Automorphism Groups of Compact Bordered Klein Surfaces

Advances in Mathematics, 2019

Let X be a (projective, geometrically irreducible, non-singular) algebraic curve of genus g ≥ 2 defined over an algebraically closed field K of odd characteristic p. Let Aut(X) be the group of all automorphisms of X which fix K element-wise. It is known that if |Aut(X)| ≥ 8g 3 then the prank (equivalently, the Hasse-Witt invariant) of X is zero. This raises the problem of determining the (minimum-value) function f (g) such that whenever |Aut(X)| ≥ f (g) then X has zero prank. For even g we prove that f (g) ≤ 900g 2. The odd genus case appears to be much more difficult although, for any genus g ≥ 2, if Aut(X) has a solvable subgroup G such that |G| > 252g 2 then X has zero prank and G fixes a point of X. Our proofs use the Hurwitz genus formula and the Deuring Shafarevich formula together with a few deep results from Group theory characterizing finite simple groups whose Sylow 2-subgroups have a cyclic subgroup of index 2. We also point out some connections with the Abhyankar conjecture and the Katz-Gabber covers.

Annales Fennici Mathematici

We classify compact Riemann surfaces of genus g, where g − 1 is a prime p, which have a group of automorphisms of order ρ(g − 1) for some integer ρ ≥ 1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ > 6, and of the first and third authors for ρ = 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p + 1, together with the non-orientable regular hypermaps of characteristic −p, with automorphism group of order divisible by the prime p; this extends results of Conder,Širáň and Tucker for maps.

2021

Let S be a p-subgroup of the K-automorphism group Aut(X ) of an algebraic curve X of genus g ≥ 2 and p-rank γ defined over an algebraically closed field K of characteristic p ≥ 3. In this paper we prove that if |S| > 2(g− 1) then one of the following cases occurs. (i) γ = 0 and the extension K(X )/K(X ) completely ramifies at a unique place, and does not ramify elsewhere. (ii) γ > 0, p = 3, X is a general curve, S attains the Nakajima’s upper bound 3(γ − 1) and K(X ) is an unramified Galois extension of the function field of a general curve of genus 2 with equation Y 2 = cX +X +X + 1 where c ∈ K∗. Case (i) was investigated by Stichtenoth, Lehr, Matignon, and Rocher, see [18, 11, 12, 15, 16].

Proyecciones (Antofagasta), 1997

In t his paper. we present tllC' uniformization of y 2 = .rP-l, with p > 5 aurl prime. i. e .. the only hyperelliptic Riemann surface of gt'nus (/-7. \\"hich admit Z j2pZ as automorphism group. This 1111ifonnization is fouud by using a fuc:hsian group which rcflects the actiou of Z/2pZ aud is coustructed starting of a triangle group of !YJW (0:¡>.p.p). I\loreover. we describe completely the action of the automorphism group in hmnology. so that we can describe the invariant subvariety for Z /2pZ in A 9 (principally polarized abelian varieties of dimension y). which is detPrmiued bv the real Abe! aplication from M 9 in A 9 .

Topology, 2000

International Journal of Number Theory, 2009

This paper examines the relationship between the automorphism group of a hyperelliptic curve defined over an algebraically closed field of characteristic two and the 2-rank of the curve. In particular, we exploit the wild ramification to use the Deuring–Shafarevich formula in order to analyze the ramification of hyperelliptic curves that admit extra automorphisms and use this data to impose restrictions on the genera and 2-ranks of such curves. We also show how some of the techniques and results carry over to the case where our base field is of characteristic p > 2.

Glasgow Mathematical Journal, 1994

In this paper we consider complex doubles of compact Klein surfaces that have large automorphism groups. It is known that a bordered Klein surface of algebraic genus g > 2 has at most 12(g − 1) automorphisms. Surfaces for which this bound is sharp are said to have maximal symmetry. The complex double of such a surface X is a compact Riemann surface X+ of genus g and it is easy to see that if G is the group of automorphisms of X then C2 × G is a group of automorphisms of X+. A natural question is whether X+ can have a group that strictly contains C2 × G. In [8] C. L. May claimed the following interesting result: there is a unique Klein surface X with maximal symmetry for which Aut X+ properly contains C2 × Aut X (where Aut X+ denotes the group of conformal and anticonformal automorphisms of X+).

Mathematical Research Letters, 2008

We show the existence of an anti-pluricanonical curve on every smooth projective rational surface X which has an infinite group G of automorphisms of either null entropy or of type Z⋉Z, provided that the pair (X, G) is minimal. This was conjectured by Curtis T. McMullen (2005) and further traced back to Marat Gizatullin and Brian Harbourne (1987). We also prove (perhaps) the strongest form of the famous Tits alternative theorem.

The Michigan Mathematical Journal, 2008

Let k be an algebraically closed field of characteristic p > 0. Suppose g ≥ 3 and 0 ≤ f ≤ g. We prove there is a smooth projective k-curve of genus g and prank f with no non-trivial automorphisms. In addition, we prove there is a smooth projective hyperelliptic k-curve of genus g and prank f whose only non-trivial automorphism is the hyperelliptic involution. The proof involves computations about the dimension of the moduli space of (hyperelliptic) k-curves of genus g and prank f with extra automorphisms.

Springer Proceedings in Mathematics & Statistics, 2014

In the last years the biregular automorphisms of the Deligne-Mumford's and Hassett's compactifications of the moduli space of n-pointed genus g smooth curves have been extensively studied by A. Bruno and the authors. In this paper we give a survey of these recent results and extend our techniques to some moduli spaces appearing as intermediate steps of the Kapranov's and Keel's realizations of M 0,n, and to the degenerations of Hassett's spaces obtained by allowing zero weights.