Herivelto Borges , Gábor Korchmáros , Pietro Speziali
{"title":"特性 p 中具有大线性自变群的平面曲线","authors":"Herivelto Borges , Gábor Korchmáros , Pietro Speziali","doi":"10.1016/j.ffa.2024.102402","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>G</em> be a subgroup of the three dimensional projective group <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> defined over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of order <em>q</em>, viewed as a subgroup of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>K</mi><mo>)</mo></math></span> where <em>K</em> is an algebraic closure of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. For <span><math><mi>G</mi><mo>≅</mo><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and for the seven nonsporadic, maximal subgroups <em>G</em> of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, we investigate the (projective, irreducible) plane curves defined over <em>K</em> that are left invariant by <em>G</em>. For each, we compute the minimum degree <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <em>G</em>-invariant curves, provide a classification of all <em>G</em>-invariant curves of degree <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, and determine the first gap <span><math><mi>ε</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> in the spectrum of the degrees of all <em>G</em>-invariant curves. We show that the curves of degree <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> belong to a pencil depending on <em>G</em>, unless they are uniquely determined by <em>G</em>. For most examples of plane curves left invariant by a large subgroup of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, the whole automorphism group of the curve is linear, i.e., a subgroup of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>K</mi><mo>)</mo></math></span>. Although this appears to be a general behavior, we show that the opposite case can also occur for some irreducible plane curves, that is, the curve has a large group of linear automorphisms, but its full automorphism group is nonlinear.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"96 ","pages":"Article 102402"},"PeriodicalIF":1.2000,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Plane curves with a large linear automorphism group in characteristic p\",\"authors\":\"Herivelto Borges , Gábor Korchmáros , Pietro Speziali\",\"doi\":\"10.1016/j.ffa.2024.102402\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <em>G</em> be a subgroup of the three dimensional projective group <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> defined over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of order <em>q</em>, viewed as a subgroup of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>K</mi><mo>)</mo></math></span> where <em>K</em> is an algebraic closure of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. For <span><math><mi>G</mi><mo>≅</mo><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and for the seven nonsporadic, maximal subgroups <em>G</em> of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, we investigate the (projective, irreducible) plane curves defined over <em>K</em> that are left invariant by <em>G</em>. For each, we compute the minimum degree <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <em>G</em>-invariant curves, provide a classification of all <em>G</em>-invariant curves of degree <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, and determine the first gap <span><math><mi>ε</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> in the spectrum of the degrees of all <em>G</em>-invariant curves. We show that the curves of degree <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> belong to a pencil depending on <em>G</em>, unless they are uniquely determined by <em>G</em>. For most examples of plane curves left invariant by a large subgroup of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, the whole automorphism group of the curve is linear, i.e., a subgroup of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>K</mi><mo>)</mo></math></span>. Although this appears to be a general behavior, we show that the opposite case can also occur for some irreducible plane curves, that is, the curve has a large group of linear automorphisms, but its full automorphism group is nonlinear.</p></div>\",\"PeriodicalId\":50446,\"journal\":{\"name\":\"Finite Fields and Their Applications\",\"volume\":\"96 \",\"pages\":\"Article 102402\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-03-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Finite Fields and Their Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1071579724000418\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579724000418","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设 G 是定义在 q 阶有限域 Fq 上的三维射影群 PGL(3,q) 的一个子群,视为 PGL(3,K) 的一个子群,其中 K 是 Fq 的代数闭包。对于 G≅PGL(3,q)和 PGL(3,q)的七个无间隔的最大子群 G,我们研究了定义在 K 上的由 G 左不变的(投影的、不可还原的)平面曲线。对于每条曲线,我们都计算了 G 不变曲线的最小度 d(G),提供了所有度为 d(G) 的 G 不变曲线的分类,并确定了所有 G 不变曲线度谱中的第一缺口 ε(G)。我们证明了阶数为 d(G) 的曲线属于取决于 G 的笔状曲线,除非它们是由 G 唯一决定的。对于大多数由 PGL(3,q) 的一个大子群保持不变的平面曲线,曲线的整个自变群是线性的,即 PGL(3,K) 的一个子群。虽然这似乎是一种普遍现象,但我们证明,对于某些不可还原平面曲线,也可能出现相反的情况,即曲线有一个大的线性自变群,但其整个自变群是非线性的。
Plane curves with a large linear automorphism group in characteristic p
Let G be a subgroup of the three dimensional projective group defined over a finite field of order q, viewed as a subgroup of where K is an algebraic closure of . For and for the seven nonsporadic, maximal subgroups G of , we investigate the (projective, irreducible) plane curves defined over K that are left invariant by G. For each, we compute the minimum degree of G-invariant curves, provide a classification of all G-invariant curves of degree , and determine the first gap in the spectrum of the degrees of all G-invariant curves. We show that the curves of degree belong to a pencil depending on G, unless they are uniquely determined by G. For most examples of plane curves left invariant by a large subgroup of , the whole automorphism group of the curve is linear, i.e., a subgroup of . Although this appears to be a general behavior, we show that the opposite case can also occur for some irreducible plane curves, that is, the curve has a large group of linear automorphisms, but its full automorphism group is nonlinear.
期刊介绍:
Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering.
For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods.
The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.