{"title":"On the p-rank of singular curves and their smooth models","authors":"Sadık Terzi","doi":"10.1016/j.ffa.2025.102578","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we are concerned with the computation of the <em>p</em>-rank and <em>a</em>-number of singular curves and their smooth models. We consider a pair <span><math><mi>X</mi><mo>,</mo><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of proper curves over an algebraically closed field <em>k</em> of characteristic <em>p</em>, where <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is a singular curve which lies on a smooth projective variety, particularly on smooth projective surfaces <em>S</em> (with <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>S</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>=</mo><mi>q</mi><mo>(</mo><mi>S</mi><mo>)</mo></math></span>) and <em>X</em> is the smooth model of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. We determine the <em>p</em>-rank of <em>X</em> by using the exact sequence of group schemes relating the Jacobians <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>J</mi></mrow><mrow><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span>. As an application, we determine a relation about the fundamental invariants <em>p</em>-rank and <em>a</em>-number of a family of singular curves and their smooth models. Moreover, we calculate <em>a</em>-number and find lower bound for <em>p</em>-rank of a family of smooth curves.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102578"},"PeriodicalIF":1.2000,"publicationDate":"2025-01-20","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/S1071579725000085","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
Abstract
In this paper, we are concerned with the computation of the p-rank and a-number of singular curves and their smooth models. We consider a pair of proper curves over an algebraically closed field k of characteristic p, where is a singular curve which lies on a smooth projective variety, particularly on smooth projective surfaces S (with ) and X is the smooth model of . We determine the p-rank of X by using the exact sequence of group schemes relating the Jacobians and . As an application, we determine a relation about the fundamental invariants p-rank and a-number of a family of singular curves and their smooth models. Moreover, we calculate a-number and find lower bound for p-rank of a family of smooth curves.
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.