Classification of Hermitian-Relative Curves up to Projective Equivalence

초록

Let q be an even power of a prime p, and let Fq be the finite field with q elements. For A=(aij)∈GL(3,Fq), CA denotes the plane curve defined by the equation (xq,yq,zq)A(x,y,z)t=0. We call CA a Hermitian-relative curve. Let A∗=(ajiq) be the conjugate transpose. Then CA is a Hermitian curve if A=A∗, which is an important curve for various reasons. Two Hermitian-relative curves CA and CB are projectively equivalent if and only if there exists T∈GL(3,Fq) such that B=T∗AT. In previous works [4, 5], we determined all possible pairs (Nq(CA),Iq(CA)), where Nq(C) and Iq(C) denote the number of rational points and rational inflexions on a curve C, respectively. The resulting pairs are: (qq+1,qq+1), (q+1,q+1), (1, 1), (q-q+1,0), (q+1,1), (q+1,2), (q+q+1,1), and (q+2q+1,0). Extending these results, we aim to categorize the curves into equivalence classes according to their projective equivalence. It is well known that the curves of type (qq+1,qq+1) are Hermitian curves, which constitute a single equivalence class. We focus on partitioning curves of other types into their respective projective equivalence classes. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.

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