Small tight sets of hyperbolic quadrics

Beukemann, Linda; Metsch, Klaus

doi:10.1007/s10623-012-9676-4

Cited by 7 publications

(5 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that if

\mathcal {O}

is an

m

‐ovoid of

\mathcal {Q}_r

and

\Pi _j

is a

j

‐space of

{\rm PG}(2r+1,q)

, then it immediately follows from Lemma 2.1(c) that

\begin{equation*} |\Pi _j^\perp \cap \mathcal {O}| + q^{r-j}|\Pi _j \cap \mathcal {O}|= m (q^{r-j}+1), \end{equation*}

and this result is a counterpart of [5, Lemma 2.1].…”

Section: Preliminary Resultsmentioning

confidence: 99%

A modular equality for m$m$‐ovoids of elliptic quadrics

Gavrilyuk

Metsch

Pavese

2023

Bulletin of London Math Soc

View full text Add to dashboard Cite

An 𝑚-ovoid of a finite polar space  is a set  of points such that every maximal subspace of  contains exactly 𝑚 points of . In the case when  is an elliptic quadric 𝑞, we prove that an 𝑚ovoid exists only if 𝑚 satisfies a certain modular equality, which depends on 𝑞 and 𝑟. This condition rules out many of the possible values of 𝑚. Previously, only a lower bound on 𝑚 was known, which we slightly improve as a byproduct of our method. We also obtain a characterization of the 𝑚-ovoids of  − (7, 𝑞) for 𝑞 = 2 and (𝑚, 𝑞) = (4, 3).

show abstract

“…Note that if

\mathcal {O}

is an

m

‐ovoid of

\mathcal {Q}_r

and

\Pi _j

is a

j

‐space of

{\rm PG}(2r+1,q)

, then it immediately follows from Lemma 2.1(c) that

\begin{equation*} |\Pi _j^\perp \cap \mathcal {O}| + q^{r-j}|\Pi _j \cap \mathcal {O}|= m (q^{r-j}+1), \end{equation*}

and this result is a counterpart of [5, Lemma 2.1].…”

Section: Preliminary Resultsmentioning

confidence: 99%

A modular equality for m$m$‐ovoids of elliptic quadrics

Gavrilyuk

Metsch

Pavese

2023

Bulletin of London Math Soc

View full text Add to dashboard Cite

show abstract

“…Further, Drudge [13, Corollary 9.1] proved that, for q ≥ 4, any x-tight set of Q + (2n + 1, q) with x ≤ √ q is the disjoint union of x generators (hence, none such exist if x > 2 and the rank n + 1 is odd). This bound was improved several times [2,11] and finally Beukemann and Metsch [4] proved that the same conclusion holds if 1 ≤ n ≤ 3 and x ≤ q, or if n ≥ 4, q ≥ 71 and x ≤ q − 1.…”

Section: Introductionmentioning

confidence: 90%

On tight sets of hyperbolic quadrics

Gavrilyuk¹

2019

Preprint

View full text Add to dashboard Cite

We prove that the parameter x of a tight set T of a hyperbolic quadric Q + (2n + 1, q) of an odd rank n + 1 satisfies x 2 + w(w − x) ≡ 0 mod q + 1, where w is the number of points of T in any generator of Q + (2n + 1, q). As this modular equation should have an integer solution in w if such a T exists, this condition rules out roughly at least one half of all possible parameters x. It generalizes a previous result by the author and K. Metsch shown for tight sets of a hyperbolic quadric Q + (5, q) (also known as Cameron-Liebler line classes in PG(3, q)).

show abstract

“…We shall denote by C the cone with vertex P and base the (2r)dimensional Baer subgeometry B P . With the hyperplanes of C, we denote the rich hyperplanes through P described above in (1), (2) and (3). The lines of C are the lines through P and a point of B P .…”

Section: Lemma 23 [1]mentioning

confidence: 99%

Tight sets in finite classical polar spaces

Nakić

Storme

2017

Advances in Geometry

View full text Add to dashboard Cite

We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries $\text{PG}(2r+1,\,\sqrt{q})$ and generators of H(2r + 1, q), if q ≥ 81 is an odd square and i < (q2/3 − 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Δ, Δ⊥}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Δ, Δ⊥} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity ⊥ of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under ⊥. This improves previous results where $i \lt q^{5/8} / \sqrt{2} +1$ was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q2/3 − 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q).

show abstract

Small tight sets of hyperbolic quadrics

Cited by 7 publications

References 10 publications

A modular equality for m$m$‐ovoids of elliptic quadrics

A modular equality for m$m$‐ovoids of elliptic quadrics

On tight sets of hyperbolic quadrics

Tight sets in finite classical polar spaces

Contact Info

Product

Resources

About