Existence of (b, d)-geprofi sets in P4 in linear general position for fixed d and large b

Determine whether, for a fixed integer d ≥ 2 and sufficiently large b, there exists a (b, d)-geprofi set Z ⊂ P4 with |Z| = bd whose points are in linear general position and whose general projection to P3 is the full intersection of a curve of degree b and a surface of degree d.

Background

The paper introduces (b, d)-geprofi sets in P4 and proves broad existence results, including nontrivial examples and many instances in linear general position (LGP), especially for fixed b and large d (see Theorem C / Corollary 6.7).

However, the authors explicitly note that the complementary direction—fixing d and letting b grow—remains unknown, and they pose it as an open question focused on LGP configurations. This question probes the asymptotic existence of bd-point configurations in LGP whose general projection to P3 becomes a full intersection of a degree-b curve and a degree-d surface.