A New Form of Smooth Cubic Surfaces with 9 Lines-Reference-Cited by-同舟云学术

A New Form of Smooth Cubic Surfaces with 9 Lines

Published:2023-09-30 Issue:44 Volume: Page:62-78
ISSN:2149-1402
Container-title:Journal of New Theory
language:
Short-container-title:

Author:

KARAOĞLU Fatma¹^ORCID

Affiliation:

1. GEBZE TECHNICAL UNIVERSITY

Abstract

A smooth cubic surface has at most 27 lines, with equality if and only if the underlying field is algebraically closed. Only a few cases are possible regarding the number of lines over fields that are not algebraically closed. The next two cases of interest are smooth cubic surfaces with 15 or 9 lines. The author has recently settled the case of 15 lines. In this paper, we address the case of smooth cubic surfaces with 9 lines. We describe a way to create some cubic surfaces with 9 or more lines based on a set of six field elements. Conditions on the six parameters are given under which the surface has exactly 9, 15, or 27 lines. However, the problem of generating all cubic surfaces with 9 lines remains open.

Publisher

Gaziosmanpasa University

Subject

Geology,Ocean Engineering,Water Science and Technology

Reference24 articles.

1. A. Cayley, On the Triple Tangent Planes of Surfaces of the Third Order, Cambridge Journal of Mathematics (4) (1849) 118--138

2. L. Schlafli, An Attempt to Determine the Twenty-Seven Lines upon a Surface of the Third Order and to Divide such 35 Surfaces into Species in Reference to the Reality of the Lines upon the Surface, The Quarterly Journal of Mathematics (2) (1858) 55--110.

3. B. Segre, Le rette delle Superficie Cubiche nei Corpi Commutativi, Bollettino dell'Unione Matematica Italiana 3 (4) (1949) 223--228.

4. L. A. Rosati, L'equazione delle 27 Rette della Superficie Cubica Generale in un Corpo Finito, Bollettino dell'Unione Matematica Italiana 3 (12) (1957) 612--626.

5. L. E. Dickson, Projective Classification of Cubic Surfaces Modulo 2, Annals of Mathematics 16 (1915) 139--157.