Authors: FATMA KARAOĞLU

Abstract: We perform an opportunistic search for cubic surfaces over small fields of characteristic two. The starting point of our work is a list of surfaces complied by Dickson over the field with two elements. We consider the nonsingular ones arising in Dickson' s work for the fields of larger orders of characteristic two. We investigate the properties such as the number of lines, singularities and automorphism groups. The problem of determining the possible numbers of lines of a nonsingular cubic surface over the fields of $\mathbb{C}, \mathbb{R}, \mathbb{Q}, \mathbb{F}_q$ where q odd, $\mathbb{F}_2$ was considered by Cayley and Salmon, Schlafli, Segre, Rosati and Dickson, respectively. Our work contributes this problem over the larger fields of even characteristic. Besides that we investigate the structure of nonsingular surfaces with $15$ and $9$ lines. This work is a contribution to the study of nonsingular cubic surfaces with less than $27$ lines.

Keywords: Geometry, finite field, cubic surface, number of lines, Dickson surfaces

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