Blocking sets of nonsecant lines to a conic in PG(2,q), q odd

Aguglia, Angela; Korchmáros, Gábor

doi:10.1002/jcd.20042

Cited by 8 publications

(13 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since x l / ∈ B, the tangent plane through l contains at least q + 1 points of B. Using the fact that |l ∩ B| = 1, it follows that all the planes through l together contain at least (q + 1)q + 1 = q 2 + q + 1 points of B, a contradiction to (2).…”

Section: Proposition 32 ([8]mentioning

confidence: 91%

Blocking sets of tangent and external lines to a hyperbolic quadric in PG(3,q)

Bruyn

Sahoo

Sahu

2018

Discrete Mathematics

View full text Add to dashboard Cite

Let H be a hyperbolic quadric in P G(3, q), where q is a prime power. Let E (respectively, T) denote the set of all lines of P G(3, q) which are external (respectively, tangent) to H. We characterize the minimum size blocking sets in P G(3, q), q = 2, with respect to the line set E ∪ T. We also give an alternate proof characterizing the minimum size blocking sets in P G(3, q) with respect to the line set E for all odd q.

show abstract

Section: Proposition 32 ([8]mentioning

confidence: 91%

Blocking sets of tangent and external lines to a hyperbolic quadric in PG(3,q)

Bruyn

Sahoo

Sahu

2018

Discrete Mathematics

View full text Add to dashboard Cite

show abstract

“…Note that Lemma 2.1 yields that every line of L is a secant line of C. The combinatorial characterization of blocking sets of non-secant lines to C, as given in [2], will be a key tool in the proof of Theorem 1.2. Proposition 2.3.…”

Section: Proposition 22mentioning

confidence: 98%

“…Then C (l) belongs to I since q ≡ 3 (mod 4). We claim that for any C s ∈ I and for any l ∈ L \L P , l not tangent to C s , (2) C s is external to C (l) if and only if l is a secant of C s .…”

Section: Proof Of Theorem 12mentioning

confidence: 98%

Line partitions of internal points to a conic in PG(2,q)

Giulietti

2009

Combinatorica

View full text Add to dashboard Cite

All sets of lines providing a partition of the set of internal points to a conic C in P G(2, q), q odd, are determined. There exist only three such linesets up to projectivities, namely the set of all non-tangent lines to C through an external point to C, the set of all nontangent lines to C through a point in C, and, for square q, the set of all non-tangent lines to C belonging to a Baer subplane P G(2, √ q) with √ q + 1 common points with C. This classification theorem is the analogous of a classical result by Segre and Korchmáros [9] characterizing the pencil of lines through an internal point to C as the unique set of lines, up to projectivities, which provides a partition of the set of all non-internal points to C. However, the proof is not analogous, since it does not rely on the famous Lemma of Tangents of Segre which was the main ingredient in [9]. The main tools in the present paper are certain partitions in conics of the set of all internal points to C, together with some recent combinatorial characterizations of blocking sets of non-secant lines, see [2], and of blocking sets of external lines, see [1].

show abstract

“…(i) each Q i is collinear with two pairs of points of E, say P σ (i, 1) , P σ (i, 2) , and P σ (i, 3) , 3) , P σ (i,4) } , then Q 1 , Q 2 and Q 3 are collinear.…”

Section: Lemma 24mentioning

confidence: 99%

“…(3) L is the set of all secants to C, q even, see [3]; (4) L is the set of all external lines and tangents to C, q odd, see [2].…”

Section: Introductionmentioning

confidence: 99%