The symmetric representation of lines in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e6806" altimg="si22.svg"><mml:mrow><mml:mi mathvariant="normal">PG</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>⊗</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:

Lavrauw, Michel; Popiel, Tomasz

doi:10.1016/j.disc.2019.111775

Cited by 13 publications

(49 citation statements)

References 11 publications

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“…If we assume that the curve

MJX-tex-caligraphicscriptF

has two irreducible factors of degrees 2, then there are 2 polynomials

f_{1} \neq 0

and

f_{2} \neq 0

such that

MJX-tex-caligraphicscriptF = f_{1} f_{2}

with

deg (f_{1}) = 2

and

deg (f_{2}) = 2

. It follows from the classification of pencils of conics in

PG (2, q)

q

odd, that the pencil

MJX-tex-caligraphicscriptP (MJX-tex-caligraphicscriptZ (f_{1}), MJX-tex-caligraphicscriptZ (f_{2}))

is equivalent to the pencil of type

o_{13}

, that is,

MJX-tex-caligraphicscriptP (2 X Y, Y^{2} - Z^{2})

, corresponding to the second column of Table 5 of [10], since this is the only pencil of conics with 3 base points (which are

(1, 0, 0)

(0, 1, 1)

and

(0, - 1, 1)

). After a suitable coordinate transformation, we may therefore assume that

MJX-tex-caligraphicscriptZ (f_{1})

corresponds to

C_{1} = MJX-tex-caligraphicscriptZ (2 X Y + Y^{2} - Z^{2})

and

MJX-tex-caligraphicscriptZ (f_{2})

…”

Section: Completeness Proofmentioning

confidence: 99%

On planar arcs of size (q+3)∕2

Günay

Lavrauw

2021

J of Combinatorial Designs

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The subject of this paper is the study of small complete arcs in PG ( 2 , q ), for q odd, with at least ( q + 1 ) ∕ 2 points on a conic. We give a short comprehensive proof of the completeness problem left open by Segre in his seminal work. This gives an alternative to Pellegrino's long proof which was obtained in a series of papers in the 1980s. As a corollary of our analysis, we obtain a counterexample to a misconception in the literature.

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“…If we assume that the curve

MJX-tex-caligraphicscriptF

has two irreducible factors of degrees 2, then there are 2 polynomials

f_{1} \neq 0

and

f_{2} \neq 0

such that

MJX-tex-caligraphicscriptF = f_{1} f_{2}

with

deg (f_{1}) = 2

and

deg (f_{2}) = 2

. It follows from the classification of pencils of conics in

PG (2, q)

q

odd, that the pencil

MJX-tex-caligraphicscriptP (MJX-tex-caligraphicscriptZ (f_{1}), MJX-tex-caligraphicscriptZ (f_{2}))

is equivalent to the pencil of type

o_{13}

, that is,

MJX-tex-caligraphicscriptP (2 X Y, Y^{2} - Z^{2})

, corresponding to the second column of Table 5 of [10], since this is the only pencil of conics with 3 base points (which are

(1, 0, 0)

(0, 1, 1)

and

(0, - 1, 1)

). After a suitable coordinate transformation, we may therefore assume that

MJX-tex-caligraphicscriptZ (f_{1})

corresponds to

C_{1} = MJX-tex-caligraphicscriptZ (2 X Y + Y^{2} - Z^{2})

and

MJX-tex-caligraphicscriptZ (f_{2})

…”

Section: Completeness Proofmentioning

confidence: 99%

On planar arcs of size (q+3)∕2

Günay

Lavrauw

2021

J of Combinatorial Designs

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“…In this section, we review some definitions and theory needed in our proofs, most of which can be found in [7,13]. We also refer the reader to [8] for an overview of the interesting properties of the Veronese surface over finite fields.…”

Section: Preliminariesmentioning

confidence: 99%

“…For example, if q is odd and U is a plane containing at least one point of V(F q ), then the line-orbit distribution of U completely determines its K-orbit [15]. The line orbits themselves were determined (for all q) in [13].…”

Section: Generating Conicsmentioning

confidence: 99%

“…In particular, k-dimensional subspaces of S 2 F 3 correspond via a suitable contraction operation to tensors in (S 2 F 3 ) ⊗ F k . For further discussion and references, we refer the reader to the recent e-seminar [12] of the second author, and to the closely related articles [1,13,14,15,16,17].…”

Section: Introductionmentioning

confidence: 99%

“…In cases (ii) and (iii), one can decide whether S and S ′ belong to the same orbit by determining whether they intersect a certain orbit of lines in PG(5, q), as explained in Remark 4.11 and Section 6. The line orbits themselves were calculated by the second and third authors in [13]. We expect that the point-and hyperplaneorbit distributions given in Table 2 will also have further applications, given that we have previously found data of this kind extremely useful in related work, both theoretical [15] and computational [1].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solids in the space of the Veronese surface in even characteristic

Alnajjarine¹,

Lavrauw²,

Popiel³

2021

Preprint

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We classify the orbits of solids in the projective space PG(5, q), q even, under the setwise stabiliser K ∼ = PGL(3, q) of the Veronese surface. For each orbit, we provide an explicit representative S and determine two combinatorial invariants: the point-orbit distribution and the hyperplane-orbit distribution. These invariants characterise the orbits except in two specific cases (in which the orbits are distinguished by their line-orbit distributions). In addition, we determine the stabiliser of S in K, thereby obtaining the size of each orbit. As a consequence, we obtain a proof of the classification of pencils of conics in PG(2, q), q even, which to the best of our knowledge has been heretofore missing in the literature.

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