On the finite field Nullstellensatz

Let X c pn be an irreducible projective variety defined over the finite field F q. We will say that X satisfies the property F F N (k, q) (Finite Field Nullstellensatz of degree k over F q) if every homogeneous polynomial over F q of degree k on pn vanishing on X (F q) vanishes on X, that is, it vanishes at all points of X defined over the algebraic closure of F q . We study when FFN(k,q) holds true for the Veronese embeddings of pm, the Segre varieties, the minimal degree rational scrolls and the linearly normal embeddings of smooth curves.