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Let X be an irreducible, reduced complex projective hypersurface of degree d. A uniform point for X is a point P such that the projection of X from P has maximal monodromy. We extend and improve some results concerning the finiteness of the locus of non-uniform points for projections of hypersurfaces obtained by the authors and Cuzzucoli only for P not contained in X., Comment: 11 pages, comments welcome!

Let S be a smooth algebraic surface in $${\mathbb {P}}^3({\mathbb {C}})$$ P 3 ( C ) . Movasati and Sertöz (Rend. Circ. Mat. Palermo 2:1–17, 2020) associate an ideal $$I_{\alpha (C)}$$ I α ( C ) to the primitive cohomology class $$\alpha (C)$$ α ( C ) of C in S. We show that the equations of C can be determined by $$I_{\alpha (C)}$$ I α ( C ) under numerical conditions. We apply this result to reconstruct rational curves and arithmetically Cohen-Macaulay curves from their cohomology classes. On the other hand, we show that the class $$\alpha (C)$$ α ( C ) of a rational quartic curve C on a smooth quartic surface S is not even perfect, that is, that $$I_{\alpha (C)}$$ I α ( C ) is bigger than the sum of the Jacobian ideal of S and of the homogeneous ideals of curves D in S for which $$I_{\alpha (D)}=I_{\alpha (C)}$$ I α ( D ) = I α ( C ) .