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The symmetric representation of lines in \(\operatorname{PG} ( \mathbb{F}^3 \otimes \mathbb{F}^3 )\). (English) Zbl 1433.05063

Summary: Let \(\mathbb{F}\) be a finite field, an algebraically closed field, or the field of real numbers. Consider the vector space \(V = \mathbb{F}^3 \otimes \mathbb{F}^3\) of \(3 \times 3\) matrices over \(\mathbb{F} \), and let \(G \leq \text{PGL} ( V )\) be the setwise stabiliser of the corresponding Segre variety \(S_{3 , 3} ( \mathbb{F} )\) in the projective space \(\operatorname{PG} ( V )\). The \(G\)-orbits of lines in \(\text{PG} ( V )\) were determined by M. Lavrauw and J. Sheekey [J. Geom. 108, No. 1, 5–23 (2017; Zbl 1361.05022)] as part of their classification of tensors in \(\mathbb{F}^2 \otimes V\). Here we solve the related problem of classifying those line orbits that may be represented by symmetric matrices, or equivalently, of classifying the line orbits in the \(\mathbb{F} \)-span of the Veronese variety \(\mathcal{V}_3 ( \mathbb{F} ) \subset S_{3 , 3} ( \mathbb{F} )\) under the natural action of \(K = \operatorname{PGL} ( 3 , \mathbb{F} )\). Interestingly, several of the \(G\)-orbits that have symmetric representatives split under the action of \(K\), and in many cases this splitting depends on the characteristic of \(\mathbb{F} \). Although our main focus is on the case where \(\mathbb{F}\) is a finite field, our methods (which are mostly geometric) are easily adapted to include the case where \(\mathbb{F}\) is an algebraically closed field, or the field of real numbers. The corresponding orbit sizes and stabiliser subgroups of \(K\) are also determined in the case where \(\mathbb{F}\) is a finite field, and connections are drawn with old work of Jordan and Dickson on the classification of pencils of conics in \(\operatorname{PG} ( 2 , \mathbb{F} )\), or equivalently, of pairs of ternary quadratic forms over \(\mathbb{F} \).

MSC:

05B25 Combinatorial aspects of finite geometries
51E20 Combinatorial structures in finite projective spaces
14N05 Projective techniques in algebraic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds

Citations:

Zbl 1361.05022
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References:

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