On the cohomology of \(N_C(-2)\) in positive characteristic (Q6608792)

If \(C\) is a general curve of genus \(g\) with a general embedding \(C \hookrightarrow \mathbb P_k^3\) of degree \(d\), \textit{G. Ellingsrud} and \textit{A. Hirschowitz} asked under what conditions is \(H^0 (N_C (-2))=0\), where \(N_C\) is the normal bundle on \(C\) [C. R. Acad. Sci. Paris Sér. I Math. 299, 245--248 (1984; Zbl 0572.14007)]. \textit{D. Perrin} used liaison to give a partial answer when char \(k=0\) [Courbes passant par m points généraux de \(P^ 3\). (Curves passing through m general points of \(P^ 3)\). Société Mathématique de France (SMF), Paris (1987; Zbl 0648.14028)]. The author later gave a complete answer when char \(k=0\), showing that the vanishing occurs except for \((d,g)\) in the exceptional list \(\{(4,1), (5,2), (6,2), (6,4), (7,5), (8, 5)\}\) [J. Lond. Math. Soc. 104, 886--925 (2021; Zbl 1480.14022)].\N\NIn the paper under review, the author extends the solution to take into account the characteristic of \(k\). To state the result, let \(f:C \to \mathbb P^3\) be a \textit{Brill-Noether curve}, meaning that it corresponds to a point in a component of \(\overline M_g (\mathbb P^3, d)\) which both dominates \(\overline M_g\) and whose generic member is a nondegenerate map from a smooth curve. If \(C\) is a general Brill-Noether curve of degree \(d\) and genus \(g\), then \(h^0 (N_C (-2))=1\) if char \(k=2\) and \(d+g\) is even; otherwise \(h^0 (N_C (-2))=0\), except for \((d,g)\) in the exceptional list. Further, for each \((d,g)\) in the exceptional list, the author computes \(h^0 (N_C (-2))\). The new behavior for char \(k=2\) and \(g=0\), \(d\) even was expected because the normal bundle is a twist of a Frobenius pullback, but for \(g > 0\), the new examples come as a surprise, which is explained by the author using additional structure beyond the fact that \(N_C\) is a Frobenius pullback.