Nets of conics containing a double line in PG(2,q), q even

2025/09/04 by Alnajjarine, Nour, Lavrauw, Michel
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.03840

Abstract

This paper completes the classification of nets of conics containing at least one double line in PG(2,q) for q even. This classification contributes to the classification of partially symmetric tensors in \mathbbF_q³ ⊗ S² \mathbbF_q³, q even. The proof is obtained using geometric and combinatorial properties of the Veronese surface in 5-dimensional projective space over the finite field of even order. In particular, the orbits of planes in PG(5,q) that intersect the nucleus plane of the Veronese surface in at least one point are classified. As a result, it is shown that there are exactly 18 equivalence classes of nets in PG(2,q), q even, containing at least one double line, 9 of which have an empty base.

Citations

• Linear complete symmetric rank-distance codes
• Webs and squabs of conics over finite fields
• 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
• The symmetric representation of lines in PG(\mathbbF³ ⊗ \mathbbF³)

Related

• Nets of conics of rank one in PG(2,q), q odd
• Webs and squabs of conics over finite fields
• A classification of planes intersecting the Veronese surface over finite fields of even order
• Combinatorial invariants for nets of conics in PG(2,q)
• Solids in the space of the Veronese surface in even characteristic