Determining the Rank of Tensors in $$\mathbb {F}_q^2\otimes \mathbb {F}_q^3\otimes \mathbb {F}_q^3$$

Alnajjarine, Nour; Lavrauw, Michel

doi:10.1007/978-3-030-43120-4_22

Cited by 2 publications

(5 citation statements)

References 3 publications

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“…Some of the results are classical and can be found in e.g. [11,12], others are more recent results from [2,3,4,6,16,17,18].…”

Section: Preliminariesmentioning

confidence: 96%

“…A conic is defined as the zero locus Z(f ) of a quadratic form f = a 00 X 2 + a 01 XY + a 02 XZ + a 11 Y 2 + a 12 Y Z + a 22 Z 2 on PG(2, q). The conic Z(f ) and the quadratic form f have discriminant ∆ f = 4a 00 a 11 a 22 + a 01 a 02 a 12 − a 00 a 2 12 − a 11 a 2 02 − a 22 a 2 01 .…”

Section: Linear Systems Of Conicsmentioning

confidence: 99%

“…2 ). The singular conics in W 6 are the zero locus of aX 0 X 2 + bX 2 1 + cX 1 X 2 + dX 2 2 with a 2 b = 0. If a = b = 0, W 6 contains the double line X 2 2 = 0 and q pairs of real lines defined by X 2 (X 1 + dX 2 ) = 0.…”

Section: Webs Of Conicsmentioning

confidence: 99%

“…Suppose q is even. If a = 0, then c(b + d) = 0 and W 14,1 contains the unique double line Z((X 0 + X 1 + X 2 ) 2 ), along with 3q 2 pairs of real lines and q 2 pairs of imaginary lines defined by X 2 (bX 0 + dX 1 ) and c(X 0 + X 1 ) 2 + bX 2 (X 0 + X 1 ) + cX 2 2 . Otherwise, when a = 0, we distinguish between the cases where d = 1 + b and d = 1 + b.…”

Section: Webs Of Conicsmentioning

confidence: 99%

“…These numbers serve as combinatorial invariants for webs and squabs of conics, and they are determined by studying the different types of hyperplanes (K-orbits) incident with their associated points and lines of PG (5, q). Note that the K-orbits on points of PG(5, q) are well-known and quite simple, while K-orbits of lines of PG (5, q) were classified in [16], based on the canonical forms of tensors of format (2,3,3) from [19]. Further, we prove in Theorem 4.31 that the intersection of a line of PG (5, q) with the secant variety of the Veronese surface completely determines the singularity of its associated web of conics in PG (2, q).…”

Section: Introductionmentioning

confidence: 99%

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Solids in the space of the Veronese surface in even characteristic

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“…Some of the results are classical and can be found in e.g. [11,12], others are more recent results from [2,3,4,6,16,17,18].…”

Section: Preliminariesmentioning

confidence: 96%

Section: Linear Systems Of Conicsmentioning

confidence: 99%

Section: Webs Of Conicsmentioning

confidence: 99%

Section: Webs Of Conicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solids in the space of the Veronese surface in even characteristic

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Combinatorial invariants for nets of conics in $$\mathrm {PG}(2,q)$$

Lavrauw

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2021

Des. Codes Cryptogr.

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The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $${\mathbb {C}}$$ C and $$\mathbb {R}$$ R in 1906–1907. The analogous problem for finite fields $$\mathbb {F}_q$$ F q with q odd was solved by Dickson in 1908. In 1914, Wilson attempted to classify nets (two-dimensional systems) of conics over finite fields of odd characteristic, but his classification was incomplete and contained some inaccuracies. In a recent article, we completed Wilson’s classification (for q odd) of nets of rank one, namely those containing a repeated line. The aim of the present paper is to introduce and calculate certain combinatorial invariants of these nets, which we expect will be of use in various applications. Our approach is geometric in the sense that we view a net of rank one as a plane in $$\mathrm {PG}(5,q)$$ PG ( 5 , q ) , q odd, that meets the quadric Veronesean in at least one point; two such nets are then equivalent if and only if the corresponding planes belong to the same orbit under the induced action of $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) viewed as a subgroup of $$\mathrm {PGL}(6,q)$$ PGL ( 6 , q ) . Since q is odd, the orbits of lines in $$\mathrm {PG}(5,q)$$ PG ( 5 , q ) under this action correspond to the aforementioned pencils of conics in $$\mathrm {PG}(2,q)$$ PG ( 2 , q ) . The main contribution of this paper is to determine the line-orbit distribution of a plane $$\pi $$ π corresponding to a net of rank one, namely, the number of lines in $$\pi $$ π belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of $$\pi $$ π , and we will use this fact in forthcoming work to develop an efficient algorithm for calculating the orbit of a given net of rank one. As a more immediate application, we also determine the stabilisers of nets of rank one in $$\mathrm {PGL}(3,q)$$ PGL ( 3 , q ) , and hence the orbit sizes.

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Determining the Rank of Tensors in $$\mathbb {F}_q^2\otimes \mathbb {F}_q^3\otimes \mathbb {F}_q^3$$

Cited by 2 publications

References 3 publications

Solids in the space of the Veronese surface in even characteristic

Solids in the space of the Veronese surface in even characteristic

Combinatorial invariants for nets of conics in $$\mathrm {PG}(2,q)$$

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