Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 21 (2026), 211 -- 218
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

CUBIC CURVES ASSOCIATED TO QUADRICS

Mircea Crâșmăreanu

Abstract. The aim of this paper is to study two associated cubic curves C1,2(Γ) to a given Euclidean quadric Γ through two polynomials generated by the symmetric 3 × 3 and 4 × 4 matrices defining Γ. More precisely, we focus here on the computation of the discriminant D of C1,2(Γ ) expressed in its Weierstrass form, with a special view toward the singular cases D=0. All quadrics with reduced equation are discussed.

2020 Mathematics Subject Classification: 51N2L05
Keywords: Quadric; invariants; (singular) cubic curve; discriminant

Full text (PDF)

References

  1. M. Crâșmăreanu, Magic conics, thier integer points and complementary ellipses, An. Stiint. Univ. Al. I. Cuza Iasi Mat. 67(2021), no. 1, 129-148. MR4275113. Zbl 1513.11093. DOI: https://doi.org/10.47743/anstim.2021.00010.
  2. M. Crâșmăreanu, The diagonalization map as submersion, the cubic equation as immersion and Euclidean polynomials, Mediterr. J. Math. 19(2022), no. 2, paper no. 65. MR4383352. Zbl 1485.15010. DOI: https://doi.org/10.1007/s00009-022-01996-6.
  3. M. Crâșmăreanu, Cubics and conics geodesically associated to the points of a geometric surface, Annales Mathematicae et Informaticae 62(2025), 46-54. MR5002476. Zbl 8175172. DOI: https://doi.org/10.33039/ami.2025.11.002.
  4. M. Crâșmăreanu, Cubic curves and conics associated to the sphere S4, Open Journal of Mathematical Sciences 10(2026), 11-16. DOI: https://doi.org/10.30538/oms2026.0267.
  5. M. Crâșmăreanu, Elliptic curves associated to a spacelike curve in the Lorentz plane, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 565(30), 2026, 159-167. MR5037395. Zbl 8162242. DOI: https://doi.org/10.21857/mjrl3urdk9.
  6. M. Crâșmăreanu, A cubic curve associated to a Menelaus-Ceva triple, Intern. Electr. J. Geom. (IEJG) 19(2026), no. 1, 12-17. MR5062776. DOI: https://doi.org/10.36890/iejg.1803403.
  7. M. Crâșmăreanu, Euclidean-Lagrange and Cantor-Lagrange quartic polynomials and associated cubic curves, Ann. Acad. Rom. Sci. Math. Appl. 18(2026), no. 2, 103-109. MR5065555. Zbl 8199115. DOI: https://doi.org/10.56082/annalsarscimath.2026.2.103.
  8. M. Crâșmăreanu, C.-L. Pripoae, G.-T. Pripoae, CR-selfdual cubic curves, Mathematics, 13(2025), no. 2, paper no. 317. DOI: https://doi.org/10.3390/math13020317.
  9. G. Glaeser, H. Stachel, B. Odehnal, The universe of quadrics, Cham: Springer, 2024. MR4886983. Zbl 1554.51001 DOI: https://doi.org/10.1007/978-3-662-70306-9.
  10. A. Pogorelov, Geometry, Mir Publishers, Moscow, 1987.


Mircea Crâșmăreanu, ORCID: https://orcid.org/0000-0002-5230-2751
Faculty of Mathematics, University "Al. I. Cuza",
Iași, 700506, Romania.
e-mail: mcrasm@uaic.ro
https://www.math.uaic.ro/~mcrasm/


Received: March 02, 2026; Accepted: May 10, 2026;
Published electronically: May 17, 2026