On fixed points on compact Riemann surfaces
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A point of a Riemann surface X is said to be its fixed point if it is a fixed point of one of its nontrivial holomorphic automorphisms. We start this note by proving that the set Fix(X) of fixed points of a Riemann surface X of genus g≥ 2 has at most 82(g-1) elements and this bound is attained just for X having a Hurwitz group of automorphisms, i.e., a group of order 84(g-1). The set of such points is invariant under the group of holomorphic automorphisms of X and we study the corresponding symmetric representation. We show that its algebraic type is an essential invariant of the topological type of the holomorphic action and we study its kernel, to find in particular some sufficient conditions for its faithfulness.
Abstract
A point of a Riemann surface X is said to be its fixed point if it is a fixed point of one of its nontrivial holomorphic automorphisms. We start this note by proving that the set Fix(X) of fixed points of a Riemann surface X of genus g≥ 2 has at most 82(g-1) elements and this bound is attained just for X having a Hurwitz group of automorphisms, i.e., a group of order 84(g-1). The set of such points is invariant under the group of holomorphic automorphisms of X and we study the corresponding symmetric representation. We show that its algebraic type is an essential invariant of the topological type of the holomorphic action and we study its kernel, to find in particular some sufficient conditions for its faithfulness.
참고문헌 (13)
- E. Bujalance, "On fixed points of involutions of compact Riemann surfaces", Math. Scand, 105(1), pp.16~24 (2009)
- G. Castelnuovo, "Ricerche de geometria sulle curve algebriche", Atti Acad. Sci. Torino, 24, pp.346~373 (1889)
- H. M. Farkas, "Riemann Surfaces, In Graduate Text in Mathematics" (1980), Springer-Verlag
- A. Garcia, "Remarks on fixed points of automorphisms and higher-order Weierstrass points in prime characteristic", Manuscripta Math, 69(3), pp.301~303 (1990)
- G. Gromadzki, "On conjugacy of p-gonal automorphisms of Riemann surfaces", Rev. Mat. Complut, 21(1), pp.83~87 (2008)
- G. Gromadzki, "On the number of p-gonal coverings of Riemann surfaces", Rocky Mountain Journal of Mathematics, 40(4), pp.1221~1226 (2010)
- R. Horiuchi, "Fixed points of automorphisms of compact Riemann surfaces and higher-order Weierstrass points", Proc. Amer. Math. Soc, 105(4), pp.856~860 (1989)
- J. Lewittes, "Automorphisms of compact Riemann surfaces", Amer. J. Math, 85(4), pp.734~752 (1963)
- A. M. Macbeath, "Action of automorphisms of a compact Riemann surface on the first homology group", Bull. London Math. Soc, 5, pp.103~108 (1973)
- K. Magaard, "On Weierstrass points of Hurwitz curves", J. Algebra, 300(2), pp.647~654 (2006)
- M. J. Moore, "Fixed points of automorphisms of compact Riemann surfaces", Canad. J. Math, 22(5), pp.922~932 (1970)
- F. Severi, "Vorlesungen uber algebraische Geometrie" (1921), Teubner
- C. L. Siegel, "Some remarks on discontinuous groups", Ann. of Math. (2), 46, pp.708~718 (1945)