On higher-order Weierstrass points and the finiteness of the automorphism group (Q1313504)

Let \(X\) be a non-singular irreducible projective curve of genus \(g \geq 2\) over an algebraically closed field of characteristic \(p>0\). Let \(\Phi_ q\) denote the morphism given by a basis for the space of holomorphic \(q\)-differentials on \(X\), and \(W(q)\) the set of \(q\)-Weierstrass points. If the order-sequence \(\{\varepsilon_ i\}\) of \(\Phi_ q\) at the general points satisfies \(\varepsilon_ i=i\) for \(0 \leq i \leq (2q-1)(g-1)-1\), then \(\Phi_ q\) is called classical. Using results of \textit{A. Neeman} [Invent. Math. 75, 359-376 (1984; Zbl 0555.14009)], the author proves the following, and deduces from it the finiteness of the automorphism group of curves in prime characteristics, which is known in characteristic zero: For \(q\geq 2\), \(W(q)\) be the set of \(q\)-Weierstrass points on \(X\). Then, given a natural number \(N\), there exists \(q = q(N) \geq 2\) such that \(\Phi_ q\) is classical and \(\# W(q)\geq N\).