Higher-dimensional intrinsic characterization of Artin–Schreier geproci configurations

Characterize intrinsically, in projective spaces of dimension greater than three, when an $F_N$-Artin–Schreier configuration on concurrent lines spanning the ambient space is $(q,N)$-geproci, generalizing the necessary-and-sufficient condition $q\leq N$ established in $P^3$.

Background

The paper proves that an FNF_N-Artin–Schreier configuration supported on qq concurrent lines spanning P3P^3 is (q,N)(q,N)-geproci exactly when q≤Nq\leq N. This characterization relies on interpolation for the direction set in P1P^1 and a Vandermonde-type argument.

In higher-dimensional projective spaces, the interpolation criterion depends on the Hilbert function and general-position properties of the direction set in a higher-dimensional projective space. The authors explicitly ask whether an analogous intrinsic criterion exists in those dimensions.

References

In Theorem~\ref{thm:P3-characterization}, the condition $q\leq N$ is both necessary and sufficient for an $F_N$-Artin--Schreier configuration on $q$ concurrent lines spanning $P3$ to be $(q,N)$-geproci. Is there an analogous intrinsic characterization in higher dimension?

— Artin-Schreier geproci configurations in projective spaces of arbitrary dimension  (2609.03024 - Chiantini et al., 2 Sep 2026) in Question 1, Section 5 (Further questions)

How sharp is the regularity bound used in the iterative construction? In particular, can the Artin--Schreier lifting procedure produce geproci sets in $Pn$ with complete-intersection degrees smaller than those in Propositions~\ref{prop:every-dimension-odd} and \ref{prop:every-dimension-char2}?

— Artin-Schreier geproci configurations in projective spaces of arbitrary dimension  (2609.03024 - Chiantini et al., 2 Sep 2026) in Question 2, Section 5 (Further questions)

To what extent is the Artin--Schreier structure forced by the geproci property? In particular, can one characterize geproci sets in positive characteristic supported on concurrent lines?

— Artin-Schreier geproci configurations in projective spaces of arbitrary dimension  (2609.03024 - Chiantini et al., 2 Sep 2026) in Question 3, Section 5 (Further questions)