{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,27]],"date-time":"2026-01-27T14:42:35Z","timestamp":1769524955244,"version":"3.49.0"},"reference-count":12,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2021,6,5]],"date-time":"2021-06-05T00:00:00Z","timestamp":1622851200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,6,5]],"date-time":"2021-06-05T00:00:00Z","timestamp":1622851200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Tubitak","award":["118F159"],"award-info":[{"award-number":["118F159"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2022,9]]},"abstract":"Abstract<\/jats:title>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 $${\\mathbb {C}}$$<\/jats:tex-math>\n C<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\mathbb {R}$$<\/jats:tex-math>\n R<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in 1906\u20131907. The analogous problem for finite fields $$\\mathbb {F}_q$$<\/jats:tex-math>\n \n F<\/mml:mi>\n q<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with q<\/jats:italic> 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\u2019s classification (for q<\/jats:italic> odd) of nets of rank one<\/jats:italic>, 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 $$\\mathrm {PG}(5,q)$$<\/jats:tex-math>\n \n PG<\/mml:mi>\n (<\/mml:mo>\n 5<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, q<\/jats:italic> 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 $$\\mathrm {PGL}(3,q)$$<\/jats:tex-math>\n \n PGL<\/mml:mi>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> viewed as a subgroup of $$\\mathrm {PGL}(6,q)$$<\/jats:tex-math>\n \n PGL<\/mml:mi>\n (<\/mml:mo>\n 6<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Since q<\/jats:italic> is odd, the orbits of lines<\/jats:italic> in $$\\mathrm {PG}(5,q)$$<\/jats:tex-math>\n \n PG<\/mml:mi>\n (<\/mml:mo>\n 5<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> under this action correspond to the aforementioned pencils of conics in $$\\mathrm {PG}(2,q)$$<\/jats:tex-math>\n \n PG<\/mml:mi>\n (<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The main contribution of this paper is to determine the line-orbit distribution<\/jats:italic> of a plane $$\\pi $$<\/jats:tex-math>\n \u03c0<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> corresponding to a net of rank one, namely, the number of lines in $$\\pi $$<\/jats:tex-math>\n \u03c0<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of $$\\pi $$<\/jats:tex-math>\n \u03c0<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, 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 $$\\mathrm {PGL}(3,q)$$<\/jats:tex-math>\n \n PGL<\/mml:mi>\n (<\/mml:mo>\n 3<\/mml:mn>\n ,<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and hence the orbit sizes.<\/jats:p>","DOI":"10.1007\/s10623-021-00881-9","type":"journal-article","created":{"date-parts":[[2021,6,5]],"date-time":"2021-06-05T13:02:33Z","timestamp":1622898153000},"page":"2021-2067","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Combinatorial invariants for nets of conics in $$\\mathrm {PG}(2,q)$$"],"prefix":"10.1007","volume":"90","author":[{"given":"Michel","family":"Lavrauw","sequence":"first","affiliation":[]},{"given":"Tomasz","family":"Popiel","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8590-0301","authenticated-orcid":false,"given":"John","family":"Sheekey","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,6,5]]},"reference":[{"key":"881_CR1","doi-asserted-by":"crossref","unstructured":"Alnajjarine N., Lavrauw M.: Determining the rank of tensors in $$\\mathbb{F}_q^2 \\otimes \\mathbb{F}_q^3 \\otimes \\mathbb{F}_q^3$$. In: Slamanig D., Tsigaridas E., Zafeirakopoulos Z. (eds.) MACIS 2019: Mathematical Aspects of Computer and Information Sciences. Lecture Notes in Computer Science, vol. 11989. Springer, Cham (2020).","DOI":"10.1007\/978-3-030-43120-4_22"},{"key":"881_CR2","doi-asserted-by":"publisher","first-page":"401","DOI":"10.2307\/2370672","volume":"49","author":"AD Campbell","year":"1927","unstructured":"Campbell A.D.: Pencils of conics in the Galois fields of order $$2^n$$. Am. J. Math. 49, 401\u2013406 (1927).","journal-title":"Am. J. Math."},{"key":"881_CR3","doi-asserted-by":"publisher","DOI":"10.4171\/138","volume-title":"Analytic Projective Geometry","author":"E Casas-Alvero","year":"2014","unstructured":"Casas-Alvero E.: Analytic Projective Geometry. 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Journal de math\u00e9matiques pures et appliqu\u00e9es 2, 403\u2013438 (1906).","journal-title":"Journal de math\u00e9matiques pures et appliqu\u00e9es"},{"key":"881_CR7","unstructured":"Jordan C.: R\u00e9duction d\u2019un r\u00e9seau de formes quadratiques ou bilin\u00e9aires: deuxi\u00e8me partie. Journal de math\u00e9matiques pures et appliqu\u00e9es, 3, 5\u201351 (1907)."},{"key":"881_CR8","doi-asserted-by":"publisher","first-page":"111775","DOI":"10.1016\/j.disc.2019.111775","volume":"343","author":"M Lavrauw","year":"2020","unstructured":"Lavrauw M., Popiel T.: The symmetric representation of lines in $$\\text{ PG }(\\mathbb{F}_q^3 \\otimes \\mathbb{F}_q^3)$$. Discret. Math. 343, 111775 (2020).","journal-title":"Discret. 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Soc. 81, 351\u2013364 (1977).","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"881_CR12","doi-asserted-by":"publisher","first-page":"187","DOI":"10.2307\/2370239","volume":"36","author":"AH Wilson","year":"1914","unstructured":"Wilson A.H.: The canonical Types of Nets of Modular Conics. Am. J. Math. 36, 187\u2013210 (1914).","journal-title":"Am. J. 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