  We consider the vanishing ideal of a projective space over a finite field. An
explicit set of generators for this ideal has been given by Mercier and
Rolland. We show that these generators form a universal Gr\"obner basis of the
ideal. Further we give a projective analogue of the footprint bound, and a
version of it that is suitable for estimating the number of points of a
projective algebraic variety over a finite field. An application to Serre's
inequality for the number of rational points of projective hypersurfaces over
finite fields is included
