Automorphisms of generalized Fermat curves

2019

RACSAM

Self Cite

If n ≥ 3, then moduli space M 0,[n+1] , of isomorphisms classes of (n+1)-marked spheres, is a complex orbifold of dimension n − 2. Its branch locus B 0,[n+1] consists of the isomorphism classes of those (n + 1)-marked spheres with non-trivial group of conformal automorphisms. We prove that B 0,[n+1] is connected if either n ≥ 4 is even or if n ≥ 6 is divisible by 3, and that it has exactly two connected components otherwise. The orbifold M 0,[n+1] also admits a natural real structure, this being induced by the complex conjugation on the Riemann sphere. The locus M 0,[n+1] (R) of its fixed points, the real points, consists of the isomorphism classes of those marked spheres admitting an anticonformal automorphism. Inside this locus is the real locus M R 0,[n+1] , consisting of those classes of marked spheres admitting an anticonformal involution. We prove that M R 0,[n+1] is connected for n ≥ 5 odd, and that it is disconnected for n = 2r with r ≥ 5 is odd.2010 Mathematics Subject Classification. 30F10, 30F60, 32G15.

Section: Generalized Fermat Curves a Closed Riemann Surface S Is Calmentioning

confidence: 99%

On the connectivity of the branch and real locus of $${\mathcal M}_{0,[n+1]}$$M0,[n+1]

Atarihuana

2019

RACSAM

Self Cite

“…In the case of generalized Fermat curves, in [7] (See Theorem 7) the ramification indexes were explicitly computed, which allows us to determine the hyperosculating points. The following theorem will be useful for this purpose.…”

Section: 2mentioning

confidence: 99%

“…In [2] it was proved that, for n = 3 and k ≥ 3, a generalized Fermat curve of the type (k, n) has a unique generalized Fermat group of type (k, n) and later, in [7], as long (k −1)(n−1) > 2 (equivalently, g k,n > 1), this uniqueness property was proved to be true in general. This fact asserts that the moduli space of generalized Fermat curves of the type (k, n) can be identified with the moduli space of orbifolds of genus zero with (n + 1) cone points, each one of order k. In particular, generalized Fermat curves of type (k, n) provide a (n−2) complex dimensional family inside the moduli space of surfaces of genus g k,n (those of type (k, 2) are exactly the classic Fermat curves of degree k).…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless, having them may help to understand the geometry of the moduli spaces of curves. In this paper we study an interesting family of non-hyperelliptic Riemann surfaces, called generalized Fermat curves, where these different descriptions have been studied and a good understanding of them have been achieved; see for instance [1,2,3,7].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Weierstrass weight of the hyperosculating points of generalized Fermat curves

Journal of Pure and Applied Algebra

Leyton-Álvarez

2019

Self Cite

Let (S, H) be a generalized Fermat pair of the type (k, n). If F ⊂ S is the set of fixed points of the non-trivial elements of the group H, then F is exactly the set of hyperosculating points of the standard embedding S ֒→ P n . We provide an optimal lower bound (this being sharp in a dense open set of the moduli space of the generalized Fermat curves) for the Weierstrass weight of these points.

“…In [11] Y. Prokhorov classified the finite simple non-abelian subgroups of the Cremona group of rank 3, that is, the group of birational automorphisms of the 3-dimensional complex projective space; these groups being isomorphic to A 5 , A 6 , A 7 , PSL 2 (7), PSL 2 (8) and PSp 4 (3) (where A n denotes the alternating group in n letters). In his classification, the group PSL 2 (8) is seen to act on some smooth Fano threefold in P 8 C of genus 7, this being the dual Fano threefold of the Fricke-Macbeath curve.…”

Section: Introductionmentioning

confidence: 99%

About the Fricke–Macbeath curve

European Journal of Mathematics

2017

A closed Riemann surface of genus g ≥ 2 is called a Hurwitz curve if its group of conformal automorphisms has order 84(g − 1). In 1895, A. Wiman noticed that there is no Hurwitz curve of genus g = 2, 4, 5, 6 and, up to isomorphisms, there is a unique Hurwitz curve of genus g = 3; this being Klein's plane quartic curve. Later, in 1965, A. Macbeath proved the existence, up to isomorphisms, of a unique Hurwitz curve of genus g = 7; this known as the Fricke-Macbeath curve. Equations were also provided; that being the fiber product of suitable three elliptic curves. In the same year, W. Edge constructed such a genus seven Hurwitz curve by elementary projective geometry. Such a construction was provided by first constructing a 4-dimensional family of genus seven closed Riemann surfaces S µ admitting a group Gµ ≅ Z 3 2 of conformal automorphisms so that S µ Gµ has genus zero. In this paper we discuss the above curves in terms of fiber products of classical Fermat curves and we provide a geometrical explanation of the three elliptic curves in Macbeath's description. We also observe that the jacobian variety of each S µ is isogenous to the product of seven elliptic curves (explicitly given) and, for the particular Fricke-Macbeath curve, we obtain the well known fact that its jacobian variety is isogenous to E 7 for a suitable elliptic curve E.