THE LÜROTH SEMIGROUP OF A PLANE CURVE C OVER 𝔽q AND THE PRIMITIVE SET OF P ∈ C(𝔽q)

Abstract

Let C ⊂ ℙ2 be a smooth and geometrically connected curve defined over 𝔽q. We study the Lüroth semigroup of C over 𝔽q, i.e. the set L(C, q) of all degrees > 0 of some spanned line bundle on C defined over 𝔽q. Fix P ∈ C(𝔽q). We also study the set of all t ∈ ℕ such that both [Formula: see text] and ωC(–tP) are spanned (the primitive set of P), when q is a square, C is the Hermitian curve and P ∈ C(𝔽q).