A classification of planes intersecting the Veronese surface over finite fields of even order

Solids in the space of the Veronese surface in even characteristic

“…Some of the results are classical and can be found in e.g. [11,12], others are more recent results from [2,3,4,6,16,17,18].…”

“…Scholars such as C. Jordan were among the first to investigate these systems, exploring their properties over both real and complex numbers [13,14], while C. Segre made significant contributions in the study of pencils of quadrics in higher dimensions [20], particularly focusing on algebraically closed fields with characteristics different from two. Their pioneering work laid the foundation for subsequent research at the start of the 20th century on linear systems of conics (including over finite fields) by Dickson, Campbell, Wilson, Wall, and others [7,8,22,21,16,17,3,4]. We direct the reader to [16] for a detailed elucidation on the insufficiency of the elementary divisor method employed by C. Segre and fellow scholars in addressing the finite field case, contrasting its efficacy in the case of algebraically closed fields.…”

“…For the odd characteristic case, we refer to [17,Section 9] for a classification of nets of rank one and for an explanation of some of the shortcomings of Wilson's treatment. For the even characteristic case we refer to [4] were we pointed out some errors in Campbell's work, and classified nets of conics with non-empty base.…”