| Let g ≥ 2 be an integer and k algebraically closed field of characteristic p ≠ 2. Let Xg be an irreducible algebraic curve defined over k. In this work we determine the list of automorphism groups of Xg for cyclic curves and all automorphism groups for g = 3 and all p. We also determine the equations of corresponding curves in each case.;The automorphism group of Xg over k, which we denoted by G = Aut(Xg), is a finite group. The order of such a group is bounded by |G| ≤ 84( g - 1) when p = 0 and |G| ≤ 16g4 when p > 0.;If there is a cyclic subgroup Cn ◃ G such that g(XCn) = 0 then such curves are called cyclic curves or superelliptic curves. Then we have covers Xg →f0 P1 →f P1. The group G¯ = G/Cn is called the reduced automorphism group of Xg. We determine groups G which occur as automorphism groups of cyclic curves in any characteristic p ≠ 2 and for any genus g ≥ 2. Furthermore, we determine the ramification signatures of the covers &phis;, &phis;0 and &phis; ∘ &phis; 0. The moduli space of covers with fixed group G and ramification signature C is a Hurwitz space H not necessarily irreducible. There is a map from H to the moduli space of genus g algebraic curves Mg. The image of this map is a subvariety of Mg and denoted by H(G,C). The dimension of H(G,C) is determined by the braid action.;Let &phis; : P1x → P1z be the cover with signature (sigma 1, sigma2, sigma3). Then G¯ be the monodromy group of &phis;. We fix coordinates in P 1 as x and z respectively. Then z is a rational function in x of the degree | G¯|. Let q1, q 2, q3 be corresponding branch points of &phis;. Let S be the set of branch points of phi: Xg → P1z and let yn = f(x) be the equation of Xg and W be the images in P1x of roots of f(x) and V:= ⋃i=1 3 &phis;-1(qi). By considering the equations z-qi for all i, we define three equations ϕ,chi,psi. Since points in the fiber &phis; -1 (S{q1,q 2,q3}) are the roots of equation Psi (x) - lambda · gamma(x) = 0, we define G( x):= leS\&cubl0;q1 ,q2,q3 &cubr0; (Psi(x) - lambda · gamma(x)). Then we determine the equations of the curve yn = f(x) for each fixed &phis; by considering the set V ∩ W. Hence we are able to determine a parametric equation X of a cyclic curve for given G such that G = Aut(X). We list those equations for each G that we have listed before.;Further we determine the list of automorphism groups of genus 3 in every characteristic. Similar techniques can be used for g > 3. It is the first time such a result is proven. Similar techniques can be used for g > 3. |
