A coordinate ring with constant filtered Hilbert polynomial has finitely many rational affine zeros. Its projective zero locus is empty, since every nonzero representative would produce arbitrarily many affine zeros by scaling.

theorem CCDN.projectiveIdealZeroLocus_eq_empty_of_hilbert_dimension_zero {n : ℕ} (I : Ideal (MvPolynomial (Fin (n + 1)) ℚ)) (hdim : (filteredHilbertPolynomial I).natDegree = 0) : projectiveIdealZeroLocus I = ∅

An ideal of ℚ[X₀, …, Xₙ] whose filtered Hilbert polynomial is constant has empty projective zero locus.