Reconstructing curves from their Hodge classes

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 ) .