The composite of a succession of blow-ups is projective

For a Noetherian scheme X, the composite X_r → X of a finite succession of blow-ups is a single blow-up of X up to isomorphism (AlgebraicGeometry.exists_blowUp_composite_iso, [TSPA, Tag 080B]), and a blow-up of a locally Noetherian scheme is projective (AlgebraicGeometry.Scheme.IdealSheafData.isProjective_blowUpπ, [TSPA, Tag 02NS]); so the composite is projective (AlgebraicGeometry.Scheme.BlowUpSequence.isProjective_composite).

theorem AlgebraicGeometry.Scheme.BlowUpSequence.isProjective_composite {X : Scheme} [IsNoetherian X] (S : X.BlowUpSequence) : IsProjective S.composite

The composite of a finite succession of blow-ups of a Noetherian scheme is projective: it is a single blow-up up to isomorphism [TSPA, Tag 080B], and blow-ups are projective [TSPA, Tag 02NS].