Non-singular plane curves with an element of “large” order in its automorphism group

Abstract

Let Mg be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by Mg(G) the subset of Mg of curves δ such that G (as a finite non-trivial group) is 2010 Mathematics Subject Classification. 14H37, 14H50, 14H45. ) >.(3) if σ has order d(d − 2) then δ is K-isomorphic to X d + Y d−1 Z + Y Z d−1 = 0 and for d = 4, 6 we haveis an irreducible locus with one element, andThe automorphism groups for d = 4, 6 are given explicitly in §3.3, Proposition 15.(4) if σ has order d 2 − 3d + 3 then δ is K-isomorphic to the Klein curve K d :) is irreducible with one element, andWe refer to Remark 18 of §3.4 for the classical case d = 4.Remark 2. The above situations do not fit with curves that have large automorphism group in the classical definition. For example, the curve δ : X d + Y d−1 Z + XZ d−1 = 0 is defined over Q, δ/Aut(δ) is a projective line and the morphism δ → δ/Aut(δ) is ramified at two points of ramification index (d − 1) 2 and at d − 1-points of

On twists of smooth plane curves

“…Remark 3.7. Indeed, it is not difficult to prove (see [3,4]) that the automorphism group of the curves in the previous family is isomorphic to GAP(54, 5) and generated by the elements [Y : Z : α 0 X] together with Consider its image by Σ inside H 1 (Gal(M/Q), PGL 3 (M )). We need to check that its image is not the trivial element, and then the result is an immediate consequence of Theorem 3.1.…”

“…Moreover, for smooth plane curves defined over k with a cyclic automorphism group generated by a diagonal matrix, we provide a general theoretical result to compute all its twists. These families of smooth plane curves have already been studied by the first two authors in [3,4]. These families have genus arbitrarily high, so the method in [17] does not work for them.…”

Riemann surfaces defined over the reals

“…We consider the following two cases: (Case d = 4). We use quite similar techniques as the ones in [3,4,5]. It is clear that ψ := [X : ζ d/2 Y : Z] ∈ Aut(S t,d ) is an homology of order d/2 ≥ 4 (Definition 4.4).…”

Linear automorphisms of smooth hypersurfaces giving Galois points

“…Theorem 1.6. ( [1]). Let X be a smooth hypersurface of degree d ≥ 4 in P n+1 , and g ∈ Aut(X) be a linear automorphism of order k(d − 1) (resp.…”

“…Theorem 1.10 does not fold for an outer Galois point (see Example 3.9). For n = 1, the automorphism groups of curves with Galois points are classified ( [1,9]). There are studies on automorphism groups of plane curves using Galois points ( [1,11,13,14,17,18]).…”