The ramification sequence for a fixed point of an automorphism of a curve and the Weierstrass gap sequence (Q926240)

Let \(X\) be a nonsingular projective complete curve defined over an algebraically closed field of positive characteristic \(p \neq 2, 3\). Let \(G\) be a subgroup of \(\text{Aut} (X)\). For a point \(P \in X\) denote by \(G (P)\) the decomposition group of \(P\), i.e. the subgroup of elements of \(G\) fixing \(P\). As it is known, it admits a filtration \(G(P) = G_0 (P) \supset G_1 (P) \supset \dots\), where \(G_i = \{ \sigma \in G(P) | \nu_P (\sigma(t)-t) \geq i+1 \}\), for a local uniformizer \(t\) at \(P\) and a valuation \(\nu_P\). Of principal interest in the paper is \(G_1 (P)\), i.e. the \(p\)-part of \(G (P)\). For a wild ramification point \(P\), the author gives a faithful representation of \(G_1 (P)\). Using this representation, he relates the jumps in the ramification filtration to the gaps of the Weierstrass semigroup at \(P\), producing in this way upper bounds for the length of the ramification filtration and the number of jumps in the filtration in terms of the genus of \(X\) and the characteristic \(p\). Given are several examples in which the length of the filtration of \(G(P)\) is calculated including the representation of \(G_1 (P)\) for \(p\)-cyclic covers of the affine line. Finally, the author discusses curves with \(2\)-dimensional representation of \(G(P)\) in which case it turns out that there is only one jump and \(G_1 (P)\) is abelian. It is indicated that there is a large number of examples of such curves.