Abstract Let F be a finite field, an algebraically closed field, or the field of real numbers. Consider the vector space V = F 3 ⊗ F 3 of 3 × 3 matrices over F , and let G ≤ PGL ( V ) be the setwise stabiliser of the corresponding Segre variety S 3 , 3 ( F ) in the projective space PG ( V ) . The G -orbits of lines in PG ( V ) were determined by the first author and Sheekey as part of their classification of tensors in F 2 ⊗ V in [15] . 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 F -span of the Veronese variety V 3 ( F ) ⊂ S 3 , 3 ( F ) under the natural action of K = PGL ( 3 , 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 F . Although our main focus is on the case where F is a finite field, our methods (which are mostly geometric) are easily adapted to include the case where 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 F is a finite field, and connections are drawn with old work of Jordan and Dickson on the classification of pencils of conics in PG ( 2 , F ) , or equivalently, of pairs of ternary quadratic forms over F .

中文翻译：

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PG(F3⊗F3)中直线的对称表示

摘要 令 F 为有限域、代数闭域或实数域。考虑在 F 上的 3 × 3 矩阵的向量空间 V = F 3 ⊗ F 3 ，并让 G ≤ PGL ( V ) 是射影空间 PG ( V ) 中相应的 Segre 变体 S 3 , 3 ( F ) 的设置稳定器)。PG ( V ) 中线的 G 轨道由第一作者和 Sheekey 确定，作为他们在 [15] 中 F 2 ⊗ V 中张量分类的一部分。在这里，我们解决了对那些可以用对称矩阵表示的线轨道进行分类的相关问题，或者等价地，对 Veronese 变体 V 3 ( F ) ⊂ S 3 , 3 ( F ) 下的 F 跨度中的线轨道进行分类K=PGL(3,F)的自然作用。有趣的是，几个具有对称代表的 G 轨道在 K 的作用下分裂，在许多情况下，这种分裂取决于 F 的特性。尽管我们主要关注 F 是有限域的情况，但我们的方法（主要是几何的）很容易适应包括 F 是代数闭域或实数域的情况。K的相应轨道尺寸和稳定子群也是在F是有限域的情况下确定的，并与Jordan和Dickson在PG(2,F)中的圆锥曲线分类的旧工作中建立联系，或等效地, 对 F 上的三元二次形式。