On the Finite Field Nullstellensatz for the intersection of two quadric hypersurfaces

braic scheme defined over a finite field GFðpÞ.X ðKÞwill denote the set of allK-points of X . For every power q of p with qd p let XðqÞ denote the set of all GFðqÞ-points of X . Hence XðqÞJX ðq 0Þ if q, q 0 are p-powers and q 0 d qd p. XðKÞ is the union of all XðqÞ, qg 0 and q a p-power. If X is reduced, then the scheme X is uniquely determined by the algebraic variety XðKÞ in the sense of Serre (Hilbert Nullstellensatz). If X is not a zero-dimensional scheme, then XðKÞ is infinite. We fix a p-power q with qd p and we would like to see up to what order the finite set XðqÞ determines the infinite set XðKÞ. Now assume that X is projective and that it is equipped with an embedding XHP defined over GFðqÞ. Let k be an integer. We say that the pair ðX ;X ðqÞÞ satisfies the Finite Field Nullstellensatz of order k (or just that FFNðkÞ is true for X and XðqÞ) if