Surfaces of section for Seifert fibrations
aa r X i v : . [ m a t h . G T ] F e b SURFACES OF SECTION FOR SEIFERT FIBRATIONS
BERNHARD ALBACH AND HANSJ ¨ORG GEIGES
Abstract.
We classify global surfaces of section for flows on 3-manifolds defin-ing Seifert fibrations. We discuss branched coverings — one way or the other— between surfaces of section for the Hopf flow and those for any other Seifertfibration of the 3-sphere, and we relate these surfaces of section to algebraiccurves in weighted complex projective planes. Introduction
Global surfaces of section are an important tool for understanding the dynamicsof non-singular vector fields on manifolds. For a given flow or class of flows, it isa basic question to understand the existence of such surfaces of section, and theirtopological properties.Here are some recent results in this direction in the context of Reeb dynam-ics; for a more comprehensive overview of the literature we refer to [1] and [16].Hryniewicz–Salom˜ao–Wysocki [9] establish sufficient (and C ∞ -generically neces-sary) conditions for a finite collection of periodic orbits of a Reeb flow on a closed3-manifold to bound a positive global surface of section of genus zero. Buildingon the work of Hryniewicz [7], who found a characterisation of the periodic Reeborbits of dynamically convex contact forms on the 3-sphere that bound disc-likeglobal surfaces of section, existence results for higher genus surfaces of section havebeen established by Hryniewicz–Salom˜ao–Siefring [8].Even for one of the most simple flows in dimension three, the Hopf flow on the3-sphere, global surfaces of section display rich features. The paper [1] determinesthe topology of such surfaces of section and describes various ways to constructthem. Furthermore, it was shown there how to relate such surfaces of section toalgebraic curves in the complex projective plane, and how to interpret them as atoy model for the elementary degenerations arising in symplectic field theory; thisled to a new proof of the degree-genus formula for complex projective curves.The aim of the present paper is to extend the results of [1] to all flows on 3-manifolds that define a Seifert fibration. We decide the existence question and givea topological classification of the global surfaces of section for such flows. For someclassification statements we restrict attention to positive surfaces of section, whichmeans that the boundary components are supposed to be oriented positively by theflow (see Section 3 for details). This is motivated by the particular interest of thesepositive sections, for instance in Reeb dynamics.We discuss branched coverings between surfaces of section for the Hopf flow andthose of any other Seifert fibration of the 3-sphere S . Rather intriguingly, these Mathematics Subject Classification.
Symplectic Structures in Geometry,Algebra and Dynamics , funded by the DFG (Project-ID 281071066 – TRR 191). branched coverings can go either way, and both approaches can be used to computethe genus of positive d -sections with a Riemann–Hurwitz argument.We also relate the surfaces of section in S to algebraic curves in weighted com-plex projective planes. On the one hand, the degree-genus formula for such algebraiccurves can then be used to compute, once again, the genus of positive d -sections.Conversely, our direct geometric arguments for surfaces of section provide an alter-native proof of this degree-genus formula. These considerations have contributed tofilling a gap in the earlier versions of [6], and they give a more geometric alternativeto the arguments in [15].We hope that the explicit descriptions of global surfaces of section in the presentpaper will prove useful for the study of Besse manifolds (cf. Section 2.3) in Finslergeometry and Reeb dynamics.2. Seifert manifolds
Let M be a closed, oriented 3-manifold with a Seifert fibration M → B overa closed, oriented surface B . Many of the results in this paper also hold, mutatismutandis , when M or B is not orientable. However, for the interpretation of theSeifert fibres as the orbits of a flow, and to be able to speak of positive surfaces ofsection, the fibres have to be oriented.We refer the reader to [10] for the basic theory of Seifert fibred spaces.2.1. Seifert invariants.
We recall the notation and conventions for the Seifertinvariants from [3] that we shall use in the present paper. In the description of M as M = M (cid:0) g ; ( α , β ) , . . . , ( α n , β n ) (cid:1) , the non-negative integer g stands for the genus of B , the α i are positive integersgiving the multiplicities of the singular fibres, and each β i is an integer coprimewith the respective α i , describing the local behaviour near the singular fibre. A pair( α i , β i ) with α i = 1 does not actually give rise to a singular fibre, but it correspondsto a modification of the fibration that contributes to the Euler number.The non-singular fibres are called regular. Near any regular fibre, the Seifertfibration is a trivial S -bundle.When we speak of a ‘Seifert manifold M ’, we always mean that a Seifert fibrationhas been chosen on the 3-manifold M . Some 3-manifolds admit non-isomorphicSeifert fibrations [3, 4].The topological interpretation of these invariants is as follows. Let B be thesurface with boundary obtained from B by removing the interior of n disjointdiscs D , . . . , D n ⊂ B . Denote by M = B × S → B the trivial S -bundleover B . Write S , . . . , S n for the boundary circles of B , with the opposite of theorientation induced as boundary of B . We write the (isotopy class of) the fibre in M as h = {∗} × S . On the boundary of M we consider the curves q i = S i × { } ,with 1 ∈ S ⊂ C . Let V i = D × S , i = 1 , . . . , n , be n copies of a solid torus withrespective meridian and longitude µ i = ∂D × { } , λ i = { } × S ⊂ ∂V i . The Seifert invariants ( α i , β i ) then encode the identifications(1) µ i = α i q i + β i h, λ i = α ′ i q i + β ′ i h, URFACES OF SECTION FOR SEIFERT FIBRATIONS 3 where integers α ′ i , β ′ i are chosen such that (cid:12)(cid:12)(cid:12)(cid:12) α i α ′ i β i β ′ i (cid:12)(cid:12)(cid:12)(cid:12) = 1 . The identifications (1) are equivalent to(2) h = − α ′ i µ i + α i λ i , q i = β ′ i µ i − β i λ i . Writing C i for the spine { } × S of the solid torus V i , we deduce from (2) thatthe fibre h is homotopic to α i C i in V i , and the boundary curve q i , to − β i C i . For α i >
1, the spine C i is the corresponding singular fibre.The Euler number of the Seifert fibration is defined as e = − n X i =1 β i α i . Equivalences of Seifert invariants.
Up to isomorphism, a Seifert fibrationis determined by the Seifert invariants. Different sets of Seifert invariants corre-spond to isomorphic Seifert fibrations if and only if they are related to each otherby the following operations, see [10, Theorem 1.5]:(o) Permute the n pairs ( α i , β i ).(i) Add or delete any pair ( α, β ) = (1 , α i , β i ) by ( α i , β i + k i α i ), where P ni =1 k i = 0.These operations allow one to write a Seifert manifold in terms of normalised Seifertinvariants M (cid:0) g ; (1 , b ) , ( α , β ) , . . . , ( α n , β n ) (cid:1) , where b ∈ Z and 0 < β i < α i . Notice that the Euler number is invariant under theseoperations. In general, we shall not be using normalised invariants. Occasionallyit is convenient to collect all pairs ( α i , β i ) with α i = 1 into a single pair (1 , b ), sothat the remaining pairs actually correspond to singular fibres.2.3. Besse flows.
Flows of non-singular vector fields where all orbits are closedare often referred to as
Besse flows , in particular in Reeb dynamics; see [5, 11, 14].In this context it is worth mentioning that, by a classical result of Epstein [2], anyBesse flow on a 3-manifold defines a Seifert fibration.3.
Global surfaces of section
The orientations of M and B in the Seifert fibration M → B induce an orien-tation on the fibres. Equivalently, we may think of the Seifert fibres as the orbitsof an effective, fixed point free S -action. A singular fibre with Seifert invariants( α, β ), α >
1, consists of points with isotropy group Z α = (cid:10) e π i /α (cid:11) < S .A global surface of section is an embedded compact surface Σ ⊂ M whoseboundary is a collection of Seifert fibres, and whose interior intersects all otherSeifert fibres transversely. The interior of Σ will intersect every regular fibre (exceptthose forming the boundary of Σ) in the same number d of points. A singular fibreof multiplicity α can also arise as a boundary component, or it will intersect theinterior of Σ in d/α points. For short, we speak of a d -section . Note 3.1.
Given a d -section Σ, the multiplicity α of a singular fibre not in theboundary of Σ divides d . B. ALBACH AND H. GEIGES
We orient Σ such that these intersection points are positive. The boundaryorientation of a component of ∂ Σ may or may not coincide with the orientation asa Seifert fibre. If the orientations coincide for every component of ∂ Σ, we call the d -section positive .See [1] for a variety of explicit surfaces of section for the Hopf fibration S → S .4. 1 -sections for Seifert fibrations We begin with a necessary criterion for the existence of a 1-section.
Lemma 4.1.
If a Seifert manifold admits a -section, then every singular fibre is aboundary fibre, and the corresponding pair ( α, β ) satisfies β ≡ ± α , dependingon whether the singular fibre is a positive or negative boundary component.Proof. The first claim follows from Note 3.1. For the second statement, we considera solid torus V around the singular fibre C as in Section 2. Let σ be the curve ∂V ∩ Σ, oriented as a boundary component of Σ \ Int( V ). The curve σ must behomotopic in V to ± C , depending on the sign of the boundary fibre C .The Seifert fibres lying in ∂V represent the fibre class h . Since these fibres inter-sect Σ positively in a single point, the intersection number σ • h on ∂V , computedwith respect to the oriented basis ( µ, λ ), equals − sic !). Since q • h = 1, this meansthat σ can be written as σ = − q + ah for some a ∈ Z , which in V is homotopic to( β + aα ) C . For this to equal ± C , we must have β ≡ ± α . (cid:3) This lemma implies that a Seifert manifold admitting a 1-section is of the form M (cid:0) g ; (1 , b ) , ( α , ± , . . . , ( α n , ± (cid:1) , where we may assume that the α i are greaterthan 1. The individual signs determine the topology of the Seifert manifold. For α i = 2, both signs are possible.We now show that manifolds of this form do admit a 1-section, and we cancharacterise those admitting a positive one. Theorem 4.2.
A Seifert manifold admits a -section with n singular boundaryfibres and b ± positive resp. negative regular boundary fibres if and only if it isisomorphic to M (cid:0) g ; (1 , b + − b − ) , ( α , ± , . . . , ( α n , ± (cid:1) . The sign in ( α i , ± ) de-termines the sign of the corresponding singular boundary fibre. Any two -sectionson a given Seifert manifold are isotopic, provided they have the same number ofpositive resp. negative regular boundary fibres. Example 4.3.
In [1, Section 2.5] the reader finds an explicit example of a pair ofpants 1-section for the Hopf fibration with two positive and one negative boundaryfibres, corresponding to the description of the Hopf fibration as the Seifert manifold M (cid:0)
0; (1 , , (1 , , (1 , − (cid:1) . Proof of Theorem 4.2.
Given a Seifert manifold of the described form, we constructa 1-section as follows. Remove (open) solid tori around the n singular fibres, and afurther b + + b − solid tori around regular fibres. The remaining part M of M is ofthe form B × S → B , and we take the constant section B × { } as the part ofour desired 1-section over B .For the gluing of a solid torus V corresponding to a singular fibre of type ( α, ± α ′ = ∓ β ′ = 0. This means we have the identifications h = ± µ + αλ, q = ∓ λ. URFACES OF SECTION FOR SEIFERT FIBRATIONS 5
Therefore, the oriented vertical annulus A in V with boundary ∂A = ± C ∓ λ = ± C + q will intersect each Seifert fibre in V \ { C } transversely in a single positivepoint, and it glues with B to extend the 1-section. (For the sign of the intersectionpoint, notice that − q = ± λ takes the role of σ in the preceding proof, and ± λ • h = − α >
1. Thus, in the same waywe may extend the 1-section over b ± solid tori glued according to the prescription(1 , ± a = 0 in the notationof the proof of Lemma 4.1, and β = ± (cid:3) d -sections for Seifert fibrations Notice that in Theorem 4.2 it was not necessary to assume α i >
1. We con-tinue to work with unnormalised Seifert invariants, although occasionally it maybe convenient to collect all pairs ( α i , β i ) with α i = 1 into a single term.5.1. A necessary criterion.
Again we start with a necessary criterion for theexistence of a section.
Lemma 5.1.
Suppose M (cid:0) g ; ( α , β ) , . . . , ( α n , β n ) (cid:1) admits a d -section. If the sin-gular fibre C = C i with invariants ( α, β ) = ( α i , β i ) , α i > , is not a boundary fibre,then α | d . If C is a boundary fibre, then α | ( dβ ∓ , where the sign depends on C being a positive or negative boundary. Remark 5.2.
For α > α and d are coprime), so it is determined a priori — provided a d -section exists — which singular fibres will be boundary fibres. Also, the sign ofthe boundary fibres is predetermined, unless α = 2. Proof of Lemma 5.1.
For C not being a boundary fibre, use Note 3.1. If C is aboundary fibre, we argue as in the proof of Lemma 4.1, except that now we have σ • h = − d . This implies that σ = − dq + ah for some a ∈ Z , which is homotopic in V to ( dβ + aα ) C , whence dβ + aα = ± (cid:3) The Z d -quotient. If we think of a Seifert fibration as a manifold M withan effective, fixed point free S -action, the subgroup Z d = (cid:10) e π i /d (cid:11) < S also actson M , and the quotient M/ Z d is again Seifert fibred. The regular fibres of M have length 2 π , a singular fibre of multiplicity α has length 2 π/α . In the quotient M/ Z d , the regular fibres have length 2 π/d , and a singular fibre of multiplicity α in M descends to a singular fibre of length 2 π gcd( α, d ) /αd , so its multiplicity is α/ gcd( α, d ).The Seifert invariants of M/ Z d can be computed from a section of M → B over B ⊂ B (in the notation of Section 2), since this descends to just such a section of M/ Z d → B , see [10, Proposition 2.5]. Proposition 5.3.
The Z d -quotient M = M/ Z d of M = M (cid:0) g ; ( α , β ) , . . . , ( α n , β n ) (cid:1) is the Seifert manifold M = M (cid:0) g ; ( α , β ) , . . . , ( α n , β n ) (cid:1) , B. ALBACH AND H. GEIGES where α i = α i / gcd( α i , d ) and β i = dβ i / gcd( α i , d ) . In particular, the Euler num-bers are related by e ( M → B ) = d · e ( M → B ) . (cid:3) The quotient map M → M is a branched covering, branched transversely to anysingular fibre with α i | d , where the branching index is α i .5.3. d -sections descend to the Z d -quotient. We now assume that M admitsa d -section, so that the conclusion of Lemma 5.1 holds. We permute the Seifertinvariants such that (for the appropriate k ∈ { , . . . , n } )(3) ( α i | ( dβ i ∓
1) and α i ∤ d for i = 1 , . . . , k,α i | d for i = k + 1 , . . . , n. The assumption α i ∤ d in the first case of (3) is not, strictly speaking, necessary. Itavoids the ambiguity where to place the α i that equal 1, but the formulas we shallderive also hold when we count some of the α i = 1 amongst the first alternative.For i = 1 , . . . , k , we write dβ i = a i α i + ε i with a i ∈ Z and ε i ∈ {± } . Then M = M (cid:0) g ; (1 , b ) , ( α , ε ) , . . . , ( α k , ε k ) (cid:1) with b = k X i =1 a i + n X i = k +1 dβ i α i . For α i = 2 and d odd, or for α i = 1 counted amongst the first alternative, thereis an ambiguity in the choice of ε i (and the corresponding a i ), but this does notinvalidate any of our statements.With Theorem 4.2 we see that M always admits a 1-section, and a positive oneprecisely when b ≥ ε i = 1 for i = 1 , . . . , k . The Euler number e of M → B equals e = − b − k X i =1 ε i /α i , so the condition for M to admit a positive 1-section can be rephrased as: all ε i equal 1 and e ≤ − P ki =1 /α i .In fact, by the same topological arguments as in [1, Proposition 3.2] one canshow that any d -section of M can be isotoped to one that is invariant under the Z d -action, and it then descends to a 1-section of M . Conversely, any 1-section of M lifts to a Z d -invariant d -section of M . Also, a d -section of M , if it exists, isdetermined up to isotopy by its boundary orientations. For simplicity, we shallrestrict the discussion of the topology of these d -sections to positive ones.5.4. The local behaviour near singular fibres.
To gain a better understandingof d -sections, we briefly describe their behaviour near singular fibres. We write( α, β ) for the invariants of the fibre C in question.If α | ( dβ ±
1) and α ∤ d , then C is a boundary component of the d -section. Near C , the d -section looks like a helicoidal surface, where the number of turns around C is controlled by the condition that any neighbouring regular fibre cuts the surfacein d (positive) points.If α | d , then the d -section near C consists of d/α transverse discs to C .The quotient map M → M restricts on any Z d -invariant d -section Σ of M toa branched covering Σ → Σ, with Σ a 1-section of M . Each singular fibre with URFACES OF SECTION FOR SEIFERT FIBRATIONS 7 α i | d gives rise to d/α i branch points of index α i over a single point in the interiorof Σ. Along the boundary components, regular or singular, the covering Σ → Σ isunbranched.5.5.
The topology of positive d -sections. Here is our main theorem concerningpositive d -sections. Theorem 5.4.
The Seifert manifold M = M (cid:0) g ; ( α , β ) , . . . , ( α n , β n ) (cid:1) admits apositive d -section if and only if, for some k ∈ { , . . . , n } (and after permuting theSeifert invariants), condition (3) is satisfied with the sign α i | ( dβ i − in the firstalternative, and the Euler number e satisfies (4) de + X α i ∤ d α i ≤ . This positive d -section, if it exists, is unique up to isotopy. Unless α i | d for all i and e = 0 , it is a connected, orientable surface of genus − d (cid:16) − g + ( d − e + n X i =1 α i − n (cid:17) − X α i ∤ d (cid:16) − α i (cid:17) with − de + X α i ∤ d (cid:16) − α i (cid:17) boundary components.Proof. The left-hand side of inequality (4) equals the integer − b , so the inequalityis simply the reformulation of the condition b ≥
0. The sign condition on the firstalternative of (3) is equivalent to requiring ε = . . . = ε k = 1.The number of boundary components of the positive d -section Σ equals that ofthe 1-section Σ of M , which by Theorem 4.2 is b + k = − de + X α i ∤ d (cid:16) − α i (cid:17) . If Σ has empty boundary, which happens when e = 0 and α i | d for all i = 1 , . . . , n ,it may be disconnected, depending on the branching of the map Σ → Σ; an obviouscase is a trivial S -bundle M = B × S → B . When the boundary is non-empty, Σis connected by the argument in the proof of [1, Theorem 4.1]. For the remainderof the proof we assume this latter case.Write ˆΣ for the closed surface obtained by gluing b + k copies of D to the d -section Σ of M along its boundary components, and ˆΣ for the closed surface ob-tained in the same way from the 1-section Σ of M . This gluing is done abstractly,that is, the surfaces ˆΣ and ˆΣ are no longer embedded surfaces in M or M , respec-tively. The Euler characteristic of ˆΣ equals 2 − g , since the interior of the 1-sectionis a copy of the base B with b + k points removed.The surface ˆΣ is a branched cover of ˆΣ as follows. Each of the discs glued toone of the b + k boundary components gives rise to a single branch point in ˆΣ ofindex d , since both a regular fibre in M as well as a singular fibre with α i coprimeto d is a d -fold cover of the corresponding fibre in M . B. ALBACH AND H. GEIGES
Each singular fibre of M with α i | d gives rise to a point in ˆΣ covered by d/α i points of branching index α i . This statement about covering points and indicesremains true for α i = 1.Thus, the Riemann–Hurwitz formula for the branched covering ˆΣ → ˆΣ gives χ ( ˆΣ) = d (cid:0) − g − ( b + k ) − ( n − k ) (cid:1) + ( b + k ) + n X i = k +1 dα i . The formula for the genus (2 − χ ( ˆΣ)) / d -section follows with some simplearithmetic. Notice that the formula is invariant under adding or removing pairs(1 ,
0) from the Seifert invariants. (cid:3)
Examples.
Here are some elementary applications of Theorem 5.45.6.1.
The Hopf flow on S . The Hopf fibration S → S has the description M (cid:0)
0; (1 , (cid:1) as a Seifert manifold, with e = −
1, see [1, Section 2.4]. The followingis [1, Theorem 4.1].
Example 5.5.
For any natural number d , the Hopf fibration admits a uniquepositive d -section, which has genus ( d − d − / d boundary components.5.6.2. Seifert fibrations of S . According to [3, Proposition 5.2], a complete list ofthe Seifert fibrations of S is provided by M (cid:0)
0; ( α , β ) , ( α , β ) (cid:1) , where α β + α β = 1 (in particular, α and α are coprime). In order to avoidduplications in this list, one may assume α ≥ α and 0 ≤ β < α . For ourpurposes, this is not relevant, and so we shall not insist on this choice, since weprefer to retain the symmetry in α and α .The corresponding S -action on S ⊂ C is given by( A , ) θ ( z , z ) = (cid:0) e i α θ z , e i α θ z (cid:1) , θ ∈ R / π Z . Notice the choice of indices. The fibre C = { z = 0 } ∩ S through the point (1 , α , the fibre C = { z = 0 } ∩ S through (0 , α .The Euler number of this fibration is e = − β α − β α = − α α . The Hopf fibration corresponds to the choice α = α = 1.We shall discuss these Seifert fibrations in more detail in Section 7. The followingexample answers a question posed to us by Christian Lange. Example 5.6.
The Seifert fibration M (cid:0)
0; ( α , β ) , ( α , β ) (cid:1) of S admits a positive α α -section, with boundary a single regular fibre, of genus ( α − α − / α α -section explicitly. A more systematictreatment, using algebraic curves in weighted projective spaces, will be given inSections 7 and 8. We replace the round 3-sphere S ⊂ C by its diffeomorphic copy S α ,α := (cid:8) ( z , z ) ∈ C : | z | α + | z | α = 1 (cid:9) , URFACES OF SECTION FOR SEIFERT FIBRATIONS 9
The S -action is as in ( A , ), and we write C α ,α , C α ,α for the fibres throughthe points (1 ,
0) and (0 , ρ α ,α : S α ,α −→ S ( z , z ) ( z α , z α ) . We have ρ α ,α (cid:0) e i α θ z , e i α θ z (cid:1) = (cid:0) e i α α θ z α , e i α α θ z α (cid:1) , so each S -orbit on S α ,α maps α α : 1 to a Hopf orbit on S , except for thetwo exceptional orbits C α ,α i , which are α i -fold coverings of the Hopf orbits C i , i = 1 ,
2. So ρ α ,α is a covering branched along (and transverse to) C α ,α , C α ,α with branching index α , α , respectively. Note 5.7.
Points with the same image under ρ α ,α do indeed come from the same S -orbit on S α ,α . For instance, the point (cid:0) e π i /α z , z (cid:1) lies on the same orbit as( z , z ), since there is a k ∈ Z with kα ≡ α , and then with θ := 2 πk/α we have θ ( z , z ) = (cid:0) e π i /α z , z (cid:1) .It follows that the quotient map π , : S α ,α → S α ,α /S = C P is given by π , ( z , z ) = [ z α : z α ] ∈ C P = S . Observe that the preimage of a point [ w : w ] ∈ C P , where we may assume that | w | + | w | = 1, consists of the points (cid:0) λ ( w ) /α , λ ( w ) /α (cid:1) ∈ S α ,α for somefixed choice of roots and λ , λ ∈ S ⊂ C with λ α = λ α , which means we canwrite λ = e i α θ and λ = e i α θ for a suitable θ ∈ R / π Z .The element e π i /α α ∈ S defines a Z α α -action on S α ,α . In other words,this is the action generated by( z , z ) (cid:0) e π i /α z , e π i /α z (cid:1) . By the Chinese remainder theorem, for k ∈ { , , . . . , α α − } we obtain the α α independent choices of roots of unity (cid:0) e π i k/α , e π i k/α (cid:1) , so the map ρ α ,α is infact the quotient map under this Z α α -action.Summing up, we have the following commutative diagram: S α ,α S = S α ,α / Z α α ρ α ,α ✲ C P π , ❄ π Hopf ✛ Remark 5.8.
The quotient S α ,α / Z α α can also be recognised as S with theHopf fibration by computing its Seifert invariants. By Proposition 5.3 we have S α ,α / Z α α = M (cid:0)
0; (1 , α β ) , (1 , α β ) (cid:1) = M (cid:0)
0; (1 , , (1 , α β + α β ) (cid:1) = M (cid:0)
0; (1 , , (1 , (cid:1) , which describes the diagonal action of S on S ⊂ C .For the Hopf fibration, it is easy to write down a disc-like 1-section, where theboundary fibre sits over the point [1 : 1] ∈ C P . Lifting this to S α ,α , one finds an α α -section as the image of the α α -valued map C ∋ z (cid:16) z − p | z − | + | z | (cid:17) /α , (cid:16) z p | z − | + | z | (cid:17) /α ! ∈ S α ,α . As z → ∞ , this becomes asymptotic to the (regular) fibre over [1 : 1] ∈ C P . For z = 0 the map is α -valued. This corresponds to the α points over [1 : 0] ofbranching index α . Similarly, the point z = 1 maps to the α branch points ofindex α .5.6.3. Existence of d -sections. As a corollary of Theorem 5.4 and the discussion inSection 5.3 we mention the following existence statement, which is in response toa question by Umberto Hryniewicz.
Corollary 5.9.
Every Seifert manifold admits a d -section for a suitable positiveinteger d . It admits a positive d -section if and only if its Euler number is non-positive.Proof. For the Seifert manifold M = M (cid:0) g ; ( α , β ) , . . . , ( α n , β n ) (cid:1) we may choose d as a common multiple of the α i . Then the Z d -quotient is of the form M (cid:0) g ; (1 , b ) (cid:1) .This admits a 1-section, which can be lifted to a d -section of M .For this choice of d , condition (4) will be satisfied if and only if e ≤
0. If e ispositive, (4) will not be satisfied for any choice of d . (cid:3) Remark 5.10.
Changing the orientation of M amounts to changing the sign ofthe Euler number. So a Seifert manifold always admits a positive d -section for atleast one of the two orientations. If e = 0, the Seifert fibration can be realised as aBesse Reeb flow [11]. 6. Weighted projective planes
In this section we recall the bare essentials of weighted projective planes that weshall need to give explicit realisations of surfaces of section for the Seifert fibrationsof S . For more details see [6].6.1. The definition of weighted projective planes.
Let a , a , a ∈ N be atriple of pairwise coprime positive integers. The quotient of C \ { (0 , , } underthe C ∗ -action given by λ ( z , z , z ) = ( λ a z , λ a z , λ a z )is called the (well-formed) weighted complex projective plane P ( a , a , a )with weights a , a , a . The equivalence class of a point ( z , z , z ) is written as[ z : z : z ] a , where a = ( a , a , a ).The space P ( a , a , a ) is a complex manifold of complex dimension 2, except forthree cyclic singularities at the points [1 : 0 : 0] a , [0 : 1 : 0] a and [0 : 0 : 1] a . Indeed,the subset U := (cid:8) [ z : z : z ] a : z = 0 (cid:9) URFACES OF SECTION FOR SEIFERT FIBRATIONS 11 can be identified with the quotient C / Z a under the action of the cyclic group Z a < S ⊂ C given by ζ ( w , w ) = ( ζ a w , ζ a w ) , ζ ∈ Z a . If we write [ w , w ] a for the equivalence class of ( w , w ) in this cyclic quotient,the claimed identification is given by U −→ C / Z a [ z : z : z ] a [ z − a /a z , z − a /a z ] a . The ambiguity in the choice of a root of order a is accounted for by the equivalencerelation on the right. This map is well defined, with inverse[ w , w ] a [1 : w : w ] a . There are analogous descriptions for the subsets U i = { z i = 0 } ⊂ P ( a , a , a ) for i = 1 , Algebraic curves in weighted projective planes.
A complex polynomial F in the variables z , z , z is called a degree - d weighted-homogeneous poly-nomial (with a choice of a , a , a understood) if F ( λ a z , λ a z , λ a z ) = λ d F ( z , z , z )for some d ∈ N and all ( z , z , z ) ∈ C and λ ∈ C ∗ . Then the zero set { F = 0 } isa well-defined subset of P ( a , a , a ).The polynomial F is called non-singular if there are no solutions to the systemof equations F = ∂F∂z = ∂F∂z = ∂F∂z = 0in C \ { (0 , , } . Proposition 6.1. If F is non-singular, then { F = 0 } ⊂ P ( a , a , a ) is a smoothRiemann surface.Proof. The proof is completely analogous to the usual proof for algebraic curves in C P = P (1 , , P ( a , a , a ). However, this does not pose a problem, since these singularitiesare cyclic. A holomorphic chart around 0 ∈ C / Z a i is given by z z a i , so on thecomplex curve { F = 0 } these points are non-singular. (cid:3) Seifert fibrations of the -sphere We now give a systematic and comprehensive description of the positive d -sections for all Seifert fibrations of S . Weighted projective lines appear naturallyas the base of these fibrations. We first give a description of global surfaces ofsection using the topological arguments from Section 5. A description in terms ofalgebraic curves in weighted projective planes will follow in Section 8. An alternative relation with the Hopf fibration.
As we shall see present-ly, it is convenient to replace the S -action ( A , ) by( A , ) θ ( z , z ) = (cid:0) e i α θ z , e i α θ z (cid:1) , θ ∈ R / π Z . As before, α , α may be any pair of coprime natural numbers, and the S -actions( A , ) constitute a complete list of the Seifert fibrations of the 3-sphere. In contrastwith Section 5.6.2, we now regard these as actions on the round S .Nonetheless, the spheres S α ,α again have to play their part, and as before weconsider the map ρ α ,α defined in (5). We now compute ρ α ,α (cid:0) e i θ z , e i θ z (cid:1) = (cid:0) e i α θ z α , e i α θ z α (cid:1) , which tells us that each Hopf S -orbit on S α ,α is mapped 1 : 1 to an orbit of( A , ) on S , except for the Hopf orbits C α ,α i , which are α i -fold coverings of theexceptional A , -orbits C i , i = 1 ,
2. Each regular A , -orbit is covered by α α Hopf orbits.
Remark 7.1.
Beware that in the set-up of Section 5.6.2, the orbits C α ,α , C α ,α were of multiplicity α , α , respectively, whereas here the orbits C , C are of mul-tiplicity α , α , respectively.Observe that the quotient S /S under the A , -action can be naturally identi-fied with the weighted projective line P ( α , α ), which is topologically a 2-sphere,and metrically an orbifold with an α - and an α -singularity. The quotient map π , : S → S /S = P ( α , α ) is given by π , ( z , z ) = [ z : z ] ( α ,α ) ∈ P ( α , α ) . In conclusion, these maps fit together into the following commutative diagram,where ρ α ,α sends [ z : z ] ∈ C P to [ z α : z α ] ( α ,α ) ∈ P ( α , α ): S α ,α ρ α ,α ✲ S = S α ,α / Z α α C P π Hopf ❄ ρ α ,α ✲ P ( α , α ) π , ❄ Remark 7.2.
Contrast this diagram with the one in Section 5.6.2. Whereas theearlier diagram was used to lift a 1-section of π Hopf to an α α -section of π , , thediagram here will allow us to lift d -sections of π , , for all admissible d , to d -sectionsof π Hopf .7.2.
Positive d -sections. Before we discuss the lifting of d -sections, we are goingto derive the classification of the positive d -sections for the Seifert fibrations of S as a corollary of Theorem 5.4. Corollary 7.3.
Table 1 constitutes a complete list of the positive d -sections Σ ofthe Seifert fibrations of S defined by ( A , ) . The expression ∂ Σ) in Table 1 stands for the number of boundary componentsof the d -section Σ. In the first line, in case α i equals 1, the regular fibre C i maywell be a component of ∂ Σ. URFACES OF SECTION FOR SEIFERT FIBRATIONS 13 d ∂ Σ) C C genus(Σ) kα α , k ∈ N k ∂ Σ ∂ Σ ( kα − kα − − k kα α + α , k ∈ N ; k + 1 ⊂ ∂ Σ ∂ Σ ( kα +1)( kα − − k α > kα α + α , k ∈ N ; k + 1 ∂ Σ ⊂ ∂ Σ ( kα − kα +1)+1 − k α > kα α + α + α , k ∈ N ; k + 2 ⊂ ∂ Σ ⊂ ∂ Σ ( kα +1)( kα +1) − − k α , α > Table 1.
Positive d -sections Σ for the Seifert fibrations of S . Proof of Corollary 7.3.
We need to check under which assumptions on d the nec-essary and sufficient conditions (3) and (4) are satisfied. Recall that we can writethe Seifert fibrations of S as M (cid:0)
0; ( α , β ) , ( α , β ) (cid:1) with α β + α β = 1, andthe Euler number of this fibration is e = − /α α .If both α and α divide d , then d = kα α for some k ∈ N , and the left-handside of (4) equals − k . Notice that one or both of the α i may equal 1.Next we consider the case that α | ( dβ −
1) (and α ∤ d ) and α | d (including thecase α = 1). The divisibility condition α | ( dβ −
1) is equivalent to α being adivisor of dβ − α β = dβ − α β = β ( d − α ) , and hence to α | ( d − α ). This means that d has to be of the form d = kα α + α for some k ∈ N = { , , , . . . } . The case with the roles of α and α exchanged isanalogous.If α i | ( dβ i − i = 1 ,
2, then the same argument as before shows that α | ( d − α )and α | ( d − α ). For α i ∤ d (and in particular α i >
1) these divisibility conditionsare equivalent to d = kα α + α + α .The remaining data in Table 1 on ∂ Σ) and the genus of Σ follow fromLemma 5.1 and Theorem 5.4. (cid:3)
Remark 7.4.
Notice that every Seifert fibration of S admits a disc-like d -section,that is, a section of genus zero with one boundary component. This underlines thesubtlety of the construction in [13] of a Reeb flow on S without any disc-like globalsurface of section.7.3. Lifting d -sections. We now want to describe an alternative method for com-puting the genus of d -sections, based on lifting a d -section of π , to a d -section of π Hopf in the diagram from Section 7.1, and then using previous results on surfacesof section for the Hopf flow [1].Thus, let Σ ⊂ S be a positive d -section for π , . Write e Σ ⊂ S α ,α for the liftof Σ under the α α -fold covering ρ α ,α : S α ,α → S . As in Table 1, we write k ∈ N for the number of regular boundary components of Σ. Additionally, one orboth of C , C may belong to the boundary. (If neither of them does, then k ≥ As observed in Section 7.1, each regular fibre of π , is covered 1 : 1 by α α Hopf fibres. This contributes kα α boundary components to e Σ. Away from theboundary, e Σ is a d -section for π Hopf for the same reason. In particular, if C , C ∂ Σ, then e Σ is a positive d -section for the Hopf flow.If C i ⊂ ∂ Σ for at least one i ∈ { , } , the situation is more complicated. A pointon C i has α i preimages on C α ,α i under ρ α ,α , whereas each point on Σ \ ( C ∪ C )has α α preimages. It follows that along C α ,α i the lifted surface e Σ is no longerembedded, but looks like a rather slim open book with spine C α ,α i and α j pages,where ( i, j ) is a permutation of (1 , ρ α ,α to ensure that the lifted surface e Σ is a d -section for the Hopf flow.7.3.1. The behaviour of e Σ near an exceptional boundary fibre. Even though this isnot relevant for the genus calculations, it is worth understanding the behaviour of e Σ near a boundary fibre C α ,α , say. Let σ be the curve ∂V ∩ Σ as in the proof ofLemma 4.1 or 5.1, with V a tubular neighbourhood of C ⊂ S . Write ˜ σ for thelift of σ to e Σ. This is a (possibly disconnected) curve on the boundary of a tubularneighbourhood e V of C α ,α ⊂ S α ,α .Write ˜ µ for the meridian on ∂ e V , and ˜ λ for the longitude determined by the classof a Hopf fibre. Since C α ,α is an α -fold cover of C , but Int( e Σ) is an α α -foldcover of Int(Σ) near the boundary, the curve ˜ σ lies in the class m ˜ µ + α ˜ λ for some m ∈ Z . The fact that e Σ is a d -section (away from the boundary) translates into − d = ˜ σ • ˜ λ = m , cf. the proof of Lemma 5.1. Thus, if α and d are coprime (whichin the global picture is equivalent to C being a boundary fibre, unless α = 1), thecurve ˜ σ is indivisible, and hence connected. Otherwise, by the alternative in (3), α is a divisor of d , and ˜ σ is made up of α parallel curves on ∂ e V .When we remove thin collar neighbourhoods of C α ,α ⊂ e Σ and C ⊂ Σ, then inthe first case we obtain a covering e Σ → Σ with a single boundary component corre-sponding to C α ,α , which is an α α -fold covering of the corresponding boundarycomponent of Σ. In the second case, C α ,α gives rise to α boundary componentsin e Σ, each of which is an α -fold cover of one and the same boundary componentof Σ.Beware, however, that this modification of e Σ into a surface with embeddedboundary need not give the correct topology of a positive d -section for the Hopfflow. Before we describe the proper desingularisation of e Σ, we analyse the casewithout exceptional fibres in the boundary.7.3.2.
The case C i ∂ Σ , i = 1 , . When C is not contained in the boundaryof Σ, the d -section Σ is near C made up of d/α discs transverse to C . Everyneighbouring regular fibre cuts these discs α times, and hence in a total of d points. Since C α ,α → C is an α -fold covering, the lifted surface e Σ cuts C α ,α in d points, and the branching index of e Σ → Σ at these points equals α .Similarly, each of the d/α intersection points of Σ with C has α preimages in e Σ of branching index α .Each of the k boundary components of Σ is covered 1 : 1 by α α boundarycomponents of e Σ, so the branched covering e Σ → Σ extends to a covering of closedsurfaces without additional branching points.
URFACES OF SECTION FOR SEIFERT FIBRATIONS 15
In [1, Theorem 4.1] it was shown that a positive d -section e Σ for the Hopf flowis a connected, orientable surface of genus ( d − d − / d boundarycomponents. Write g for the genus of Σ. Then the Riemann–Hurwitz formula gives2 − ( d − d −
2) = α α (cid:16) − g − dα − dα (cid:17) + 2 d. With d = kα α this yields the formula in the first line of Table 1.7.3.3. The case C i ⊂ ∂ Σ . We illustrate the situation when one or both of C , C are boundary fibres by considering the case C ⊂ ∂ Σ and C ∂ Σ. The othercases can be treated in an analogous fashion.In order to desingularise the lifted surface e Σ near C α ,α , we modify the copy S α ,α of the 3-sphere and the quotient map ρ α ,α . For some small ε >
0, we define S α ,α ,ε := (cid:8) ( z , z ) ∈ C : | z | α + | z α − εz α | = 1 (cid:9) . This is still a (strictly) starshaped hypersurface with respect to the origin, andhence a diffeomorphic copy of the 3-sphere.In the commutative diagram of Section 7.1, we replace S α ,α by S α ,α ,ε , andthe map ρ α ,α by ρ εα ,α ( z , z ) := (cid:0) z α , z α − εz α (cid:1) . The preimage of a point ( w , w ) ∈ S under ρ εα ,α consists of the points( z , z ) = (cid:0) w /α , ( w + εw α /α ) /α (cid:1) , where the choice of root w /α is the same in both components. If w = 0 and w + εw α /α = 0 for all choices of w /α , this gives us α α preimages. Hence,generically a fibre of π , is covered 1 : 1 by α α Hopf fibres, as for ρ α ,α . (i) The desingularised lift of C . While under ρ α ,α the fibre C of π , was covered α times by C α ,α , and in particular each point on C had only α preimages, under ρ εα ,α the point ( w , ∈ C is covered by α α points (cid:0) w /α , ε /α w /α (cid:1) . Noticethat the α different choices of w /α yield points on the same Hopf fibre, so thepreimage of C consists of α Hopf fibres in the boundary of e Σ, each of which is an α -fold cover of C . This allows us to pass to closed surface ˆ e Σ , ˆΣ as in the proof ofTheorem 5.4, and the boundary component C ⊂ ∂ Σ contributes α points in ˆ e Σ ofbranching index α sitting over a single point in ˆΣ. (ii) Branch points with z = 0 . One case when a point in S has fewer than α α preimages is when w = 0, that is, points on C . The fibre C of π , is covered α times by the Hopf fibre C α ,α . Since C is a fibre of multiplicity α , the d -sectionΣ cuts it in d/α points. As in Section 7.3.2 this gives us d branch points upstairson C α ,α of branching index α . (iii) Branch points with z = 0 . The only other case when a point ( w , w ) ∈ S has fewer than α α preimages under ρ εα ,α is when w + εw α /α = 0 for somechoice of w /α . These correspond to points upstairs with z = 0, that is, pointson C α ,α . The restriction of ρ εα ,α to C α ,α is given by (cid:0) e i θ , (cid:1) (cid:0) e i α θ , − ε e i α θ (cid:1) , so C α ,α maps injectively to the π , -fibre through the point (1 , − ε ). We choose ε > λ = e i θ . We now ask: what are the preimages of( w , w ) = ( λ α , − ελ α ), except for the point ( λ, ∈ C α ,α ? The equations (cid:26) z α = λ α ,z α − εz α = − ελ α have the solutions ( z = λ e π i k/α , k ∈ { , . . . , α − } z = α √ ελ (cid:0) e π i k/α − (cid:1) /α , where α √ ε denotes the root in R + . For each k ∈ { , . . . , α − } ( sic !) and given λ we obtain α solutions on different Hopf fibres. For k = 0 we obtain the uniquesolution ( λ, α .The d -section Σ cuts the π , -fibre through the point (1 , − ε ) in d points. Foreach of these d points we obtain a single branch point upstairs of index α . Computing the genus of e Σ . We can now compute the genus of e Σ by counting thebranch points of e Σ → Σ upstairs. We have(i) α branch points of index α ,(ii) d branch points of index α ,(iii) d branch points of index α .The Riemann–Hurwitz formula then tells us that2 − ( d − d −
2) + α ( α −
1) + d ( α −
1) + d ( α −
1) = α α (2 − g ) . With d = kα α + α we obtain the formula in the second line of Table 1.8. Surfaces of section and algebraic curves
We now wish to relate surfaces of section for Seifert fibrations of S to algebraiccurves in weighted projective planes P (1 , α , α ). The degree-genus formula forsuch algebraic curves [15] then allows us to confirm Corollary 7.3. Conversely, ourresults on positive d -sections can be read as an alternative proof of the degree-genusformula. Moreover, the lifting of d -sections as described in Section 7.3 permits anatural interpretation as the lifting of algebraic curves from P (1 , α , α ) to C P .8.1. Weighted homogeneous polynomials.
As explained in Section 6.2, an al-gebraic curve in P (1 , α , α ) is defined by a (1 , α , α )-weighted homogeneous poly-nomial. We want to consider such polynomials of the form(6) F ( z , z , z ) = f ( z , z ) − z d , The following lemma is the analogue of the factorisation of an unweighted homo-geneous polynomial in two variables into linear factors [12, Lemma 2.8].
Lemma 8.1. If F is non-singular, then f is of the form f ( z , z ) = ( b α z α − a α z α ) · . . . · ( b α k z α − a α k z α ) · z ε · z ε with ε , ε ∈ { , } and [ a i : b i ] ( α ,α ) ∈ P ( α , α ) , i = 1 , . . . , k , pairwise distinctpoints different from [1 : 0] ( α ,α ) and [0 : 1] ( α ,α ) . If at least one ε i equals , onemay have k = 0 .In particular, the degree d of F equals d = kα α + ε α + ε α . URFACES OF SECTION FOR SEIFERT FIBRATIONS 17
Proof.
We have f (0 ,
1) = 0 if and only if f is divisible by z . Similarly, f (1 ,
0) = 0holds if and only if f is divisible by z . In either case, there can be at most onesuch factor each of z or z , lest F be singular.After dividing by these factors, in case they are present, we have a sum ofcomplex monomials c mn z m z n with α m + α n = const . and at least one non-zero monomial each without a z - or z -factor, respectively. From the identity α m + α n = α m ′ + α n ′ we deduce α | ( n − n ′ ) and α | ( m − m ′ ), which impliesthat f is of the form f ( z , z ) = z ε · z ε · k X i =0 c i z iα z ( k − i ) α with c , c k = 0.The sum in this description is a homogeneous polynomial of degree k in z α and z α , and hence, by [12, Lemma 2.8], has a factorisation into linear factors in z α and z α . The condition on the [ a i : b i ] ( α ,α ) is equivalent to f having no repeatedfactors. (cid:3) Constructing d -sections from algebraic curves. The following is the gen-eralisation of [1, Proposition 5.1] from the Hopf flow to all Seifert fibrations of S . Proposition 8.2.
The complex algebraic curve C = { F = 0 } in the weightedcomplex plane P (1 , α , α ) , where F is a weighted homogeneous polynomial of theform (6) , defines a positive d -section for the flow ( z , z ) (cid:0) e i α θ z , e i α θ z (cid:1) on S if and only if F is non-singular. The proof of this proposition is analogous to that of [1, Proposition 5.1], and weonly sketch the necessary modifications.In C \{ (0 , } we consider the weighted ‘radial’ projection to S given by sending( z , z ) to the unique point on S of the form ( z /r α , z /r α ), r ∈ R + . Thisprojection map is smooth. We define weighted ‘punctured radial complex planes’for ( a, b ) ∈ S ⊂ C as P a,b := (cid:8) ( az α , bz α ) : z ∈ C ∗ (cid:9) . Under the radial projection we have, for z = r e i θ , (cid:0) az α , bz α (cid:1) (cid:0) e i α θ a, e i α θ b (cid:1) , that is, P a,b projects to the A , -orbit through ( a, b ) on S .The affine part (cid:8) ( z , z ) ∈ C : f ( z , z ) = 1 (cid:9) can then be shown to intersecteach P a,b in d points that do not lie on a real radial ray, and hence project to d distinct points on the corresponding A , -orbit on S , except in the following cases.- Solutions of the equation f ( z ,
0) = 1 or f (0 , z ) = 1 correspond to C or C not being a boundary fibre, and they give rise to d/α or d/α intersectionpoints, respectively.- Solutions to the equation f ( z , z ) = 0, that is, points at infinity on C ,correspond to boundary fibres.The correct asymptotic behaviour near the boundary fibres can be demonstratedwith the explicit factorisation of f given by Lemma 8.1. Branched coverings of algebraic curves.
The map ρ α ,α : S α ,α → S has a natural extension to ρ α ,α : C P −→ P (1 , α , α )[ z : z : z ] [ z : z α : z α ] (1 ,α ,α ) . A weighted algebraic curve C = { F = 0 } ⊂ P (1 , α , α ) lifts to the algebraic curve e C = { e F = 0 } ⊂ C P , where e F is the homogeneous polynomial of degree d definedby e F ( z , z , z ) = F ( z , z α , z α ). The lift e C of a non-singular algebraic curve C may well be singular.Under this extended map ρ α ,α , the standard punctured plane (cid:8) ( az, bz ) : z ∈ C ∗ (cid:9) ⊂ C , with ( a, b ) ∈ S α ,α , maps to P a α ,b α .The discussion in Section 7.3 carries over to algebraic curves C = { F = 0 } with F as in (6), only changing notation to (weighted) homogeneous coordinates. Thus,we see the same branching behaviour in e C → C as for the (extended) surfaces ofsection, and the same process of desingularisation in case C contains one or bothof the points [0 : 1 : 0] (1 ,α ,α ) or [0 : 0 : 1] (1 ,α ,α ) at infinity.8.4. Computing the genus of algebraic curves via surfaces of section.
Hereis the general formula for computing the genus of a non-singular algebraic curve ina weighted projective plane, see [15, Corollary 3.5], [6, Theorem 5.3.7].
Theorem 8.3.
The genus of a non-singular algebraic curve of degree d in theweighted projective plane P ( a , a , a ) is given by g = 12 (cid:16) d a a a − d X i We are grateful to Christian Lange and Umberto Hryniewiczfor their interest in this work. Their questions and comments have contributed sig-nificantly to the writing of this paper. References [1] P. Albers, H. Geiges and K. Zehmisch , A symplectic dynamics proof of the degree-genusformula, arXiv:1905.03054 .[2] D. B. A. Epstein , Periodic flows on 3-manifolds, Ann. of Math. (2) (1972), 66–82.[3] H. Geiges and C. Lange , Seifert fibrations of lens spaces, Abh. Math. Sem. Univ. Hambg. (2018), 1–22.[4] H. Geiges and C. Lange , Erratum to “Seifert fibrations of lens spaces” – Fibrations overnon-orientable bases, arXiv:2009.08767 .[5] V. L. Ginzburg, B. Z. Gurel and M. Mazzucchelli , On the spectral characterization ofBesse and Zoll Reeb flows, arXiv:1909.03310 .[6] T. Hosgood , An introduction to varieties in weighted projective space, arXiv:1604.02441v5 . URFACES OF SECTION FOR SEIFERT FIBRATIONS 19 [7] U. L. Hryniewicz , Systems of global surfaces of section for dynamically convex Reeb flowson the 3-sphere, J. Symplectic Geom. (2014), 791–862.[8] U. L. Hryniewicz, P. A. S. Salom˜ao and R. Siefring , Global surfaces of section withpositive genus for dynamically convex Reeb flows, arXiv:2012.12055 .[9] U. L. Hryniewicz, P. A. S. Salom˜ao and K. Wysocki , Genus zero global surfaces of sectionfor Reeb flows and a result of Birkhoff, arXiv:1912.01078 .[10] M. Jankins and W. D. Neumann , Lectures on Seifert Manifolds , Brandeis Lecture Notes ,Brandeis University, Waltham, MA (1983); available at [11] M. Kegel and C. Lange , A Boothby–Wang theorem for Besse contact manifolds, ArnoldMath. J. , to appear.[12] F. Kirwan , Complex Algebraic Curves , London Math. Soc. Stud. Texts , Cambridge Uni-versity Press, Cambridge (1992).[13] O. van Koert , A Reeb flow on the three-sphere without a disk-like global surface of section, Qual. Theory Dyn. Systs. (2020), Paper No. 36, 16 pp.[14] M. Mazzucchelli and M. Radeschi , On the structure of Besse contact spheres, arXiv:2012.05389 .[15] P. Orlik and P. Wagreich , Equivariant resolution of singularities with C ∗ action, in: Pro-ceedings of the Second Conference on Compact Transformation Groups (Amherst, 1971),Part I, Lecture Notes in Math. , Springer-Verlag, Berlin (1972), 270–290.[16] P. A. S. Salom˜ao and U. L. Hryniewicz , Global surfaces of section for Reeb flows indimension three and beyond, Proceedings of the International Congress of Mathematicians (Rio de Janeiro, 2018), Vol. II, World Sci. Publ., Hackensack, NJ (2018), 941–967. Mathematisches Institut, Universit¨at zu K¨oln, Weyertal 86–90, 50931 K¨oln, Ger-many Email address ::