Identifying the Canonical Component for the Whitehead Link
aa r X i v : . [ m a t h . G T ] S e p Identifying the Canonical Component for the Whitehead Link
Emily LandesAugust 20, 2018
Abstract
In this paper we determine topologically the canonical component of the SL ( C ) character variety ofthe Whitehead link complement. Since the seminal work of Culler and Shalen, character varieties have proven to be a powerful tool forstudying hyperbolic 3-manifolds; for exampe, they provide efficient means of detecting essential surfaces inhyperbolic knot complements ([3], [2], [9]). For such a useful tool, the character variety itself is a rather basicconcept and yet determining explicit models for even the simplest hyperbolic knot complements, let alonelink complements, is a difficult problem. We are particularly interested in the canonical component (i.e thatcontaining a character of a discrete faithful representation) of these character varieties. Only recently haveexplicit models for the canonical components of a full family of hyperbolic knots been determined ([7]).In beginning to understand the explicit models for hyperbolic knots, our attention extends to constructingcanonical components of hyperbolic 2-component links. In [7] the authors determine the SL C charactervarieties for the twist knots. As the twist knots can be obtained by Dehn filling one of the cusps of theWhitehead link complement, we are naturally interested in constructing the particular character variety ofthe Whitehead link and studying the effect Dehn surgery has on the character variety. The focus of thispaper is to construct canonical component for the SL C character variety of the Whitehead link complementwhich we will do as the following theorem Theorem 1.
The canonical component of the character variety of the Whitehead link complement is arational surface isomorphic to P blown up at points. Understanding an algebro-geometric surface usually means understanding a surface up to birational equiv-alence. One reason for this is that a lot of the information about a variety is carried by the birationalequivalence class. For complex curves the birational equivalence class contains a unique smooth model (upto isomorphism). However, for complex surfaces, although there may be more than one smooth model ina birational equivalence class, there is a notion of a minimal smooth model. That is the smooth birationalmodel which has no ( −
1) curves.The minimal smooth model for the canonical component of the Whitehead link is P . We are able todetermine this minimal model from a particular projective model, S, for the canonical component. Thedefining polynomial for the canonical component of the Whitehead link cuts out an affine surface in C .1ompactifying this surface in P × P gives the singular surface, S . Although S is a singular surface, it isbirational to a conic bundle which in turn is birational to P .There are three models for the components of the algebraic sets we discuss in this paper. There is theaffine model defined by an ideal of polynomials in C . There is the projective model, which may or may notbe smooth, obtained by compactifying the affine models in P × P . Finally, if the projective model is notsmooth, there is the smooth projective model obtained resolving singular points of the projective model. Inthis paper when we refer to components of the character variety we mean the smooth projective model. Wewill specify when speaking of an affine model or a singular projective model.Although knowing the birational equivalence class is helpful in understanding how the variety behaves,determining the variety topologically requires understanding the isomorphism class. In this case, since S isrational, we can use the minimal model to determine the isomorphism class. Smooth surfaces birational to P are isomorphic either to P × P or to P blown-up at n points. For smooth surfaces, this isomorphism classcan be determined directly from the Euler characteristic. Although we can calculate the Euler characteristicfor S , it does not determine the isomorphism class since S is not smooth. Rather than work with the singularsurface S we will work with the smooth surface obtained by resolving the singularities of S .Away from the four singular points, S looks like a conic bundle in the sense that it is a bundle over P whose fibers are conics i.e. curves in P cut out by degree 2 polynomials. While the total space of a conicbundle is smooth a given fiber may not be. The model, S , for the canonical component of the Whiteheadlink has six fibers which are not smooth. Five of these fibers are degenerate i.e. have exactly one singularitywhereas the sixth fiber is a double line i.e. every point is a singularity. It is worth remarking that doublelines are a fairly rare feature to conic bundles. More precisely, all conics can be parameterized by P andthe double lines correspond to a codimension 3 subvariety ([4]). Hence a conic bundle with a double linefiber corresponds to a line which passes through a particular codimension 3 subvariety in P which is a rareoccurrence.Since S is birational to a conic bundle and so birational to P , it is isomorphic either to P × P or P blown-up at n points. For n ≤
8, the surfaces P blown-up at n points are nice algebro-geometric objets inthe sense that they exhibit only finitely many ( −
1) curves that is curves with self-intersection number − P blown-up at 10 points, has infintely many ( −
1) curves.The Whitehead link complement can be obtained by 1 / M br ). The manifold which results upon 1 /n Dehn filling on one ofthe cusps of M br is a hyperbolic two component 2-bridge link complement ([6]). For n = 1 , . . . , M br (1 /n ). For n = 1 , . . . ,
4, the character variety of M br (1 /n ) has a component which is a rational surface. Aside from theWhitehead link, the rational component(s) of these character varieties are not canonical components. Theorem 2.
For n = 2 , . . . , , the character variety of M br (1 /n ) has a component which is a rational surfaceisomorphic to P blown-up at 7 points. Similar to the canonical component of the Whitehead link, all of these rational surfaces exhibit a doubleline fiber. Unlike the Whitehead link, these rational surfaces have finitely many ( −
1) curves. The surface P blown-up at 7 points has exactly 47 ( −
1) curves. There is something to be said for the fact that examplesof this surface come from canonical components of character varieties of hyperbolic link complements. It isa start to understanding how topology and algebraic geometry fit together.We start, in Section 2, by defining the character variety. In Section 3 we provide some background inalgebraic geometry. The main theorem will be proved in Section 4. In Section 5 we discuss the charactervarieties for similar hyperbolic 2 component 2-bridge link complements.2
Preliminaries
Here we briefly describe the SL ( C ) representation and character varieties. Standard references on thisinclude [3] and [9]. For any finitely generated group
Γ = h g , . . . , g n | r , . . . , r m i , the set of SL ( C ) representations R (Γ) = Hom (Γ , SL C ) has the structure of an affine algebraic set [3]. We view this space as R (Γ) = { ( x , . . . , x ) ∈ ( SL ( C )) n | r j ( x , . . . , x n ) = I, j = 1 , . . . , m } . Notice that R (Γ) can be identified with r − ( I, . . . , I ) where r : ( SL ( C )) n → ( SL ( C )) m is the map r ( x ) = ( r ( x ) , . . . , r m ( x )). That R (Γ) is analgebraic set follows from the fact that r is a regular map. Identifying SL ( C ) n with a subset of C n , we canview R (Γ) as an algebraic set over C . We should note that the isomorphism class of R (Γ) does not dependon the group presentation and in general, R (Γ) is not irreducible. In fact the abelian representations (i.e.representations with abelian image) comprise a component of this algebraic set [3]. The character of a representation ρ : Γ → SL ( C ) is a map χ ρ : Γ → C defined by χ ρ ( γ ) = tr ( ρ ( γ )). The setof characters is defined to be X (Γ) = { χ ρ | ρ ∈ R (Γ)) } . For each g ∈ Γ there is a regular map τ g : R (Γ) → C defined by τ g ( ρ ) = χ ρ ( g ). Let T be the subring of the coordinate ring on R (Γ) generated by 1 and τ g , g ∈ Γ.In [3] it is shown that the ring T is finitely generated, for example by { τ g i g i ...g ik | ≤ i < i < · · · < i k ≤ n } .In particular any character χ ∈ X (Γ) is determined by its value on finitely many elements of Γ. As a result,for t , . . . , t s generators of T , the map t = ( t , . . . , t s ) : R (Γ) → C s defined by ρ ( t ( ρ ) , . . . , t s ( ρ )) inducesa map X (Γ) → C s . Culler and Shalen use the fact that this map is injective to show that X (Γ) inherits thestructure of an algebraic set ([3]).Let M be a hyperbolic manifold and let Γ = π ( M ). We refer to the affine algebraic set ˜ X ( π ( M )) asthe affine SL ( C ) character variety of M . The affine canonical component of ˜ X ( π ( M )) is the componentcontaining a character, χ , of a discrete faithful representation and is denoted by ˜ X ( π ( M )). We areinterested not so much in the affine model but in a closed projective model. By SL ( C ) character variety andcanonical component we mean the projective models X ( π ( M )) and X ( π ( M )) respectively. For hyperbolicknots and links as χ is a smooth point, X is unique [10]. In this context Thurston’s Hyperbolic DehnSurgery Theorem states that for an orientable, hyperbolic 3-manifold of finite volume, with n -cusps, X hascomplex dimension n .We will be particularly interested in studying X for hyperbolic two component two-bridge link comple-ments. These will be complex surfaces. A more detailed account of the character variety in this case will bediscussed in § The purpose of this section is to review the algebro-geometric concepts relevant to the main proof of the ispaper. For more details see [5] or [8] . 3 .1 Conic bundles
The character varieties of all of our examples have a component which is a conic bundle. A conic is a curvedefined by a polynomial over P of degree 2. Smooth conics have the genus zero ([8]) so are spheres. Adegenerate conic consists of two spheres intersecting one one point. In this paper the term conic bundle willbe used to mean a conic bundle over P i.e. over a sphere. Conic bundles are nice algebro-geometric objects.Whilst there is no classification of complex surfaces, there is a classification for the subclass of P bundlesover P which are slightly different than conic bundles in the sense that conic bundles may can have fiberswith singularities. Any P bundle over P comes from a projectivized rank 2 vector bundle over P . As therank 2 vector bundles are parametrized by Z , the P bundles over P are parameterized by Z . Each vectorbundle over P can be written as E = O ⊕ O ( − e ) ([1], [5]). Here O denotes the trivial rank 2 vector bundleover P and O ( − e ) denotes the vector bundle whose section has self-intersection number e . Proposition 3.1.
A conic bundle is a rational surface.Proof.
Any conic bundle T can be realized as a hypersurface defined by a polynomial f T of bidegree (2 , m )over P × P . In particular a generic fiber of the coordinate projection of T to P is a nondegenerate conic.This means that T is locally, and hence birationally, equivalent to P × P which is birationally equivalentto P .Another way to see that T is rational is by looking at the canonical divisor. The canonical divisor K T of T is the canonical divisor K P × P of P × P twisted by the divisor class of T , all restricted to T . Namely K T = ( O P × P ( − , − ⊗ O P × P (2 , m )) | S = O P × P ( − , m − | S . In particular, the canonical divisor K T corresponds to the line bundle O P × P ( − , m − | S the number of global sections of which are characterizedby the number of polynomials of bidgree ( − , m − − , m − T . The only surfaces in which the canonical bundle has no global sections arerational and ruled (i.e. birational to P and a fibration over a curve with P fibers). Corollary 3.1.
A conic bundle is isomorphic to either P × P or P blown-up at n points for some integer n . It is a known fact ([5] Chapter V) that for two birational varieties the birational equivalence betweenthem can be written as sequence of blow-ups and blow-downs. In particular P is birational to either P × P or to P blown-up at n points. Hence any rational surface is isomorphic to P × P or to P blown-up at n points. Blowing-up varieties at points is a standard tool for resolving singularities and determining isomorphismclasses of surfaces and we make repeated use of such in this paper.Since blowing-up is a local process, we can do all of our blow-ups in affine neighborhoods. For ourpurposes, understanding what it means to blow-up subvarieties of A and A at a point should be sufficient.For more details refer to [5] or [8].Intuitively blowing-up A at a point can be described as replacing a point in A by an exceptionaldivisor (i.e. a copy of P ). To understand this more concretely, we will describe the blow-up of A at theorigin. Consider the product A × P . Take x, y as the affine coordinates of A and t, u as the homogeneouscoordinates of P . The blow-up of A at (0 ,
0) is the closed subset Y = { [ x, y : t, u ] | xu = ty } in A × P . The4low-up comes with a natural map γ : Y → A which is just projection onto the first factor. Notice that thefiber over any point ( x, y ) = (0 , ∈ A is precisely one point in Y . However, the fiber over ( x, y ) = (0 , P worth of points in Y (i.e. { (0 , , t, u ) } ⊂ Y ). Since A − { (0 , } ≃ Y − γ − (0 , γ is a birational mapand A is birational to Y . Blowing-up A at a point p = 0 simply amounts to a change in coordinates.Suppose we want to blow up a subvariety X ⊂ A at a point, p . Take the blow-up Y of A at p . Thenthe blow-up Bl | p ( X ) of X at p is the closure γ − ( X − p ) in Y where γ is as described above. We note that Bl | p ( X ) is birational to X − p and if Bl | p ( X ) is smooth, γ − ( p ) will intersect Y in a zero dimensional variety.For our paper we need to understand how blowing-up a surface at a smooth point affects the Eulercharacteristic. Proposition 3.2.
The Euler characteristic of a surface X blown-up at a smooth point p is χ ( Bl | p ( X )) = χ ( X ) + 1 .Proof. To blow-up X at a smooth point p we work locally in an affine neighborhood about p . Near p , X is locally A at 0. Hence the result of blowing-up X at p is the same as blowing-up A at 0. Interms of the Euler characteristic this amounts to replacing a point with an exceptional P . In particular χ ( Bl p ( X )) = χ ( X − { p } ) + χ ( P ) = χ ( X ) + 1 .In order to resolve singularities we will need to blow-up subvarieties of A at a point. Taking x , x , x as affine coordinates for A and y , y , y as projective coordinates for P , the blow-up of A at the origin isclosed subvariety, Y ′ = { [ x , x , x : y , y , y ] | x y = x y , x y = x y , x y = x y } in A × P . Just asin the case of A , this blow-up comes with a natural map γ : Y ′ → A which is simply projection onto thefirst factor. Just as before, the fiber over any point ( x , x , x ) = (0 , , ∈ A is precisely one point in Y ′ .However, the fiber over ( x , x , x ) = (0 , , P worth of points in Y ′ (i.e. { (0 , , , y , y , y ) } ⊂ Y ′ ).Since A − { (0 , , } ≃ Y ′ − γ − (0 , , γ is a birational map and A is birational to Y ′ . Blowing-up A at a point p = 0 simply amounts to a change in coordinates. To blow up a subvariety X ⊂ A at a point, p . Take the blow-up Y ′ of A at p . Then the blow-up Bl | p ( X ) of X at p is the closure γ − ( X − p ) in Y ′ .We note that Bl | p ( X ) is birational to X − p and if Bl | p ( X ) is smooth, γ − ( p ) will intersect Y ′ in a smoothcurve.In this paper we obtain smooth surfaces by resolving singularities. As the Euler characteristic of thesesmooth surfaces helps us determine the isomorphism class we keep track of how blow-up singular pointsaffects the Euler characteristic. Proposition 3.3.
If the blow-up Bl | p ( X ) of a surface X at a singular point p is smooth, then the Eulercharacteristic of Bl | p ( X ) is χ ( Bl | p ( X )) = χ ( X ) + 2 g + 1 where g is the genus of the curve γ − ( p ) in Bl | p ( X ) .Proof. Away from the point p , X is isomorphic to Bl p ( X ) \ γ − ( p ). Hence, χ ( Bl p ( X )) = χ ( X − p )+ χ ( γ − ( p )).The preimage γ − ( p ) in Bl p ( X ) is a smooth codimension-1 subvariety of the fiber over p in Bl p ( A ). Sincethe fiber over p in Bl p ( A ) is a P , γ − ( p ) in Bl p ( X ) is a smooth curve of genus g . Hence χ ( γ − ( p )) = 2 g + 2and χ ( Bl p ( X )) = χ ( X − { p } ) + χ ( γ − ( p )) = χ ( X ) + 2 g + 1 . For the proof of propsition 4.2 we will use a total transform to extend a map φ between projective varieties.The description we provide here comes from [5] (pg 410). We begin by setting up some notation. Let X and Y be projective varieties. 5 efinition 3.1. A birational transformation T from X to Y is an open subset U ⊂ X and a morphism φ : U → Y which induces an isomorphism on the function fields of X and YSince different maps must agree on the overlap for different open sets, we take the largest open set U forwhich there is such a morphism φ . It is common to say that T is defined at the points of U and Definition 3.2.
The fundamental points of T are those in the set X − U .For G the graph of φ in U × Y , let G be the closure of G in X × Y . Let ρ : G → X and ρ : G → Y beprojections onto the first and second factors respectively. Definition 3.3.
For any subset Z ⊂ X the total transform of Z is T ( Z ) := ρ ( ρ − ( Z )) .For a point p ∈ U , T ( p ) is consistent with φ ( p ); while for a point p ∈ X − U , T ( p ) is generally larger than asingle point (in our examples it will be a copy of P ). A smooth curve C in P × P is cut out by a polynomial g which is homogenous in each of the P coordinates.We say g has bidgree ( a, b ) where a is the degree of g viewed as polynomial over the first factor and b isthe degree of g viewed as a polynomial over the second factor. In the proof of Theorem 1 we will determinethe number of intersections of two smooth curves in P × P based solely on the bidegrees of their definingpolynomials. Suppose C and C are two smooth curves cut out by irreducible polynomials g and g ofbidegrees ( a , b ) and ( a , b ) respectively. Counting multiplicities, C and C intersect in a b + a b points[5] 5.1). The geometric genus, p g , of a projective variety, S , is the dimension of the vector space of global sectionsΓ( X, ω k ) of the canonical divisor w k . For a complex curve, the geometric genus coincides with the topologicalgenus and can thus be used to topologically determine the character varieties of hyperbolic knot complements.Unfortunately for complex surfaces, the geometric genus does not carry as direct topological information (forinstance it appears as h , in the Hodge decomposition [4]). However, as it may still be helpful in determiningwhich varieties can arise as the character varieties of hyperbolic two component link complements, it it worthkeeping track of this value. For a hypersuface, Z , in P × P defined by a polynomial f of bidegree ( a, b ) thegeometric genus is p g ( Z ) = ( a − a − b − .We give a brief description of this here. As the group of linear equivalence classes of divisor of P × P is P ic ( P × P ) ∼ = Z × Z , we think of the divisors of P × P as elements of Z × Z . For a linear class withrepresentative divisor D on P × P , there is an associated vector space, L ( D ) of principal divisors E suchthat D+E is effective. The vector space L ( D ) is in one-to-one correspondence with the vector space of globalsections of the line bundle L ( D ) on P × P . As the vector space of global sections of L ( D ) corresponds to thespace of polynomials over P × P with the same bidegree as that which cuts out D , the restrictions of thesepolynomials to S which are nonzero on S , correspond to the vector space of global sections of D on S . Thatis to say the kernel of the surjective map L ( D ) ։ L( D ) | S is those polynomials which vanish on S . When D is the the canonical divisor K S of the surface S , assuming all the restricted polynomials are nonzero on S ,the geometric genus of the surface g g ( S ) is then just the dimension of the vector space of these polynomials.For the hypersurface S defined by f , we can use the adjunction formula to determine K S . Namely, K S = [ K P × P ⊗ O ( S )] | S . The canonical divisor K P × P of P × P is ( − , − ∈ Z × Z . and the divisor class6 ( S ) = ( a, b ) ∈ P ic ( P × P ) ∼ = Z × Z since f has bidgeree ( a, b ). Hence, K S = ( a − , b − K S = ( a − , b −
2) corresponds to a polynomials of bidgree ( a − , b − K S = ( a − , b − | S correspond to polynomials of bidgree ( a − , b − S is a hypersuface defined by the irreducible polynomial f , no polynomial of bidgeree ( a − , b −
2) canvanish on all of S . Hence, the geometric genus of the surface g g ( S ) is then just the dimension of the vectorspace of polynomials over P × P of bidegree ( a, b ). Determining this dimension is a matter of countingmonomials of bidegree ( a − , b −
3) for which there are ( a − a − b − . The affine varieties with which we are concerned are all hypersurfaces in C i.e. they are zero sets Z ( ˜ f )of a single smooth polynomial ˜ f ∈ C [ x, y, z ]. Finding the right projective completion is tricky, especiallywith complex surfaces since different projective completions may result in non-isomorphic models. It mightseem natural to take projective closures in P . One problem with compactifying in P is that, generally, thisprojective model has singularities which take more than one blow-up to resolve. Following the work of [7] itis more natural to consider the compactification in P × P . This compactification does result in a singularsurface. However, the singularities are manageable and away from the singularities this model has the nicestructure of a conic bundle. Hence, for these reasons, this is the projective model we choose to use for ourexamples.Given an affine variety Z ( ˜ f ) defined by a polynomial ˜ f ∈ C [ x, y, z ], we construct the projective closureby homogenizing ˜ f . Let a be the degree of ˜ f when viewed as a polynomial in variables x and y . Let b be the degree of ˜ f when viewed as a polynomial in the variable z . The projective model in P × P ofthe affine variety Z ( ˜ f ) is cut out by the homogenous polynomial f = u a w b ˜ f ( xu , yu , zw ) where x, y, u are P coordinates and z, w are P coordinates. Notice that every monomial which appears in f has degree a in the P coordinates and degree b in the P coordinates so f has bidgereee ( a, b ). Let W denote the complement of the Whithead link in S and let Γ W = π ( W ). Then Γ W = h a, b | aw = wa i where w is the word w = bab − a − b − ab . a b Figure 1: Whitehead link
Proposition 4.1. ˜ X ( W ) is a hypersurface in C . roof. To determine the defining polynomial for ˜ X ( W ) in C we look at the image of R ( W ) under the map t = ( t , . . . , t s ) : R ( W ) → C s as defined in §
2. We begin by establishing the defining ideal for R ( W ). Anyrepresentation of ρ ∈ R ( W ) can be conjugated so that¯ a = ρ ( a ) = (cid:18) m m − (cid:19) ¯ b = ρ ( b ) = (cid:18) s r s − (cid:19) The polynomials which define R ( W ) then come from the relation ¯ w ¯ a − ¯ a ¯ w = 0. Writing ρ ( w ) = (cid:18) w w w w (cid:19) ,we see that ¯ w ¯ a − ¯ a ¯ w = (cid:18) − w w + w ( m − − m ) − w w ( m − m − ) w (cid:19) Hence, the representation variety is cut out by the ideal h p , p i ⊂ C [ m, m − r, s, s − ] where p = w and p = w + w ( m − − m ) − w .For the Whitehead link p = m − s − r ( r − m r + ms − m s + 2 mr s − m r s − rs +4 m rs − m rs + m r s − ms + m s − mr s + 2 m r s − m rs + m rs ) p = m − s − ( − s )(1 + s )( r − m r + ms − m s + 2 mr s − m r s − rs + 4 m rs − m rs + m r s − ms + m s − mr s + 2 m r s − m rs + m rs )Neither p nor p are irreducible. In fact their GCD is nontrivial. Let p = GCD ( p , p ). That is p = m s ( r − m r + ms − m s + 2 mr s − m r s − rs + 4 m rs − m rs + m r s − ms + m s − mr s + 2 m r s − m rs + m rs )Setting g = p p = rs and g = p p = s −
1, we can view the representation variety as Z ( h g p, g p i ) = Z ( h g , g i ) ∪ Z ( h p i ). The ideal h g , g i defines the affine variety R a = { ( a, /a, ± , ± , } ⊂ C which is justtwo copies of A . The variety R a is precisely all the abelian representations of R ( W ). With the r coordinatezero and the s coordinate ±
1, any representation in R a sends b to ± I . We are interested the componentsof the representation variety which contain discrete faithful representations. All of these representations arein the subvariety R = Z ( h p i ) of R ( W ). Hence we are concerned with with the image of R under t in thecharacter variety.The coordinate ring T w for the Whitehead link character variety is generated by the trace maps { τ a , τ b , τ ab } With these generators the map t = ( τ a , τ b , τ ab ) : R → C is t ( ρ ) = ( m + m − , s + s − , ms + m − s − + r ) =( x, y, z ). Let X ′ denote the image of R under t . Then the map t : R → X ′ induces an injective map,8 ∗ : C [ X ′ ] → C [ R ] on the coordinates rings of X ′ and R . The coordinate ring of R is C [ R ] = C [ m, m − , s, s − , r ] / < p > so the image of C [ X ′ ] under t ∗ is C [ m, m − , s, s − , r ] / < p, x = m + m − , y = s + s − , z = ms + m − s − + r > which is isomorphic to C [ x, y, z ] / < ˜ f > where ˜ f = − xy − z + x z + y z − xyz + z . Since ˜ f is smooth, X ′ is the affine variety Z ( ˜ f ). Now X ′ is a smooth affine surface in C containing the surface ˜ X . Hence X ′ = ˜ X and so ˜ X is the hypersurface Z ( ˜ f ).Throughout the rest of this section we will denote compact model X ( W ) for the canonical componentof the Whitehead link by S . We use the compact model obtained by taking the projective closure in P × P .With x, y, u the P coordinates and z, w the P coordinates, this compactification for the canonical component S is defined by f = − w xy − u w z + w x z + w y z − wxyz + u z . This surface, S is not smooth. Ithas singularities at the four points: s , , , , s , , , , s , − , , , − s , , , , S obtained by resolving the singularities of S .We do this in the following theorem. Theorem 1.
The surface ˜ S is a rational surface isomorphic to P blown-up at points. The Euler characteristic of ˜ S together with the fact that ˜ S is rational is enough to determine ˜ S up toisomorphism. Lemma 4.1. ˜ S is birational to a conic bundle.Proof. Consider the projection π P : S → P . The fiber over [ z , w ] ∈ P is the set of points [ x, y, u : z , w ]which satisfy − w xy − u w z + w x z + w y z − w xyz + u z = 0This is the zero set of a degree 2 polynomial in P which is a conic. Away from the four singularities, S is isomorphic to a conic bundle. Hence, S is birational to a conic bundle. Since ˜ S is obtained from S by aseries of blow-ups, ˜ S is birational to S and so birational to a conic bundle.9pplying Proposition 3.1 we now have that ˜ S is rational surface. Since S has degenerate fibers, ˜ S isnot isomorphic to P × P (see figure 2). So, by corollary 3.1 ˜ S is topologically P blown-up at n points. P P double line Figure 2: Canonical component of the Whitehead link.It follows from proposition 3.2 that χ ( ˜ S ) = χ ( P ) + n = 3 + n . Thus we can determine n from the Eulercharacteristic of ˜ S .To calculate the Euler characteristic of ˜ S we use the Euler characteristic of S . Since the smooth surface˜ S is obtained from S by a series of blow-ups, we can use proposition 3.3 to write χ ( ˜ S ) in terms of χ ( S ). Lemma 4.2. χ ( ˜ S ) = χ ( S ) + 4 Proof.
The smooth surface, ˜ S , is obtained by resolving the four singularities, s i , of S listed above. Abovethe singularities, a local model for ˜ S can be obtained by blowing-up S in an affine neighborhood of each ofthe singular points. Away from the singularities we can take the local model for S as a local model for ˜ S since S and ˜ S are locally isomorphic there. Each of the singularities is nice in the sense that it takes onlyone blow-up to resolve them. Hence, in terms of the Euler characteristic, we haves χ ( ˜ S ) = χ ( S − { s i } ) + Σ i =1 χ ( ˜ s i ) (1)where for i = 1 . . .
4, ˜ s i denotes the preimage of s i in ˜ S . Determining the Euler characteristic of ˜ S in termsof that for S reduces to determining ˜ s i .To blow-up S at s = [1 , , , ,
0] we consider the affine open set A ′ where x = 0 and z = 0. Noticing thatthe singularities s and s are in A ′ , we look at the blow-up of S at s in the affine open set A = A ′ \{ s , s } .Local affine coordinates for A ∼ = A are y, u, w . So to blow-up S at s we blow-up X = Z ( f | x =1 ,z =1 ) at[ y, u, w ] = [0 , ,
0] in A . As described in section 3.2, the blow-up of X at [0 , ,
0] is the closure of the10reimage of X − [0 , ,
0] in Bl | [0 , , ( A ). Using coordinates a, b, c for P , the blow-up Y of X at [0 , ,
0] isthe closed subset in A × P defined by the equations f = f | x =1 ,z =1 = u + w − u w − wy − w y + w y (2) e = y ∗ b − u ∗ a (3) e = y ∗ c − w ∗ a (4) e = u ∗ c − w ∗ b (5)We determine the local model above s and check for smoothness by looking at Y in the affine open setsdefine by a = 0, b = 0, and c = 0.First we look at Y in the affine open set defined by a = 0 (i.e. we can set a = 1). In this open set thedefining equations for Y become f = u + w − u w − wy − w y + w y (6) e = y ∗ b − u (7) e = y ∗ c − w (8) e = u ∗ c − w ∗ b (9)Using equations e e u and w in f we obtain the local model, y ( − b + c − c − c y + 2 b c y + c y ). The first factor is the exceptional plane, E and the other factor is the local modelfor Y . Notice that E and Y meet in the smooth conic − b + c − c . So, in this affine open set, the localmodel above the singularity s is a conic; a P . Since the only places all the partial derivatives of the secondfactor vanish are over the singular points s and s , this model is smooth in A × P .Next we look at Y in the affine open set defined by b = 0. In this open set the defining equations for Y become f = u + w − u w − wy − w y + w y (10) e = y − u ∗ a (11) e = y ∗ c − w ∗ a (12) e = u ∗ c − w (13)Substituting into f , we obtain the local model, u (1 − ac + c − c u + a c u − ac u ). Again, the firstfactor is the exceptional plane, E and the other factor is the local model for Y . Notice that E and Y meet in the smooth conic 1 − ac + c . So, in this affine open set, the local model above the singularity s is a conic. Since all the partial derivatives of the second factor do not simultaneously vanish, this model issmooth in A × P .Finally we look at Y in the affine open set defined by c = 0. In this open set the defining equations for Y become f = u + w − u w − wy − w y + w y (14) e = y ∗ b − u ∗ a (15) e = y − w ∗ a (16) e = u − w ∗ b (17)11ubstituting into f , we obtain the local model, w (1 − a + b − aw + a w − b w ). The first factor isthe exceptional plane, E and the other factor is the local model for Y . Notice that E and Y meet in thesmooth conic 1 − a + b . So, in this affine open set, the local model above the singularity s is a conic. Sincethe only places all the partial derivatives of the second factor vanish simultaneously are s and s , this modelis smooth in A × P .Rehomogenizing we see that blowing-up yields a smooth local model which intersects the exceptionalplane above s in the conic defined by c − a + b . Hence χ ( ˜ s ) = 2.Blowing-up S at s , s and s is similar to blowing-up S at s . For detailed calculations, we refer thereader to the Thesis (what’s the proper way to reference this?).In each case local model for Bl ( S ) | s i intersects the exceptional plane above s i in a smooth conic. Hence χ ( ˜ s i ) = 2 for i = 1 , . . . , χ ( ˜ S ) = χ ( S − { s i } ) + Σ i =1 χ ( ˜ s i ) (18)= χ ( S ) − Σ i =1 χ ( s i ) + Σ i =1 χ ( ˜ s i ) (19)= χ ( S ) − χ ( S ) + 4 (21) Proposition 4.2.
The Euler characteristic of the surface S is χ ( S ) = 9 . To calculate the Euler characteristic we will appeal to the map φ : S → P × P defined by [ x, y, u : z, w ] → [ x, y : z, w ] on a dense open set of S. That the map φ is generically 2-to-1 makes it an attractive tool indetermining the Euler characteristic of S . However, in order to calculate the Euler characteristic of S wemust understand the map φ everywhere not just generically. To this affect, there are four aspects we need toconsider. The map φ is neither surjective nor defined at the three points P = { (0 , , , , , (0 , , , , ± √ ) } .Over six points in the P × P the fiber is a copy of P . Finally, the map is branched over three copies of P .We explain how to alter the Euler characteristic calculation to account for each of these situations. Lemma 4.3.
The image of φ on U = S − P is P × P − Q where Q = P × { [0 , } (cid:31) { [1 , , , , [0 , , , }∪ P × { [1 , √ ] } (cid:31) { [ √ , , , √ ] , [ √ , , , √ ] }∪ P × { [1 , − √ ] } (cid:31) { [ − √ , , , − √ ] , [ −√ , , , − √ ] } Proof.
We can see that this is in fact the image by viewing f as a polynomial in u with coefficients in C [ x, y, z, w ]. Namely f = g + u h where g = − w xy + w x z + w y z − wxyz and h = z ( z − w ). Theimage of φ is the collection of all points [ x, y, z, w ] ∈ P × P except those for which f ( x, y, z, w ) ∈ C [ u ] is a12onzero constant. The polynomial f ( x, y, z, w ) is a nonzero constant whenever h = 0 and g = 0. It is easy tosee that h = 0 whenever [ z, w ] = { [0 , , [1 , ± √ ] } . For each of the z, w coordinates which satisfy h , there aretwo x, y coordinates which satisfy g ( z, w ). Hence the image of φ on U is all of P × P less the three twicepunctured spheres as listed above. Lemma 4.4.
The map φ smoothly extends to all of S .Proof. We can extend the map φ to all of S by using a total transformation. Let U = S − P . Then U isthe largest open set in S on which φ is defined. Let G ( φ, U ) be the closure of the graph of φ on U . Wecan then smoothly extend the map φ to all of S by defining φ at each p i ∈ P to be φ ( p i ) := ρ ρ − ( p i )where ρ : G → S and ρ : G → P × P are the natural projections. Note, that the for s ∈ U , ρ ρ − ( s )coincides with the original map so that this extension makes sense on all of S . Now, the closure of the graphis G = { [ x, y, u, z, w : a, b, c, d ] | f = 0 , ay = bx, cw = dz } . So, φ extends to S as follows: φ ((0 , , , , { [ a, b, , } φ ((0 , , , , √ )) = { ( a, b, , √ ) } φ ((0 , , , , − √ )) = { ( a, b, , − √ ) } Notice that the set Q ⊂ P × P , which is not contained in the image of φ on U , is contained in the image of φ on P . That the extension φ maps three points in S to not just three disjoint P ’s in P × P but to the threedisjoint P ’s which are are missing from the image of φ on U will be important for the Euler characteristiccalculation. Lemma 4.5.
There are six points in P × P , the collection of which we will call L , whose fiber in S isinfinite.Proof. Thinking of f as a polynomial in the variable u with coefficients in C [ x, y, z, w ], we see that the pointsin P × P which are simultaneously zeros of these coefficient polynomials are precisely the points in P × P whose fiber is infinite. We note here that the points of L are precisely the punctures of the three puncturedspheres which are not in the image of φ | U . The preimage of L in S is the union of six P ’s each intersectingexactly one other P in one point. These three points of intersection are the points on the P ’s where thecoordinate u goes to infinity which is equivalent to the points where the x and y coordinates go to zero.Thus these intersection points are precisely the points in P. The points in L along with their infinite fibersin P × P are listed below. 131 , , ,
1] has fiber { [1 , , u, , } ⊃ [0 , , , , , , ,
1] has fiber { [0 , , u, , } ⊃ [0 , , , , , √ , , √ ] has fiber { [1 , √ , u, , √ ] } ⊃ [0 , , , , √ ][1 , √ , , √ ] has fiber { [1 , √ , u, , √ ] } ⊂ [0 , , , , √ ][1 , −√ , , − √ ] has fiber { [1 , −√ , u, , − √ ] } ⊃ [0 , , , , − √ ][1 , − √ , , − √ ] has fiber { [1 , − √ , u, , − √ ] } ⊃ [0 , , , , − √ ]In calculating the Euler characteristic we will use the fact that the preimage of L in S are six P ’s whichintersect in pairs at ideal points in the set P ⊂ S . In fact, each point in P appears as the intersection of twoof these fibers and the image of P under φ is precisely L .Let B denote the branch set of φ in P × P . We have the following lemma. Lemma 4.6. χ ( B ) = 2 .Proof. The branch set, or at least the places where φ is not one-to-one, consists of the points in S whichalso satisfy the coordinate equation u = 0. The image, B ⊂ P × P , of this branched set, is the union ofthe three varieties, B , B , and B defined by the respective three polynomials f = wy − xz , f = wx − yz ,and f = w which are all P ’s. From the bidegrees of the f i we know that B intersects each of B and B in one point ([0 , , ,
0] and [1 , , ,
0] respectively) while B and B intersect in two points ( [1 , − , − , , , ,
1] ). Again thinking of f as a polynomial in u we can write f as f = A + u B where A and B are polynomials in C [ x, y, z, w ]. Since L cut out by the ideal < A, B > and B is cut out by the ideal < A > , L is a subvariety of B . That each of six points in P × P whose fiber is infinite is also a branched point isnecessary for the Euler characteristic calculation.Now that we understand the map φ everywhere we can calculate the Euler characteristic of S ; proveProposition 4.2. Proof. (Proposition 4.2) Since the set of points in P × P whose fibers are infinite coincide with the image L of the fundamental set P , and L is the intersection of Q and the branched set B , χ ( s ) = 2 χ ( P × P − B − Q ) + χ ( Q + B − L ) + χ ( φ − ( L ))= 2 χ ( P × P ) − χ ( Q ) − χ ( B ) − χ ( L ) + χ ( φ − ( L ))14he Euler characteristic of P × P is χ ( P × P ) = 4. As Q is the disjoint union of three twice-puncturedspheres, χ ( Q ) = 3( χ ( P ) −
2) = 0. Since B is three P ’s which intersect at four points, χ ( B ) = 3 χ ( P ) − χ ( point ) = 2. Now L is just six points so χ ( L ) = 6. That φ − ( L ) is the union of six P ’s which intersect inpairs at a point implies that χ ( φ − ( L )) = 6 χ ( P ) − χ ( points ) = 9. All together this gives χ ( S ) = 9. Corollary 4.1.
The Euler characteristic of ˜ S is χ ( ˜ S ) = 13 Proof.
We have χ ( ˜ S ) = χ ( S ) + 4 = 9 + 4 = 13.We are now ready to prove Theorem 1. Proof. (Theorem 1) It follows from lemma 4.1 and corollary 3.2 that χ ( ˜ S ) = χ ( P ) + n . By corollary 4.1, n must be 10 and ˜ S must be P blown-up at 10 points. The Whitehead link complement can be obtained by 1 / M br ). Figure 3: 1 /n Dehn surgery on the Borromean rings.The manifolds which results from 1 /n Dehn filling on one of the cusps of M br are ([6]) two component 2-bridgelink with Schubert normal form S (8 n, n + 1). The fundamental group of these two component 2-bridge linkshas a presentation of the form Γ = h a, b | aw = wa i with w = b ǫ a ǫ . . . b ǫ n − where ǫ i = ( − ⌊ i (4 n − n ⌋ .For n = 1 , . . . , M br (1 /n ). For n ≥ n is still of interest. Below wesummarize this information.Character Varieties for M br (1 /n )Manifold component canonical component bidegree M br (1 /
1) 1 √ (2,3) M br (1 /
2) 1 (2,2)2 √ (4,5) M br (1 /
3) 1 (2,2)2 (2,2)3 √ (6,7) M br (1 /
4) 1 (2,2)2 (4,4)3 √ (8,9)Possibly the most interesting characteristic these character varieties share is the existence of a componentwhich is defined by a polynomial of bidegree (2 , k ) where k = { , } . All of these components are P bundlesover P . Although they are not conic bundles due to the existence of singularities they all share a commonfeature. Over the same P coordinate ([ z, w ] = [1 , P P double line Figure 4: component birational to a conic bundle for M br (1 /n )Away from a few points all of these components look like conic bundles. All conics are parameterized by P and so we can think of conic bundles as curves in P . All the degeneracies live in a hypersurface in P and all16he double lines live in a codimension two subvariety inside this hypersurface. Hence, it is fairly uncommonfor a curve in P to intersect the subvariety which corresponds to double line fibers.All of these components are defined by polynomials that have singularities (four for the Whitehead linkand two for each of the other Dehn surgery compoents). While these P bundles are not isomorphic to conicbundles, they are birational to such. Since surfaces birational to conic bundles are rational, all of thesecomponents are rational surfaces and thus isomorphic either to P × P or P blown-up at some number ofpoints. As we did for the Whitehead link complement, we can use the Euler characteristic to determine thesecharacter varieties components topologically. Aside from the Whitehead link all of these components of thesecharacter varieties are hypersurfaces defined by a singular polynomial of bidegree (2 ,
2) in P × P . Each ofthese defining polynomials has two singularities which each resolve into a conic after a single blow-up. Hencethe Euler characteristic of the smooth models are equal to that of the singular models plus two. The way wecalculate the Euler characteristic of these singular models is very similar to way to we calculated such for theWhitehead link. It turns out that all of these singular (and so smooth) models have Euler characteristic 10and hence are all isomorphic to P blown-up at 7 points. From an algebro-geometric perspective P blown-upat 7 points is interesting in the sense that is has only finitely many (precisely 49) ( −
1) curves.What we have just described is a brief outline of the proof of Theorem 2.
Theorem 2.
For n = 2 , . . . , , the character variety of M br (1 /n ) has a component which is a rational surfaceisomorphic to P blown-up at 7 points. The proof is very similar to that for 1. Hence we omit the details here. Below we have listed the singulardefining polynomials for these conic bundle components.Conic bundle components for M br (1 /n )Manifold singular defining polynomial Euler Smooth Complexfor conic bundle component characteristic Surface M br (1 / − w xy + w x z + w y z − wxyz + u ( z − w z ) 13 P blown-up at 10 points M br (1 / w x + w y − wxyz + u ( z − w ) 10 P blown-up at 7 points M br (1 / w x + w y − wxyz + u ( z − w ) 10 P blown-up at 7 points w x + w y − wxyz + u ( z − w ) 10 P blown-up at 7 points M br (1 / w x + w y − wxyz + u ( z − w ) 10 P blown-up at 7 points References [1] A. Beauville.
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