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The augmented deformation space of rational maps

The Epstein deformation space parameterizes marked rational maps with prescribed combinatorial and dynamical structure. For the family of quadratic rational maps with a periodic critical cycle of order 4 and an extra critical point not lying in this cycle, S. Koch and I recently showed that the deformation space has infinitely many connected components. In the present paper we study the augmented deformation space for this example, and show, in particular, that the closure of deformation space in augmented deformation space is also disconnected in this case.


Introduction
Let A and B be two finite subsets of the 2-dimensional sphere S 2 such that |A|, |B| ≥ 3, and let f, ι : (S 2 , A) → (S 2 , B) be two maps of pairs: In unpublished work from the 1990s, A. Epstein showed that if f is not Lattès 1 , then D f,ι is either empty or a complex |B| − |A| dimensional submanifold of T (S 2 ,B) [7]. This generalizes a seminal result of W. Thurston who showed that in the postcritically finite case, when A = B, either D f,ι is empty, f is a Lattès example, or D f,ι contains exactly one point [6]. In particular, D f,ι is always connected in the postcritically finite case. When f is not post-critically finite, then D f,ι need not be connected [11], but so far there is only one known class of examples.
The counter-example to connectedness in [11] is part of a class of quadratic rational maps studied by Milnor in [14]. These are denoted Per n (0) and have one periodic n-cycle containing a single critical point, and the behavior of the other critical point under iteration is unspecified. If in addition the extra critical point does not lie in the first critical n-cycle, the rational map is said to be in Per n (0) * . To an element F ∈ Per n (0) * let f, ι : (S 2 , A) → (S 2 , B) be defined so that f is the topological covering underlying F , ι is a identification of domain and range of f so that A ⊂ B, A is the critical n-cycle, and B is the union of A and the extra critical point. For n = 3, T (S 2 ,A) and AT (S 2 ,A) are the same singleton set, and hence D f,ι is the entire space T (S 2 ,B) and AD f,ι equals AT (S 2 ,B) . For n = 4, the work in [11] shows that D f,ι is not connected, and in fact has infinitely many connected components.
Inspired by Thurston's proof for the postcritically finite case, there has been work in constructing a contracting flow on Teichmüller space whose limit set is D f,ι , (see, for example, [17]). This leads to the following question. The points in D f,ι are sometimes called the dynamical points in the space of complex structures on T (S 2 ,B) . The augmented deformation space AD f,ι , or ideal dynamical points, is the subset of the Bers augmented deformation space AT (S 2 ,B) (see [2]) defined as the equalizer of extensions f * , ι * : AT (S 2 ,B) → AT (S 2 ,A) of f * and ι * .
In this paper, we show the following.
Theorem 1. 3. For (f, ι) associated to an element of Per 4 (0) * , the closure of D f,ι in augmented deformation space AD f,ι is not connected.
At the time of this writing, it is not known whether AD f,ι is connected in the Per 4 (0) * case. Our goal in this paper is to extend the techniques of [11], and apply them to give a partial description of the structure of AD f,ι and the closure of D f,ι within it.
1.1. Some background and sketch of proof of Theorem 1.3. The question of whether D f,ι is connected has roots in work of Thurston from the 1980s, in which he showed that if F : P 1 → P 1 is a non-Lattès rational map from the complex projective line to itself with a finite post-critical set on A. If such a flow exists, there are two possibilities: one is that D f,ι is connected, and the other is that some points may be pushed out to the boundary of T (S 2 ,B) . This suggests looking at dynamical elements of the augmented Teichmüller space to find a natural connected completion of D f,ι .
The proof in [11], that D f,ι can be disconnected translates the problem about flows on Teichmüller spaces to one about the topology of algebraic varieties. A key ingredient is to define an intermediate covering M f of T (S 2 ,B) → M (S 2 ,B) that is a natural quotient of the space of marked rational maps combinatorially equal to f . The projection M f → M (S 2 ,B) is a finite covering, and hence has the structure of a quasi-projective variety.
In the case when F ∈ Per 4 (0) * , the space M f is isomorphic to a Zariski dense subset of P 1 × P 1 : , and Z indicates the locus where the both critical points of f lie in the same periodic cycle. With respect to this parameterization, the image of D f,ι in M f is the diagonal Let L f be the group of covering automorphisms of T (S 2 ,B) over M f,ι . It was shown in [11] that the projection of D f,ι on V f,ι is a regular covering with covering automorphism group a proper subgroup S f,ι ⊂ L f . In particular, S f,ι acts transitively on fibers of D f,ι → V f,ι .
Choose a basepoint d 0 ∈ D f,ι and let v 0 be its image in V f,ι . Then we can identify L f with π 1 (M f , v 0 ) and S f,ι with a subgroup of the fundamental group. Let E f,ι be the image of π 1 (V f,ι , v 0 ) → π 1 (M f , v 0 ) induced by the inclusion map. Then E f,ι is the subgroup of S f,ι that fixes the component of D f,ι containing d 0 ; E f,ι has infinite index in S f,ι ; and hence D f,ι has infinitely many components. In other words, the image of π 1 (V f,ι , v 0 ) is not sufficiently large in π 1 (M f , v 0 ). One way to try to rectify this is to put both M f and V f,ι into a larger ambient space.
Let AT (S 2 ,B) be the augmented Teichmüller space of (S 2 , B). Then L f extends to an action on AT (S 2 ,B) giving a quotient AM f and a commutative diagram Similarly, S f,ι acts on AD f,ι with quotient denoted AV f,ι . We show that AV f,ι contains a connected pure 1-dimensional algebraic subset X containing V f,ι so that AV f,ι \ X is a finite union of points (see Proposition 4.2). A study of X \ V f,ι leads to a proof of Theorem 1.3.

Organization.
In Section 2.1, we give necessary definitions of Teichmüller and moduli spaces for rational maps, and their augmented versions. In Section 3, we prove some general properties of complements of plane algebraic curves and their blowups. In Section 4, we apply these ideas to the Per 4 (0) * case, and prove Theorem 1.3.

Further questions.
There are still many open questions along the lines of this investigation. So far, the Per 4 (0) * example is the only known case when D f,ι is disconnected. Are there others? Is AD f,ι connected in the Per 4 (0) * case, and is AD f,ι connected in general? The analysis in this paper goes further than what is necessary to prove Theorem 1.3, and it is the author's hope that they can eventually be extended to answer some of these questions.
1.4. Some comments on notation. This paper grew out of the ideas in [11], but some of the notation has changed. Most notably, we refer to the Teichmüller space T f of rational maps combinatorially equivalent to f , and its corresponding Moduli space M f . This is meant to simplify the discussion. The space M f may be more familiarly known as a connected component of a Hurwitz space of rational maps [16] and was denoted by W f in [11]. A smaller change is the use of D f,ι for the deformation space D f to emphasize the dependence on both the topological branched covering of pairs f and the identification ι. Similarly, we changed the notation of S f to S f,ι . These ease our transition from a discussion of deformation space to augmented deformation space.
1.5. Acknowledgments. I would like to thank X. Buff, S. Koch, C. Mc-Mullen, R. Ramadas, and L. D. Tráng for helpful references and discussions, and the anonymous referee for their careful reading and useful comments.

Background definitions
In this section, we recall definitions and properties of Teichmüller space, moduli space and deformation space for rational maps and their augmented versions. For more details about the general theory see, for example, [5], [2], [12], [16] and [18]. We also recall definitions of liftables L f and special liftables S f,ι from [11], and examine their extensions to the augmented spaces.
where F is a rational map, φ and ψ are homeomorphisms, and (φ, ψ, F ) is equivalent to (φ 1 , ψ 1 , F ) if and only if they both give rise to a commutative diagram as above, φ and φ 1 are equivalent in T (S 2 ,B) , ψ and ψ 1 are equivalent in T (S 2 ,A) , and F and F differ by at most a pre-or post-composition by Möbius transformations. By these definitions, T f comes with natural surjections The elements of T f are holomorphic markings of the branched covering f , and p U and p L record the induced structure on domain and range after forgetting the points in B \ A. Epstein's deformation space for f, ι, seen as a subspace of T f is the equalizer of the maps p U and p L We think of D f,ι as the dynamical Teichmüller space for (f, ι). with covering automorphism group equal to the mapping class group Mod(S 2 , A) of orientation preserving homeomorphisms h : S 2 → S 2 that fix the points of A up to isotopy rel A. (Unlike the case for Teichmüller spaces and moduli space of general surfaces, Mod(S 2 , A) on T (S 2 ,A) acts without fixed points.) Similarly let M f be the space of commutative diagrams where (i, j, F ) is defined up to modifications of the domain and range of F by Möbius transformations. Then the map (φ, ψ, F ) → (φ| B , ψ| A , F ) defines a covering map has no non-trivial covering automorphisms, then f defines a unique lifting map Remark 2.1. If A contains only branch points of f and f is of generic degree two, it is possible that there is an automorphism α : S 2 → S 2 that preserves A such that f • α = f . For example we could take the double covering of S 2 to itself branched at the poles, and let A be the set of poles and B = A ∪ {q} for any additional point q. Then the half rotation r : S 2 → S 2 fixing the poles defines a covering automorphism, and f would not be uniquely defined. If on the other hand A contains at least one non-branch point q, then we can think of q as a basepoint for the unbranched part of the covering. Thus, there can be no covering of automorphisms of f that fix A.
The following Proposition was shown using slightly different language in [13] (cf. [11] Proposition 2.2). For the convenience of the reader, we include a proof below.
Proposition 2.2. If f : (S 2 , A) → (S 2 , B) has no non-trivial covering automorphisms, then the map T f → M f is a regular covering with automorphism group L f .

Proof.
Let h ∈ L f . Take any (φ, ψ, F ) ∈ T f . Then we have the commutative diagram (2.1) Since h and f h fix B and A, respectively, Putting the definitions together, we have a commutative diagram I  I  I  I  I  I  I  I  I I  I  I  I  I  I  I  I  I T If f has no automorphisms, then all vertical arrows and the left three diagonal arrows are unbranched covering maps. The right diagonal arrows may not be surjective (see [3]).

Stabilizer of deformation space.
Assume throughout this section that f has no covering automorphisms. Fix a basepoint d 0 ∈ D f,ι , and let m 0 ∈ M f,ι be the image of d 0 under the map ρ. Then we have an identification defined by the path-lifting theorem for coverings. That is, for each γ ∈ π 1 (M f,ι , m 0 ), we lift γ to a path γ on T (S 2 ,B) based at d 0 , and let (γ) be the (unique, since the covering is regular) mapping class that takes d 0 to the end point of γ . The following proposition is a consequence of the definitions.
Since f has no covering automorphisms, f : L f → Mod(S 2 , A) is uniquely defined and satisfies the commutative diagram By the identification of L f with π 1 (M f,ι , m 0 ), S f,ι can also be thought of as the equalizer of the homomorphisms The stabilizer in L f of D f,ι equals S f,ι , and S f,ι acts transitively on the fibers of the covering Thus the covering is regular and S f,ι is the group of covering automorphisms.

Augmented spaces.
A pre-stable rational curve (C, A) is a nodal curve C together with a finite set A contained in the complement of the nodes of C such that (a) the irreducible components of C are isomorphic to P 1 , and (b) the fundamental group of C is trivial. The set of nodes of C union the points of A form the distinguished points of C. A stable rational curve is a pre-stable rational curve with the following additional property: (c) the number of distinguished points on each irreducible component of C is greater than or equal to 3. For each component C of a pre-stable rational curve (C, A) there are three possibilities: (1) C is stable, i.e., it contains at least 3 distinguished points; (2) C is unstable, and contains two nodes; or (3) C is unstable, and contains one node and zero or one point in A.
Let Σ pre (S 2 ,A) be the set of pre-stable rational curves and Σ (S 2 ,A) the space of stable rational curves, and let s : Σ pre (S 2 ,A) → Σ (S 2 ,A) be the map sending (C, A) to the result of contracting components of C using the following rule: in case (1) leave C alone; and in case (2) and (3) contract C to a point. If C contains a point of A, then mark the image of the contraction by that point. This map is well-defined since in case (2) there is at most one point of A in C. We call s the stabilization map.
A marking of a pre-stable or stable rational curve is a quotient map such that φ is a homeomorphism when restricted to S 2 \ γ for some multi-curve γ ⊂ S 2 \ A, φ restricts to a bijection A → A, and the components of γ are in oneto-one correspondence with the nodes of C. The curve γ is called the contracting curve for φ. We consider two markings equivalent if they are the same up to post-composition by automorphisms of C and pre-composition by homeomorphisms (S 2 , A) → (S 2 , A) that are isotopic to the identity rel A.
The collection of markings of pre-stable and stable rational curves by (S 2 , A) is denoted by AT pre (S 2 ,A) and AT (S 2 ,A) , respectively. Post-composition by s defines a surjection AT pre (S 2 ,A) → AT (S 2 ,A) . The space AT (S 2 ,A) is called the augmented Teichmüller space of (S 2 , A). There is a natural topology on AT (S 2 ,A) such that points on AT (S 2 ,A) \ T (S 2 ,A) are the limits of sequences points on T (S 2 ,A) for which the length of γ tends to zero (see [2] for more precise definitions).
The mapping class group Mod(S 2 , A) extends to actions on AT pre (S 2 ,A) and AT (S 2 ,A) . For the former, the action can have infinite point stabilizers, but for the latter all stabilizers of points are finite. The action of Mod(S 2 , A) on AT (S 2 ,A) defines a branched covering up to holomorphic automorphism of C that do not permute components (cf. [5]).
We now define the augmented Teichmüller and moduli spaces of f . A quadratic rational map F : (C U , A) → (C L , B) is pre-admissible if (a) F defines a surjective map from C U to C L of generically constant degree that maps nodes to nodes; (b) locally near each node of C U , F has generically constant degree; and (c) (C L , B) is a stable rational curve and (C U , A) is pre-stable. We say that F is admissible if in addition (b) (C U , A) is stable. The augmented Teichmüller space AT f for f is the collection of holomorphic markings where the horizontal maps are markings in AT (S 2 ,A) and AT (S 2 ,B) respectively, and F is a pre-admissible covering. Here, as in the definition of T f , we take (φ, ψ, F ) up to the natural equivalences. With this definition, the projection q : AT f → AT (S 2 ,B) defines an isomorphism. Let p U , p L : AT f → AT (S 2 ,A) be defined by Proposition 2.6. The subgroup of liftables L f ⊂ Mod(S 2 , B) extends to an action on AT f .
Proof. Let φ : (S 2 , B) → (C L , B) be an element of AT (S 2 ,B) , and let γ be the contracting multi-curve. Since f is unbranched outside of B, it follows that f −1 (γ) is a multi-curve on S 2 \ A, and since h is liftable is pre-stable, and the inclusions i, j are defined up to holomorphic automorphisms of C U and C L . We denote an element by (i, j, F ).
AM f q y y s s s s s s s s s s The augmented deformation space is defined to be the equalizer and contains the deformation space D f,ι . Let AV f,ι = ρ(AD f,ι ). Then we have AV f,ι = Eq(p U , p L ), 2.5. Stabilizer of augmented deformation space. In this section we give a sufficient condition for D f,ι to be contained in a connected component of AD f,ι .
Proposition 2.7. The stabilizer in L f of AD f,ι equals S f,ι .
Proof. The stabilizer in L f of AD f,ι must be contained in S f,ι , since D f,ι ⊂ AD f,ι . Let h ∈ S f,ι and let (φ, ψ, F ) ∈ AD f,ι . Then by definition f h = i h. Thus, we have a commutative diagram and we have We give here a sufficient condition so that D f,ι is contained in a connected component of AD f,ι .
Proposition 2.8. Suppose there is a connected quasi-projective variety X ⊂ AV f,ι containing m 0 with the following properties (1) V f,ι ⊂ X, and (2) X has a regular neighborhood N (X) ⊂ AM f , so that N (X) ∩ M f is connected, and the image of the homomorphism induced by inclusion contains S f,ι . Then D f,ι is contained in a connected subset of AD f,ι .
Proof. Let Y 0 be the connected component of the preimage of X in AT f containing d 0 . Let U be the connected component of the preimage of N (X) in AT f containing d 0 . Then since N (X) has a retract to X, the set U has a corresponding retraction to a component of σ −1 (X). This component is necessarily Y 0 , since Y 0 ∩ U = ∅. Since the image of π 1 (N (X) ∩ M f , m 0 ) in L f = π 1 (M f , m 0 ) contains S f,ι , it follows that the action of S f,ι on AT f preserves U and hence Y 0 . Thus, we have D f,ι ⊂ Y 0 ⊂ AD f,ι .

Blowups and topology of curve complements
In this section we study the topology of surface/curve pairs and the effect of blowups (see for example [8] or [9] for a review of elementary blowup theory for surfaces).

Regular neighborhoods of algebraic curves on surfaces.
Recall that, for a simplicial subcomplex V embedded in a manifold X, a regular neighborhood N (V ) of V is an open subset of X containing V that has a deformation retract to V .
Let X be a smooth complex projective surface, and let V ⊂ X be a pure codimensional one subvariety. We first observe that we may assume that V is a finite union of smooth curves with normal crossings. Let Q be the set of singular points on V . Then V \Q is a finite union of smoothly embedded punctured Riemann surfaces in X \ Q. By successively blowing up X at the points of Q (and at points of the preimages of Q), it is possible to obtain a new projective surface X and a surjective morphism σ : X → X such that the preimage (or total transform) V = σ −1 (V ) is a union of smoooth curves with normal crossings. That is ( X, V ) is locally isomorphic near a point of intersection on V to (C 2 , {x = 0} ∪ {y = 0}).
Hereafter in this section we assume that V has smooth components intersecting in normal crossing. As before, let Q be the set of intersections of V , and let V 1 , . . . , V k be the irreducible components of V . Since each V i is smooth there is an embedded tubular neighborhood T (V i ) ⊂ X so that for i = j, T (V i ) only intersects T (V j ) near points in Q where V i and V j intersect, and if V i and V j intersect at q, then is a neighborhood of q so that (N (q), V, q) is homeomorphic to and let T (V c i ) be the tubular neighborhood given by This manifold and its embedding in X \ V is uniquely determined up to homeomorphisms of X that are isotopic to the identity rel V . Proof. For each i, T (V c i ) \ V c i has a deformation retract to S(V c i ) corresponding from the retraction of a punctured disk to its boundary circle. Thus, we have only to consider what happens near the intersection points q ∈ Q. In N (q) it is enough to show that has a deformation retract to {|x| = 1} × {|y| = 1}. Such a retraction is defined by the map .
Remark 3.2. By its construction, S(V ) is naturally homeomorphic to a graph manifold (see, for example, [10] for definitions) over the incidence graph Γ of the components of V ; this is the bipartite graph swith vertices corresponding to the components of V , and edges between v i and v j for each q ∈ Q such that V i and V j intersect at q. To each vertex v i associate the manifold S(V c i ) and to each edge of Γ between v i and v j associate the common torus boundary component of their associated vertex manifolds.

Regular neighborhoods and fundamental groups.
In this section we prove an easy variation of the Lefschetz hyperplane theorem [1] and a useful corollary.
Lemma 3.3. Let p : X → P 1 be a smooth projective surface fibered over the complex projective line, and let V be a fiber. Let C ⊂ X be a pure codimension one algebraic subset none of whose components are fibers of p. Then V has a regular neighborhood N (V ) so that induced by inclusion is surjective.
Proof. Let P ⊂ P 1 be the set of points p where p restricted to C drops in cardinality, i.e., at least one intersection point of C and the fiber above p has higher multiplicity. Let p o be the restriction of p to X o = p −1 (P 1 \ P ) \ C. Then is a fiber bundle, and the Zariski-Van Kampen theorem [19] [4] implies that for a general fiber V of p o π 1 (X \ C) π 1 (V )/K where K is the subgroup of π 1 (V ) generated by the relations γ −1 β(γ), where β ranges over automorphisms of π 1 (V ) determined by the action of π 1 (P 1 \ P ) on V . In particular, the homomorphism Then N ( V ) \ C) contains a general fiber of p o , and the claim follows.
Corollary 3.4. Let C ⊂ C 2 be an algebraic curve, and let V ⊂ C 2 be a line not contained in C. Then we can include C 2 as a Zariski open subset of a smooth projective surface X, and find a sequence of blowups σ : X → X such that (1) σ is an isomorphism over C 2 \ C; and (2) the total transform V of the closure of V in X has a regular neighborhood N ( V ) in X such that the map on fundamental groups induced by inclusion is surjective.
Proof. Let p : C 2 → C be a linear projection so that V is a fiber. Then we can define completions C 2 ⊂ P 1 × P 1 and C ⊂ P 1 , so that p extends to a projection p : P 1 × P 1 → P 1 so that the closure V of V in P 1 × P 1 is a fiber. Let C be the union of the closure of C in P 1 × P 1 , and the two lines in P 1 × P 1 \ C 2 .
Let σ : P 1 × P 1 → P 1 be sequence of blowups over points of C ∩ V such that the total transform V = σ −1 (V ) and the proper transform C over C meet in normal crossings. Let N ( V ) be a regular neighborhood of V as in Lemma 3.3. Then the homomorphism induced by inclusion is surjective.
Since σ is an isomorphism outside the preimage of C∩V , we have a commutative diagram where the vertical arrows are quotient maps that contract the exceptional curves (or 1-dimensional fibers) over points on V ∩ C. It follows that the retraction of N ( V ) \ C to V \ C descends to a retraction of N (V ) \ C to V \ C, and hence N (V ) \ C is a regular neighborhood of V \ C. Finally, P 1 × P 1 \ C = C 2 \ C, so setting N (V ) = N (V ) ∩ C 2 , it follows that N (V ) is a regular neighborhood of V and the homomorphism defined by inclusion is surjective.
Remark 3.5. The boundary manifold S( V ) associated to N ( V ) in the previous proof has the structure of a boundary manifold over the incidence graph of the irreducible components of V (cf. Remark 3.2). Furthermore, each component of V is isomorphic to a line, each vertex manifold is an S 1 fiber bundle over S 2 with a finite set of thickened fibers removed.

Application to the Main Example
Let F ∈ Per 4 (0) * , and let f, ι : (S 2 , A) → (S 2 , B) be the underlying branched covering and identification of domain and range so that A is the periodic 4 cycle, and B = A ∪ {v} where v is the extra critical point.
We study the inclusion D f,ι ⊂ AD f,ι by looking at their images V f,ι and AV f,ι in M f and AM f . Recall that in the diagram below the vertical maps on the left are unbranched coverings, while the vertical arrows on the right are branched.
Parameterization of moduli space. We begin by embedding M f in P 1 × P 1 as follows. Consider an element of M f represented as a commutative diagram By applying automorphisms of P 1 on the right side, we can assume that The above data completely determines F : P 1 × P 1 as a rational function in the variable t: It follows that in this example z is determined by x and y: We have the following (see also, [11]).
and Z is the closure in P 1 × P 1 of the affine union of curves   4.2. The quotient of augmented deformation space. Let AM f be the quotient of AT f by the action of L f , and let AV f,ι be the image of AD f,ι in AM f . Our goal in this section is to concretely describe a subspace X ⊂ AV f,ι satisfying the properties in Proposition 2.8.
First we recall that the elements of AV f,ι are the elements of AM f that equalize the two maps p L , p U : AM f → AM (S 2 ,A) . Each stratum of AM f is described by a partition of {0, 1, ∞, y, z} into two or three sets by an admissible multi-curve γ, as in Figure 3. Here, the empty multi-curve corresponds to the principal stratum M f ⊂ AM f . The corresponding stable curves are shown in Figure 4, with each P 1 , homeomorphic to S 2 , is drawn as a line.
For AM (S 2 ,A) there are only two isomorphism types (shown in Figure 5). The left picture depicts points belonging to the main component M (S 2 ,A) ⊂ AM (S 2 ,A) , which is isomorphic to a thrice punctured sphere, while the right picture depicts one of the three single point boundary points. (1) (2) (3) Figure 4. The three types of stable curves for (S 2 , B) where B has five elements. The two left define one-dimensional strata, and the right defines a point stratum.
(I) (II) Figure 5. The two topological homeomorphism types of stable curves for (S 2 , A) where A has four elements.
The elements α ∈ AM f that lie in positive dimensional strata must be of the form (1) or (2) in Figure 4. Those of type (1) lie in M f = P 1 × P 1 \ L ∪ Z and map under both p U and p L to elements of AM (S 2 ,A) of type (I). Those of type (2) divide into four subtypes: those that correspond to a partition of the form For types (2a) and (2b), p L maps α to an element in AM (S 2 ,A) of type (II), while for types (2c) and (2d), p L maps α to one of type (I). For types (2a) and (2d) p U maps α to an element of type (II), while for type (2b) and (2c) α could apriori map to an element of type (I) or (II). This is because the critical values ∞ and z lie in the same component. Thus the rational map F : C U → C L must have two isomorphic irreducible components in C U lying over the unramified component in C L upon which the distinguished points will be distributed.
From this we can reduce the types that can be in AV to (2a), (2d), as well as (2b) and (2c) under the condition that the preimage under F of the distinguished points in the unramified component lie on the same component of C U . In the allowable case of types (2a) and (2b), we see that for p L (α) and p U (α) to be equal the images of the two maps in AM (S 2 ,A) must give the partition Thus, the partition given in (2a)  We have shown the following.
Proposition 4.2. The pure 1-dimensional algebraic set X = V ∪ A 1 ∪ A 2 ⊂ AM f is contained in AV f,ι , and its complement is a finite set of points (possibly empty). Remark 4.3. We leave the question of whether AV f,ι is connected (in this case, and in general) to future study.

Blowups.
In this section, we find a connected pure 1-dimensional algebraic subset X ⊂ AV that contains V, and whose complement in AV is finite.
Let σ : P 1 × P 1 → P 1 × P 1 be sequence of blowups defined as follows. First blowup the points (0, 0), (1,1) and (∞, ∞) to get the exceptional curves E 0 , E 1 , E ∞ . Next blowup the point of intersection q ∈ E ∞ ∩ L y , where L y is the proper transform of {y = ∞}. Let E q be the exceptional divisor. The union of curves is drawn in Figure 6 (compare Figure 1).
Lemma 4.4. The map σ has the following properties: (1) σ restricts to an isomorphism on P 1 × P 1 \ σ −1 (Q); and (2) the total transform V and the proper transform L ∪ Z meet in normal crossing singularities. (3) inclusion induces a surjection on fundamental groups Proof. Properties (1) and (2) Figure 6. The proper transform of V is drawn as a dotted line, and the exceptional curves E 0 , E 1 , E ∞ and E q are drawn in red. Intersections with L are indicated as thickened black line, and intersections with Z are indicated with a thin black line Then X is connected since the punctures of V at intersections with L ∪ Z occur only in smooth points of V. Since σ is an isomorphism over V, there is a natural inclusion ν : V → X. We choose local coordinates for a neighborhood of E ∞ as follows. Let x = 1/x and y = 1/y be coordinates for an open neighborhood of p ∞ = (∞, ∞) in P 1 × P 1 , so that in the coordinates (x, y) we have p ∞ = (0, 0). Let u (and u = 1 u ) be coordinates for E ∞ . Then a neighborhood of E ∞ in P 1 × P 1 is isomorphic to the algebraic subset of C 2 × P 1 defined by {(x, y) × u ∈ C 2 × P 1 |x = uy}, and E ∞ = {(x, y) × u ∈ C 2 × P 1 | x = y = 0}.
Here u corresponds to a line through (∞, ∞) in P 1 × P 1 , and we can think of the contracting curve γ as a small loop on this line around (∞, ∞). The intersection of E ∞ with E q occurs at the point on E ∞ corresponding to (y, u) = (0, 0), and E q has a neighborhood parameterized by ((y, u), v) ∈ C 2 × P 1 where y = vu. Then x = uy = v and and as u goes to 0, Then we have Define Then κ ∞ and κ q extend to a morphism ξ : X → AV.
Proof of Theorem 1.3. Fix d 0 ∈ D f,ι , and let D 0 be the connected component of D f,ι that contains d 0 . Let D 0 be the closure of D 0 in AD f,ι . Then D 0 maps to V under the projection from AT f → AM f . Let D 0 ⊂ AT f be the connected component of the preimage of V that intersects D 0 . Then, since V is closed in X and X is connected, we can find a connected component Y 0 of the preimage of X in AT f that contains D 0 , and D 0 is the closure in Y 0 of D 0 . That is, D 0 = D 0 .
Let v 0 be the image of d 0 in V, and let γ be a close path starting at v 0 and passing along X to a point near an intersection of Z with E ∞ ∪ E q , forming a small loop around that intersection point, and returning along the original path back to v 0 . Then γ is not homotopic in AM f to an closed path on V since Z and V do not intersect. Let g be the image of γ in π 1 (M f ) after pushing off the boundary of AM f into M f . Since g / ∈ Image(π 1 ( V ) → π 1 (M f )), g does not preserve D 0 , and hence g( D 0 ) and D 0 are disjoint. These are the closures of D 0 and g(D 0 ) in AD f,ι and the claim follows.