The rank of the Cartier operator and linear systems on curves (Q5929442)

Let \(C\) be a smooth projective curve of genus \(g\) over an algebraically closed field \(k\) of characteristic \(p>0\). The Cartier operator [\textit{C. Cartier}, C. R. Acad. Sci., Paris 244, 426-428 (1957; Zbl 0077.04502)] is defined as follows: Let \(K\) be the field of rational functions on \(C\), \(\Omega_{K/k}\) the \(K\)-vector space of Kähler differentials and \(d:K\to \Omega_{K/k}\) the universal \(k\)-derivation. Let \(\Omega_{K/k}'\) be the \(K\)-vector space obtained from \(\Omega_{K/k}\) by restricting scalars via the Frobenius morphism \(F:K\to K\). \(\text{Im} (d)\) is a \(K\)-vector subspace of \(\Omega_{K/k}'\). Then the \(k\)-derivation: \(K\to \Omega_{K/k}'\), \(f\mapsto f^{p-1}df\) induces a \(K\)-linear isomorphism \({\mathcal C}^{-1}:\Omega_{K/k} t\to\text{Coker} (d)\). The Cartier operator \({\mathcal C}:\Omega_{K/k} \to\Omega_{K/k}\) is the inverse of \({\mathcal C}^{-1}\) composed with the canonical surjection \(\Omega_{K/k} \to\text{Coker}(d)\). It induces a morphism of sheaves: \(\Omega_C \to\Omega_C\) and then an operator \({\mathcal C}:H^0 (\Omega_C)\to H^0(\Omega_C)\) (such that \({\mathcal C}(c^p \omega)= c{\mathcal C}(\omega)\) for \(c\in k)\). Concerning the last operator, the author proves the following results:NEWLINENEWLINENEWLINE(1) If \(\text{rk} {\mathcal C}=m <g\) then \(g\leq(m+1) p(p-1)/2 +mp\) (but, conjecturally, \(g\leq mp+f(p))\). NEWLINENEWLINENEWLINE(2) If \({\mathcal C}^r=0\) for some \(r\geq 1\) then \(g\leq p^r(p^r-1)/2\) and this bound is sharp.NEWLINENEWLINENEWLINEThe only property of the Cartier operator used by the author is that if \(\omega\in H^0(\Omega_C)\) vanishes of order \(np+p-1\) at \(x\in C\) then \({\mathcal C}(\omega)\) vanishes of order \(n\) at \(x\). The rest of the proof consists of manipulating linear series on \(C\).NEWLINENEWLINENEWLINEBounds on the rank of \({\mathcal C}\), depending on the ramification behaviour of the canonical linear system at a point \(x\in C\), were previously given by \textit{K.-O. Stöhr} and \textit{P. Viana} [Math. Z. 200, No. 3, 397-407 (1989; Zbl 0637.14017)].