Geprofi sets in P4 that are not contained in any curve of degree b

Ascertain whether there exists a (b, d)-geprofi set Z ⊂ P4 with |Z| = bd such that Z does not lie on any curve of degree b in P4, despite the fact that its general projection to P3 is the full intersection of a degree-b curve and a degree-d surface.

Background

Most constructions in the paper yield (b, d)-geprofi sets that lie on curves of degree b in P4, particularly in LGP (e.g., on rational normal quartics or other smooth curves).

The authors explicitly pose the question of whether such containment is necessary, asking for the existence of geprofi sets whose projected intersection structure in P3 does not force membership in a degree-b curve in P4.

References

We remark that we do not know what happens in the other direction. Indeed, our work suggests the following two interesting open questions. Question 1.4. (2) Does there exist a set of points Z ⊂ P4 such that Z is (b, d)-geprofi, but yet Z does not lie on a curve of degree b in P4? (This question is not limited to LGP sets.)

— Finite sets of points in $\mathbb{P}^4$ with special projection properties (2407.01744 - Chiantini et al., 1 Jul 2024) in Question 1.4 (2), Section 1