A helly type theorem for hypersurfaces

Let r be a commutative field (finite or infinite) and let P = P(n, r) be the n-dimensional projective space over ZY Then every point x E P can be expressed by n + 1 homogene coordinates x = (x,,..., x,), not all zero and (x0,..., x,) = @x0,..., Ax,) for OflET. By a hypersurface of degree d we simply mean the set of all points x E P with p(x) = 0, where p(x) is a homogenous polynomial of degree d in the n + 1 variables x0, x1 ,..., x,.