THE LUROTH SEMIGROUP OF A PLANE CURVE C OVER F-q AND THE PRIMITIVE SET OF P is an element of C(F-q)

Let C is an element of P-2 be a smooth and geometrically connected curve defined over F-q. We study the Luroth semigroup of C over F-q, i.e. the set L(C, q) of all degrees > 0 of some spanned line bundle on C defined over F-q. Fix P is an element of C(F-q). We also study the set of all t. N such that both Omicron(C)(tP) and omega(C)(-tP) are spanned (the primitive set of P), when q is a square, C is the Hermitian curve and P is an element of C(Fq).