Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 85–90

GEOMETRIC CHARACTERIZATION OF HYPERELLIPTIC RIEMANN SURFACES

Paul Schmutz Schaller

Université de Genève, Section de mathématiques Case postale 240, CH-1211 Genève 24, Switzerland; paul.schmutz@math.unige.ch

Abstract. A new geometric characterization of hyperelliptic Riemann surfaces is given. It is proved that a closed Riemann surface of genus g ≥ 2 is hyperelliptic if and only if M has a set of at least 2g − 2 simple closed geodesics which all intersect in the same point (and mutually intersect in no other point).

1. Introduction

From a classical point of view the hyperelliptic surfaces are the most simple Riemann surfaces. They can be defined by an algebraic curve

y^{2} = F (x)

where F (x) is a polynomial of degree 2g + 1 or 2g + 2 with distinct roots (g is the genus of the surface); the automorphism (x, y) → (x, −y) then defines the so-called hyperelliptic involution.

Equivalently, we can define that a Riemann surface M is hyperelliptic if there exists a holomorphic map M → P^{1} of degree 2.

Here is a more geometric, again equivalent definition. A Riemann surface M of genus g is hyperelliptic if M has an involution with exactly 2g +2 fixed points.

Finally, hyperelliptic surfaces of genus g are characterized by the fact that the number of different Weierstrass points is minimal, namely 2g + 2 (the fixed points of the hyperelliptic involution), while on the other hand, the weight of each Weierstrass point is maximal, namely \frac{1}{2}g(g − 1).

For basic information on hyperelliptic surfaces the reader may consult for example Farkas and Kra [2], Miranda [3].

The main result of this paper is the following theorem (in the sequel, a surface will be a Riemann surface equipped with a metric of constant curvature −1).

Theorem A. A closed surface M of genus g ≥ 2 is hyperelliptic if and only if M contains 2g − 2 different simple closed geodesics which all intersect in the same point and mutually intersect in no other point.

1991 Mathematics Subject Classification: Primary 30F20; Secondary 30F10.

This is an example that a local property can imply a global property of a surface. The proof will be given in Section 3. In Section 2, some results of topological properties of sets of simple closed curves are proved, in particular the following result which generalizes the fact that a closed surface has non-separating and separating simple closed geodesics.

Theorem B. Let M be a closed surface of genus g ≥ 2. Let k be an integer with 1 ≤ k ≤ 2g. Let S_{k} be a set of k simple closed curves which intersect (transversally) in the same point such that the elements of S_{k} have no further intersection point. Then there is only one equivalence class of sets S_{k} if and only if k is even (S_{k} and S′_{k} are in the same equivalence class if M has a homeomorphism mapping the homotopy classes of the elements of S_{k} to those of S′_{k}).

2. On sets of simple closed curves

Definition. (i) A surface is a Riemann surface equipped with a metric of constant curvature −1.

(ii) A (g, n)-surface is a surface of genus g with n boundary components which are simple closed geodesics, called boundary geodesics.

(iii) A star set is a set S_{k} of k simple closed curves on a surface M which all intersect (transversally) in the same point such that among the elements of S_{k} there are no further intersection points. S_{k} is called a geodesic star set if the k simple closed curves are simple closed geodesics.

(iv) Let M and M′ be two (g, n)-surfaces. Let F be a set of simple closed curves on M, let F′ be a set of simple closed curves on M′. Then F and F′ are called equivalent if there exists a homeomorphism ϕ from M to M′ which maps the homotopy classes of the elements of F to those of F′. This relation gives rise to equivalence classes of sets of simple closed curves on M.

(v) Let M be a surface and u a simple closed geodesic of M. Then u is called separating if M \ u has two connected components, otherwise, u is called non-separating.

Lemma 1. Let M be a (g, 2)-surface with boundary geodesics u and v, u ≠ v. Let t be a simple geodesic in M relying u and v. Then M has a unique simple closed geodesic z and a unique embedded (0, 3)-surface Y such that u, v, z are the boundary geodesics of Y and t lies in Y .

Proof. Obvious.

Figure 1. On the left hand side, a (g, 1)-surface M with a simple geodesic t such that M \ t is not connected. On the right hand side, M \ t is connected.

Lemma 2. Let g ≥ 2 and let M be a (g, 1)-surface with boundary geodesic z. Let t be a simple geodesic in M, starting and ending on z. Then either M \ t is not connected or M \ t is connected and has a unique pair (u, v) of simple closed geodesics and a unique embedded (0, 3)-surface Y such that z, u, v are the boundary geodesics of Y and t lies in Y . Moreover, both cases are possible for every M.

Proof. Obvious by Figure 1.

Theorem 1. Let M be a closed surface of genus g ≥ 2. Let k be an integer with 1 ≤ k ≤ 2g. Then M has only one equivalence class of star sets S_{k} if and only if k is even.

Proof. (i) Recall that two (g, n)-surfaces are homeomorphic (by the classification of compact surfaces).

(ii) The case k = 1 is obvious since M has separating and non-separating simple closed curves. So assume that k ≥ 2. It follows that all elements of S_{k} are non-separating. We therefore may assume that the elements of S_{k} are geodesic loops. Let u ∈ S_{k}. Cut M along u, the result is a surface M_{1} of genus g −1 with two boundary components (in the proof of Theorem 1 we allow that the boundary components are only piecewise geodesic). Let v ∈ S_{k}, v ≠ u. Then, by Lemma 1, M_{1} \ v is a surface M_{2} of genus g − 1 with one boundary component. By (i), this proves the theorem for k = 2. Assume that k ≥ 3. Let w ∈ S_{k} \ {u, v}. If k = 3, then, by Lemma 2, M_{3} = M_{2} \ w is connected or not connected which proves the theorem for k = 3 (Lemma 2 proves this if the genus of M_{2} is at least 2; if M_{2} has genus 1, then the fact remains true since the boundary component of M_{2} is only piecewise geodesic). If k > 3, then M_{3} must be connected and is therefore (by Lemma 2) a surface of genus g − 2 with two boundary components.

(iii) The general case follows by induction.

Corollary 1. Let M be a closed surface of genus g. Let k be an integer with 1 ≤ k ≤ 2g. Let C_{k} = {u_{1}, u_{2}, …, u_{k}} be a set of k simple closed curves of M such that u_{i} and u_{i+1} intersect exactly once, i = 1, …, k − 1, and such that u_{i} and u_{j} are disjoint if j ∉ {i − 1, i, i + 1}. Then M has only one equivalence class of sets C_{k} if and only if k is even.

Proof. The proof is completely analogous to the proof of Theorem 1.

3. Hyperelliptic surfaces

Definition. (i) An involution ϕ ≠ id on a surface M is an isometry with ϕ^{2} = id where id is the identity.

(ii) A (g, n)-surface M is hyperelliptic if M has an involution with exactly 2g + 2 fixed points if n is even, and with exactly 2g + 1 fixed points if n is odd. Such an involution will be called a hyperelliptic involution.

Remarks. (i) It is well known that for closed surfaces the above given definition is equivalent to the usual one.

(ii) Let M be a hyperelliptic (g, n)-surface. It is easy to show that M has a unique hyperelliptic involution if n ≤ 2. However, if n ≥ 3, then a hyperelliptic involution is not unique, in general.

(iii) In the cases (g, n) = (1, 1) and (g, n) = (2, 0) all (g, n)-surfaces are hyperelliptic. If all boundary geodesics have equal length, then also in the cases (g, n) ∈ {(0, 3), (0, 4), (1, 2)} all (g, n)-surfaces are hyperelliptic.

(iv) Let M be a hyperelliptic (g, n)-surface and ψ a hyperelliptic involution. If n is odd, then ψ(u) = u for a unique boundary geodesic of M. If n ≥ 2 is even, then ψ(u) ≠ u for all boundary geodesics of M.

Theorem 2. Let M be a closed surface of genus g. Then M is hyperelliptic if and only if M has a geodesic star set S_{k} with k ≥ 2g − 2.

Proof. (i) Let M be hyperelliptic. Let A_{i}, i = 1, …, 2g + 2, be the fixed points of the hyperelliptic involution ψ of M. Let A = A_{1} . Let h_{1} be a simple geodesic from A to A_{2} . Then u_{1} = h_{1} ∪ ψ(h_{1}) is a simple closed non-separating geodesic of M. Cut M along u_{1} , the result is a (g − 1, 2)-surface M_{1} with boundary geodesics u_{1}(L) and u_{1}(R). Let A(L) be the copy of A on u_{1}(L) and let A_{2}(R) be the copy of A_{2} on u_{1}(R). Then M_{1} has a simple geodesic h_{2} from A(L) to A_{2}(R). It follows that u_{2} = h_{2} ∪ ψ(h_{2}) is a simple closed geodesic of M which intersects u_{1} only in A. Cut M_{1} along h_{2} , the result is a surface M_{2} of genus g − 1 with one boundary component b, see Lemma 1. It follows that M_{2} has a simple geodesic h_{3} between a copy of A on b and a copy of A_{3} on b such that M_{2} \ h_{3} is connected (this is possible by Lemma 2). It follows that u_{3} = h_{3}∪ψ(h_{3}) is a simple closed geodesic of M and that {u_{1}, u_{2}, u_{3}} is a geodesic star set of M. Repeating this argument we can construct a geodesic star set of order 2g + 1 on M.

(ii) Assume now that M has a geodesic star set S_{k} with k = 2g−2. It follows by Theorem 1 and its proof that M has an embedded (g − 1, 1)-surface N which contains all elements of S_{k}. Denote by z the boundary geodesic of N. In M, z is separating and is also the boundary geodesic of an embedded (1, 1)-surface X. As noted above, X is hyperelliptic with a unique hyperelliptic involution ψ_{X}. In order to prove that M is hyperelliptic, it is therefore sufficient to show that N is hyperelliptic since N then has a unique hyperelliptic involution ψ_{N} which acts on z in the same manner as ψ_{X}.

We now prove that N is hyperelliptic by induction with respect to g. If g = 2, then N is hyperelliptic since all (1, 1)-surfaces are hyperelliptic. Let u and v be two different elements of S_{k} and let S_{k−2} = S_{k} \ {u, v}. By Theorem 1, N has an embedded (g − 2, 1)-surface P which contains all elements of S_{k−2} . By induction, P is hyperelliptic with a unique hyperelliptic involution ψ_{P} . Denote by z′ the boundary geodesic of P. Let S_{k−1} = S_{k−2} ∪ {u}. Again by Theorem 1 and its proof it follows that M has an embedded (g−2, 2)-surface R containing S_{k−1} ; denote by x and y the two boundary geodesics of R. Denote by Y the, in M, embedded (0, 3)-surface with boundary geodesics z′, x, y. Then Y is separated by u into two connected components, denote them by Y (x) and Y (y). Remember that P is hyperelliptic and that u passes through a fixed point of ψ_{P} (namely, the intersection point of the elements of S_{k}). It follows that the two angles in the two intersection points of u with z′ are equal and that z′ ∩ Y (x) and z′ ∩ Y (y) have equal length. This implies by hyperbolic trigonometry (see for example [1]) that Y (x) and Y (y) are isometric. In particular, x and y have the same length. Therefore, Y has a hyperelliptic involution ψ_{Y} which interchanges x and y. This implies that ψ_{P} extends to a hyperelliptic involution ψ_{R} on R (since the action on z′ is the same for ψ_{P} and for ψ_{Y} ).

Let T = N \ R. T is an embedded (0, 3)-surface with boundary geodesics z, x, y. Since x and y have the same length, T has a hyperelliptic involution ψ_{T} which interchanges x and y. It remains thus to prove that the action of ψ_{T} and ψ_{R} is the same on x ∪ y. This again follows by the argument that v passes through a fixed point of ψ_{R} and therefore intersects x and y by the same angle and such that ψ_{T} (v ∩ T) = v ∩ T (by Theorem 1 and its proof v intersects x and y only once). This proves that the involutions ψ_{R} and ψ_{T} induce a hyperelliptic involution on N.

Corollary 2. (i) Let M be a (g, 1)-surface. Then M is hyperelliptic if and only if M has a geodesic star set S_{k} with k = 2g.

(ii) Let M be a (g, 2)-surface. Then M is hyperelliptic if and only if M has a geodesic star set S_{k} with k = 2g + 1.

Proof. The proof of Theorem 2 applies to this situation.

90 Paul Schmutz Schaller

Corollary 3. All closed surfaces of genus 2 are hyperelliptic.

Proof. This was shown during the proof of Theorem 2 using the fact that all (1, 1)-surfaces are hyperelliptic. After the proof of Theorem 2, we could also argue that all closed surfaces have a star set S_{k} with k = 2.

References

[1] Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. - Birkhäuser, 1992.

[2] Farkas, H.M., and I. Kra: Riemann Surfaces, 2nd ed. - Springer-Verlag, 1992.

[3] Miranda, R.: Algebraic Curves and Riemann Surfaces. - Grad. Stud. Math. 5, Amer. Math. Soc., 1995.

Received 7 April 1998