We introduced a new concept, the mutual conic, in order to give a complete classification of the group $ G $ generated by the images of two or three total sextactic points in the Jacobian $ J_{\mathcal{C}} $ of a smooth projective plane quartic curve $ \mathcal{C} $ (3-genus curves that are non-hyperelliptic), and we determined the geometric configuration of these points associated with each case. We supported the validity of the results with a variety of examples.
Citation: Alwaleed Kamel, Eman Alluqmani, Mohammed A. Saleem, Waleed Khaled Elshareef. On classification of groups generated by total sextactic points of smooth quartic curves[J]. AIMS Mathematics, 2025, 10(12): 28221-28242. doi: 10.3934/math.20251241
We introduced a new concept, the mutual conic, in order to give a complete classification of the group $ G $ generated by the images of two or three total sextactic points in the Jacobian $ J_{\mathcal{C}} $ of a smooth projective plane quartic curve $ \mathcal{C} $ (3-genus curves that are non-hyperelliptic), and we determined the geometric configuration of these points associated with each case. We supported the validity of the results with a variety of examples.
| [1] |
A. Kamel, W. K. Elshareef, Weierstrass points of order three on smooth quartic curves, J. Algebra Appl., 18 (2019), 1950020. https://doi.org/10.1142/S0219498819500208 doi: 10.1142/S0219498819500208
|
| [2] | A. Kamel, F. Sakai, Geometry and computation of 2-Weierstrass points on Kuribayashi quartic curves, Saitama Math. J., 26 (2009), 67–82. |
| [3] | A. Vermeulen, Weierstrass points of weight two on curves of genus three, Thesis, University of Amsterdam, 1983. |
| [4] |
A. Kamel, Group generated by total sextactic points of Kuribayashi quartic curve, J. Algebra Appl., 20 (2021), 2150184. https://doi.org/10.1142/S021949882150184X doi: 10.1142/S021949882150184X
|
| [5] |
A. Kamel, W. K. Elshareef, On the Jacobian of Kuribayashi curves, Commun. Algebra, 48 (2019), 291–298. https://doi.org/10.1080/00927872.2019.1640241 doi: 10.1080/00927872.2019.1640241
|
| [6] |
A. Kamel, M. A. Saleem, W. K. Elshareef, Group generated by total sextactic points of a family of quartic curves, Commun. Algebra, 50 (2021), 1342–1362. https://doi.org/10.1080/00927872.2021.1980796 doi: 10.1080/00927872.2021.1980796
|
| [7] | D. Eisenbud, J. Harris, The practice of algebraic curves: A second course in algebraic geometry, American Mathematical Society, 2024. |
| [8] | R. Miranda, Algebraic curves and Riemann surfaces, American Mathematical Society, 1995. |
| [9] |
L. Merta, M. Zieliński, On quartics with the maximal number of maximal tangency lines, Period. Math. Hung., 91 (2025), 309–319. https://doi.org/10.1007/s10998-025-00648-y doi: 10.1007/s10998-025-00648-y
|
| [10] |
G. Thorbergsson, M. Umehara, Sextactic points on a simple closed curve, Nagoya Math. J., 167 (2002), 55–94. https://doi.org/10.1017/S0027763000025435 doi: 10.1017/S0027763000025435
|
| [11] | R. Hartshorne, Algebraic geometry, In: Graduate texts in mathematics, Springer, 2010. |
| [12] |
Y. Liao, Y. Lin, Z. Xing, X. Yuan, Privacy image secrecy scheme based on chaos-driven fractal sorting matrix and fibonacci q-matrix, Vis. Comput., 41 (2025), 6931–6941. https://doi.org/10.1007/s00371-025-04014-4 doi: 10.1007/s00371-025-04014-4
|
Alwaleed Kamel, Eman Alluqmani, Mohammed A. Saleem, Waleed Khaled Elshareef. On classification of groups generated by total sextactic points of smooth quartic curves[J]. AIMS Mathematics, 2025, 10(12): 28221-28242. doi: 10.3934/math.20251241
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