exactly q points in O.

[1]

Explanation:

Lemma 3 shows that when the ovoid is defined by W(3,q) and q is en even defines an arc K that can be extended to be a hyperoval. This extension is possible because the arc has points either equal to q or q+1. When a second generator is used that does not equal g it creates a new line. This new line has an order of q-1. This leads to the intersections of q+ẟ being greater than the intersections q2-q+1 which contradicts the first half of lemma 3. This contradiction means that one or more planes must contain exactly q points.

Corollary:

Lemma 1, 2, and 3 are all based on results found in [3]. Examples of maximal spreads can also be found in [3]. The results from lemma 2 were originally from [22]. Partial ovoids can then be expanded to partial spreads. Maximal partial ovoid size is based on the polar space that defines it and the lower and upper bounds related to the polar space. Explained below is how lower and upper bounds on the size of maximal partial ovoids are defined based on their polar space. These polar spaces are separated into two categories sharp and not sharp. A polar space defined as sharp is a polar space that has at least one case that holds with equality.

Related Information:

Polar Spaces

Lower bounds of low rank maximal in polar spaces:


 * Polar spaces with lower bounds defined as sharp:
 * For a polar space defined by:
 * W(2n+1,q) the lower bound is defined as q+1. [14] [6]
 * H(3,q2) and q is odd or even the lower bound is defined as q2+1 + (4/9)q; q2+1. [15] [16]


 * Polar spaces with lower bounds defined as not sharp:
 * For a polar space defined by:
 * Q(4,q) and q is odd the lower bound is defined as 1.419q. [6]
 * Q(6,q), q belongs to {3,5,7}, and q ≥ 9 and odd the lower bound is defined as 2q; 2q-1. [6]
 * Q(2n,q), n ≥ 4, q is odd; Q(8,3) the lower bound is defined as 2q+1; 2q. [6]
 * Qᐨ(5,q), q = 2; 3; q ≥ 4 the lower bound is defined as 6; 16; 2q+2. [6]
 * Qᐨ(2n+1,q) the lower bound is defined as 2q+1. [6]
 * Qᐩ(2n+1,q), n = 2; n ≥ 3 the lower bound is defined as 2q; 2q+1. [6]
 * H(2n+1, q2), n ≥ 2 the lower bound is defined as q2+q+1 [5]
 * H(2n, q2), n = 2; n ≥ 3 the lower bound is defined as q2+q+1 [5] [17]

Upper bounds of low rank maximal in a polar space:


 * Maximal with upper bounds defined as sharp:
 * For a polar space defined by:
 * W(3,q) the upper bound is defined as q2+q+1. Note this polar space is sharp when q is even. [3]
 * Qᐨ(5,q) the upper bound is defined as (1/2)(q3+q+2). Note this polar space is sharp when q = 2 or 3. [5] [8] [9]
 * H(3,q2) the upper bound is defined as q3-q+1. [20]


 * Maximal with upper bounds defined as not sharp:
 * For a polar space defined by:
 * W(5,q) the upper bound is defined as 1+(q/2)$$\sqrt(5q^4+6q^3+7q^2+6q+1)$$-q2-q-1. [6]
 * Q(4,q) and q is odd the upper bound is defined as q2. [6]
 * Q(6,q), q > 13, and q is prime the upper bound is defined as q3-2q+1. [6]
 * Q(8,q) and q is odd the upper bound is defined as $$q^4-q \sqrt q$$. [6]
 * H(5,q2) the upper bound is defined as q5+1-(q2+(1/4)q-1)/$$\sqrt 2$$. [5]
 * H(4,q2) the upper bound is defined as q5-q4+q3+1. [5]

Inductive Bounds:

The inductive bounds below are based on the polar space to determine the size of the partial ovoid.


 * For a polar space defined by:
 * W(2n+1,q) the recursion is defined as Xn,q ≤ 2+(q-1)Xn-1, q. [6]
 * Qᐨ(2n+1,q) the recursion is defined as Xn,q ≤ 2+$$(q^n+1)/(q^(n-1) + 1)$$(Xn-1, q -2). [19]
 * Q(2n,q) the recursion is defined as Xn,q ≤ 1+q(Xn-1, q -1). [6]
 * Q+(2n+1,q) the recursion is defined as Xn,q ≤ 2+$$(q^n+1)/(q^(n-1) - 1)$$(Xn-1, q -2).  [6]
 * H(2n,q2) the recursion is defined as Xn,q2 ≤ q2(Xn-1, q2-q2+1). [5]
 * H(2n+1,q2) the recursion is defined as Xn,q2 ≤ q2(Xn-1, q2-q2+1). [5]

Ovoids:

An ovoid is part of projective geometry over the Galois field. [21] Ovoids are sets of three points in which no three points are collinear. Ovoids in PG(3,q) are known as elliptic quadrics. [21] An ovoid is a egg shaped graph defined in a polar space.

References:

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Ovoid - Encyclopedia of Mathematics

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