On a class of quasi-Hermitian surfaces in even characteristic

2025/08/05 by Aguglia, Angela, Montinaro, Alessandro
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.03907

Abstract

In [1], a new quasi-Hermitian variety H_ε^r in PG(r, q²), with q = 2^e and e ≥ 3 an odd integer, was constructed. The variety depends on a primitive element ε of the underlying field GF(q²).11 In the present paper, we first provide a classification of such varieties up to projective equivalence in finite projective spaces of arbitrary dimension. Then, we focus on the case r = 3 and study the structure of the lines contained in H_ε³; as a consequence, we determine the full automorphism group of H_ε³ . Finally, as a byproduct, we prove the equivalence of certain minimal codes introduced in [3].

Citations

• Minimal codes from hypersurfaces in even characteristic
• On the geometry of the Hermitian Veronese curve and its quasi-Hermitian surfaces
• On the Equivalence, Stabilisers, and Feet of Buekenhout-Tits Unitals
• Quasi-polar spaces
• A Class of Two-Weight and Three-Weight Codes and Their Applications in Secret Sharing

Related

• A characterization of Hermitian varieties as codewords
• Hermitian Varieties in a Finite Projective Space PG(N, q2)
• Characterization of non-singular hyperplanes of H(s,q²) in P G(s, q²)
• On the geometry of the Hermitian Veronese curve and its quasi-Hermitian surfaces
• Codes defined by forms of degree 2 on hermitian surfaces and Sørensen's conjecture