A homogeneous finite birational linear projection preserves the Hilbert multiplicity. Both the conductor and the dimension equality are consequences of the algebraic hypotheses.

theorem CCDN.homogeneous_finite_birational_projection_leadingCoeff_eq {K : Type u_1} {F : Type u_2} {σ : Type u_3} {τ : Type u_4} [Field K] [Field F] [Algebra K F] [Fintype σ] [Fintype τ] [LinearOrder σ] [LinearOrder τ] (I : Ideal (MvPolynomial τ K)) [I.IsPrime] [Algebra (MvPolynomial τ K ⧸ I) F] [IsScalarTower K (MvPolynomial τ K ⧸ I) F] [IsFractionRing (MvPolynomial τ K ⧸ I) F] (hI : IsHomogeneousIdeal I) (a : σ → MvPolynomial τ K) (ha : ∀ (i : σ), (a i).IsHomogeneous 1) (hint : (projectionCoordinateMap I a).IsIntegral) (hfield : IntermediateField.adjoin K (Set.range fun (i : σ) => (algebraMap (MvPolynomial τ K ⧸ I) F) ((Ideal.Quotient.mk I) (a i))) = ⊤) : (filteredHilbertPolynomial (RingHom.ker (projectionCoordinateMap I a))).leadingCoeff = (filteredHilbertPolynomial I).leadingCoeff

A homogeneous linear projection of a homogeneous prime quotient that is integral and birational (its coordinates generate the function field) preserves the leading coefficient of the filtered Hilbert polynomial, and hence the degree.