Extend the expected-dimension theorem for inclusion spaces to arbitrary characteristic

Prove that the moduli space of inclusion of linear series $G^{r,r-s-1}_{d,k}(C)$, equivalently the moduli space of non-degenerate maps $M_{(d,k)}(C,X_{r,s})^{\circ}$ to the blow-up $X_{r,s}=\operatorname{Bl}_{\mathbb P^s}(\mathbb P^r)$, has the expected dimension in arbitrary characteristic.

Background

The paper uses Farkas's characteristic-zero theorem asserting that the inclusion-of-linear-series space Gd,kr,r−s−1(C)G^{r,r-s-1}_{d,k}(C) is pure of its expected dimension. This result is needed to control the dimension of the corresponding moduli space of non-degenerate maps to Bl⁡Ps(Pr)\operatorname{Bl}_{\mathbb P^s}(\mathbb P^r).

The authors explain that Farkas's degeneration argument relies essentially on characteristic-zero transversality for linear series on P1\mathbb P^1. Their main existence theorem is proved in arbitrary characteristic by other methods, but a general arbitrary-characteristic expected-dimension theorem remains unresolved.

References

While a more satisfying resolution would be to extend Theorem \ref{thm:farkas} to arbitrary characteristic, we do not currently have a proof.

— Brill-Noether existence with secants  (2609.26713 - Cela et al., 22 Sep 2026) in Section 1, immediately following Remark 2.14-style discussion of Theorem 0.1; Section 1 introduction

We also do not know whether the natural analog of the Gieseker-Petri theorem, that $G{r,r-s-1}_{d,k}(C)$ and $M_{(d,k)}(C,X_{r,s})\circ$ are smooth, holds, in any characteristic.

— Brill-Noether existence with secants  (2609.26713 - Cela et al., 22 Sep 2026) in Section 2.3, immediately after Theorem \ref{thm:farkas}