The augmented deformation space of rational maps
Eriko Hironaka




This work was partially supported by a grant from the Simons Foundation #426722. 
  
It was recently shown that the
Epstein deformation space of marked rational maps with prescribed combinatorial and dynamical
structure can be disconnected. For example, 
the family of quadratic rational maps with a periodic critical cycle of order 4 and an extra critical
value  not lying in this cycle has a deformation space with infinitely many components.
We study the structure of 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. 


The augmented deformation space of rational maps
    Eriko Hironaka
    December 30, 2023
================================================



In celebration of Lê Dũng Tráng's 70th birthday







 


§ 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:

    f : (S^2, A) → (S^2,B)

a branched covering with branch values contained in B, and 

    ι : (S^2,A) ↪ (S^2,B)

a homeomorphism identifying domain and range, and A with a subset of B.


The Epstein deformation space _f,ι  is defined as the equalizer of the induced maps on Teichmüller spaces

    f^*,ι^* : _(S^2,B)→_(S^2,A).

 In unpublished work from the 1990s, A. Epstein showed that if f is not Lattès[See <cit.> for definitions.], then  _f,ι is either empty or 
a complex |B| - |A| dimensional  submanifold of  _(S^2,B) <cit.>. 
This generalizes a seminal result of W. Thurston who showed that in the postcritically finite case, when A = B, 
either _f,ι is empty, f is a Lattès example, or _f,ι contains exactly one point <cit.>.  
In particular, _f,ι is always connected in the postcritically finite case. 
When f is not post-critically finite, then _f,ι need not be connected <cit.>, but so far there is only
one known class of examples.

  What are necessary and sufficient conditions for _f,ι to be connected?


The counter-example to connectedness in <cit.> is part of a class of quadratic rational maps studied by
Milnor in <cit.>.   These are denoted _n(0) and
have one periodic n-cycle 
containing a single critical point, and the behavior of the other critical value under iteration is unspecified.  If in addition the extra critical
value does not lie in the first critical n-cycle, the rational map is said to be in _n(0)^*.
To an element F ∈_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, _(S^2,A) is a singleton set, and hence
_f,ι is the entire space _(S^2,B).
For n=4, the work in <cit.> shows that _f,ι is not connected, and in fact has infinitely many
connected components.  This leads to the following question.

 Does _f,ι have a natural connected closure?


The points in _f,ι are sometimes called the dynamical points in the space of complex structures
on _(S^2,B).  The augmented deformation space _f,ι,
or ideal dynamical points, is the subset  of the Bers
augmented deformation space _(S^2,B)  (see <cit.>) defined as the equalizer of extensions

    f^*,ι^* : _(S^2,B)→_(S^2,A)

of f^* and ι^*.  

At the time of this writing, it is not known whether _f,ι is connected in the _4(0)^* case.   Our goal in this paper is to 
extend the techniques of <cit.>, and apply them 
to give a partial description of the structure of _f,ι and the closure of _f,ι
within it.  We prove the following theorem.

   For (f,ι) associated to an element  of _4(0)^*,
the closure of _f,ι in augmented deformation space _f,ι is not connected.






 §.§ Some background and ideas behind the proofs

The question of whether and when _f,ι is connected has roots in work of Thurston from the 1980s, in which he
showed that if 

    F : ^1 →^1

is a non-Lattès rational map from the complex projective line 
to itself with a finite post-critical set

    ⋃_n=1^∞ F^(n)(_F),

and f : (S^2,P) → (S^2,P) is the corresponding branched covering of pointed spheres with domain and range identified, 
and postcritical set P,
then the lifting map on holomorphic markings defines a contracting map on Teichmüller space

    f^* : _(S^2,P)→_(S^2,P).

Thus, f^* has a unique fixed point, and hence _f,ι is a singleton set.

This classical result suggests that there could be a contracting flow in the general case
when the identification map ι: (S^2,A) → (S^2,B) is a strict inclusion on A.  
If such a flow exists,  there are two possibilities: one is that _f,ι is connected, and the other is that some
points may be pushed out to the boundary of _(S^2,B).  This suggests looking at dynamical elements
of the augmented Teichmüller space to find a natural connected completion of _f,ι.

The proof in <cit.>, that _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 _f
of _(S^2,B)→_(S^2,B) that is a natural quotient of the space of marked rational maps
combinatorially equal to f.    The projection  _f →_(S^2,B) is a finite covering, and 
hence has the structure of a quasi-projective variety.  

 In the case when F ∈_4(0)^*, the space _f is isomorphic to a Zariski
dense subset of ^1 ×^1:

    _f = ^1 ×^1 ∖∪

where ^1 ×^1 ∖ = _(S^2,A)×_(S^2,A), and  indicates the locus
where the both critical points of f lie in the same periodic cycle.
With respect to this parameterization, the  image of _f,ι in _f is the diagonal

    _f,ι= {(x,x) ∈^1 ×^1  |  (x,x) ∉∪}.

Let L_f be the group of covering automorphisms of _(S^2,B) over _f,ι.  It was shown in <cit.> that
the projection of _f,ι on _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 _f,ι→_f,ι.

Choose a basepoint d_0 ∈_f,ι and  let v_0 be its image in _f,ι.  Then 
we can identify L_f with π_1(_f,v_0) and S_f,ι with a subgroup of the fundamental group.  Let E_f,ι be
the image of π_1(_f,ι,v_0) →π_1(_f,v_0) induced by the inclusion map.  Then
E_f,ι is the subgroup of S_f,ι that fixes the component of _f,ι containing d_0;
E_f,ι has infinite index in S_f,ι; and hence _f,ι has infinitely many components.
In other words, the image of π_1(_f,ι,v_0) is not sufficiently large in π_1(_f,v_0). 
One way to try to rectify 
this is to put both _f and  _f,ι into a larger ambient space.


Let _(S^2,B) be the augmented Teichmüller space of (S^2,B).  Then   L_f 
extends to an action on _(S^2,B) giving a quotient _f and 
 a commutative diagram

    _(S^2,B)[d]_/L_f@^(->[r]    _(S^2,B)[d]^/L_f
    _f @^(->[r]    _f.

Similarly, S_f,ι acts on _f,ι with quotient denoted _f,ι.  The spaces 
_f,ι and _f have the advantage of being algebraic geometric sets, and can be studied 
via singularity theory.  

We define a connected pure 1-dimensional algebraic subset X of a
blowup ^1 ×^1
of ^1 ×^1 and an embedding  X →_f,ι that is surjective except possibly
a finite set of points (Proposition <ref>).   By studying
properties of X and its embedding in ^1 ×^1 we prove Theorem <ref>.






 §.§ Organization
  In Section <ref>, we give necessary definitions of Teichmüller and moduli spaces for
rational maps, and their augmented versions.  In Section <ref>,
we prove some general properties of complements of plane algebraic curves 
and their blowups. 
In Section <ref>, we apply these ideas to the _4(0)^* case, and prove Theorem <ref>.  




 §.§ Further questions


There are still many open questions along the lines of this investigation. So far, the _4(0)^* example is the only known case when _f,ι is disconnected.  Are there others?
Is _f,ι connected in the _4(0)^* case, and is _f,ι connected in general?  
The analysis in this paper suggest general approaches to these questions, which we leave for further investigation.



 §.§ Some comments on notation
  This paper grew out of the ideas in <cit.>, but some of the notation 
has changed.
Most notably, we refer to the Teichmüller space _f of
rational maps combinatorially equivalent to f, and its corresponding Moduli space _f. 
Thus, Epstein's deformation space _f,ι will be considered as a subspace of _f rather than of the isomorphic
space _(S^2,B) (also known as the parameter space).
The space _f may be more familiarly known as a connected component of a Hurwitz space of rational maps <cit.> and was
denoted by _f in <cit.>.   
A smaller change is the use of _f,ι for the deformation space _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.




 §.§ Acknowledgments
 I would like to thank 
X. Buff, S. Koch, C. McMullen, 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, <cit.>, <cit.>,
<cit.>, <cit.> and <cit.>.  We also recall definitions of liftables L_f and special liftables
S_f,ι  from <cit.>,  and examine their extensions to the augmented spaces.



 §.§ Teichmüller spaces for rational maps

Let A be a finite subset of 3 or more points of the topological 2-sphere S^2.  The
Teichmüller space 
_(S^2,A)
of holomorphic markings on (S^2,A) is the collection of orientation preserving homeomorphisms

    ϕ : (S^2,A) → (^1,ϕ(A))

defined up to post-composition by automorphisms of ^1 (i.e., Möbius transformations) and pre-composition by self-homeomorphisms of S^2 that are isotopic rel A to the identity.

Similarly, given a finite branched covering of pairs f : (S^2, A) → (S^2, B), where ∞ > |A|, |B| ≥ 3 and
B contains the branch values (or critical values) of f, we define the Teichmüller space _f of holomorphic markings 
on (f,A,B) as the set of  commutative diagrams

    (S^2,A) [r]^-ψ[d]_f    (^1,ψ(A))[d]^F
    
    (S^2,B) [r]^-ϕ    (^1,ϕ(B))

also denoted by (ϕ,ψ,F),
where F is a rational map, ϕ and ψ are homeomorphisms.  Two triples (ϕ,ψ,F) and (ϕ_1,ψ_1,F_1) in _f are
 equivalent if there  are homeomorphisms α:  (S^2,A) → (S^2,A)  isotopic to the
 identity map rel A and β:  (S^2,B) → (S^2,B) isotopic to the identity rel B, and biholomorphic maps μ,ν : ^1 →^1
 so that the diagram
 
    (^1,ψ_1(A)) [d]_F_1   (S^2,A) [l]_-ψ_1[d]_f[r]^-α   (S^2,A)[d]_f[r]^ψ   (^1,ψ(A))[d]_F
    
    (^1,ϕ_1(B)   (S^2,B) [l] _-ϕ_1[r]^-β   (S^2,B) [r]^ϕ   (^1,ϕ(B))

commutes, and μ = ψ∘α∘ψ_1^-1 and ν = ϕ∘β∘ϕ_1^-1.

 
 By these definitions, _f comes with natural surjections
 
    _f[dl]_q[dr]^p_U
    _(S^2,B)      _(S^2,A)

recording the holomorphic markings of the domain space
(p_U) and the range space (q).  Furthermore, _f can be thought of as the graph of a lifting map

    f^* : _(S^2,B)→_(S^2A)

 defined by f.  Thus, q is an isomorphism of holomorphic spaces.

We create an iteration scheme from f by partially identifying the domain (S^2,A) and range (S^2,B) of f.  That is, 
we fix a homeomorphism ι : S^2 → S^2 that restricts to an inclusion ι |_A : A ↪ B.  
Then we have a map of pairs

    ι : (S^2,A) → (S^2,B).

Let p_L = ι^* ∘ q, where ι^* : _(S^2,B)→_(S^2,A) is the map that
takes ϕ∈_(S^2,B) to the class in _(S^2,A) defined by 

    ϕ∘ι : (S^2,A) → (^1,(ϕ∘ι)(A)) = (^1, ϕ(A)).

Then the elements of _f can be thought of as the  holomorphic markings of the branched covering f, and p_U and p_L record
the induced marked  holomorphic structures (S^2,A) by the domain and range.  

The 
Epstein's deformation space _f,ι is 
the subspace of _f 
consisting of holomorphic structures on the covering and base of f that are equivalent relative to A.  That 
is,  _f,ι consists of the triples (ϕ,ψ,F) so that
ϕ^-1∘ψ is  isotopic to the identity relative to A, or equivalently 

    f^*ϕ = ι^*ϕ.

Another way to say this is that _f,ι is the equalizer in _f of the maps p_U and p_L, that is

    _f,ι = {ϕ∈_f  |  p_U(ϕ, i) = p_L(ϕ)}.

We think of _f,ι as the dynamical Teichmüller space for (f,ι).  



 §.§ Moduli spaces

Let _(S^2,A) be the space of
embeddings

    A ↪^1

up to post-composition by a Möbius transformation.  Then the restriction map [ϕ] ↦ [ϕ |_A]
defines a regular covering map

    _(S^2,A)→_(S^2,A)

with covering automorphism group equal to the 
 mapping class group  (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, (S^2,A) on _(S^2,A) acts without fixed points.)

Similarly let _f be the space of commutative diagrams
 
    A[d]_f |_A @^(->^j [r]    ^1[d]^F
    
     B @^(->^i [r]   ^1.

 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
 
    _f →_f,

with covering automorphism group L_f ⊂(S^2,B) called the subgroup of liftables
consisting of elements h ∈(S^2,B) such that for some h' ∈(S^2,A) the diagram

    (S^2,A)[d]_-f [r]^h'   (S^2,A)[d]^-f 
    
    (S^2,B) [r]^h    (S^2,B)

commutes.  



Since A and B are assumed to contain at least three points, f: (S^2,A) → (S^2,B) can have no non-trivial covering automorphisms.
In the degree 2 case, this follows from the fact that f must have exactly two branch points and two branch values.  Any other marked point would
lie in the unbranched part of the covering.   Even in the case when the degree of f is greater than 2, the fact that f may be realized as a 
rational map implies that all covering automorphisms must be conjugate to a Möbius transformation.  If there are at least three points in A,
then the Möbius transformation must be the identity.


By the definition of L_f and since f has no non-trivial covering automorphisms,  f defines a 
unique lifting map

    f^♯: L_f →(S^2,A)

where f^♯ h = h'.

 The following Proposition was shown using slightly different language in <cit.> (cf. <cit.> Proposition 2.2).
For the convenience of the reader, we include a proof below.

 The map _f→_f is a regular covering with automorphism group L_f.



Let h ∈ L_f.   Take any (ϕ,ψ,F) ∈_f.  Then we have
the commutative diagram

    (S^2,A)[d]_f    (S^2,A) [l]^f^♯ h[r]^ψ[d]_f   (^1, ψ(A))[d]^F
    
    (S^2,B)      (S^2,B)[l]^h [r]^ϕ   (^1,).

Since h and f^♯ h fix B and A, respectively,
 (ϕ _B, ψ _A, F) and (ϕ∘ h, ψ∘ f^♯ h, F) map to the same element in _f.

Conversely, suppose (ϕ,ψ,F) and (ϕ_1,ψ_1, F_1) both map to equivalent elements in _f.
Then by definition, there are embeddings i : B →^1 and j : A →^1 so
that i = ϕ |_B = ϕ_1 |_B, j = ψ_1 |_A = ψ |_A and there are Möbius transformations μ and ν such that
F = μ^-1∘ F_1 ∘ν.  We can assume for simplicity that F = F_1 by replacing (ϕ_1,ψ_1,F_1) by 
the equivalent element (μ^-1∘ϕ_1,ν^-1∘ψ_1 , F) in _f.
Then the common image of (ϕ,ψ,F) and (ϕ_1,ψ_1,F) is some (i,j,F) in _f.
Thus the images of ϕ and ϕ_1 are the same in _(S^2,B),  the images of ψ and ψ_1
are the same in _(S^2,A), and hence there are mapping classes h ∈(S^2,B) and f^♯ h ∈(S^2,A)
such that ϕ = h ∘ϕ_1 and ψ = f^♯ h ∘ψ_1.  The maps h and f^♯ h complete a diagram of the form (<ref>),
and thus h ∈ L_f.


Let q : _f →_(S^2,B) be the map sending (i,j,F) to the equivalence class containing i in _(S^2,B);
let p_U: _f →_(S^2,A) be the map sending (i,j,F) to the equivalence class containing j in _(S^2,A);
let ι^* : _(S^2,B)→_(S^2,A) be the forgetful map sending an inclusion i : B →^1
to i ∘ι |_A; and
let p_L = ι^* ∘q.
Then we have the commutative diagram

    _f[dl]_≃^q [dr]^p_U[dd]^ρ
    _(S^2,B)[dd]_/ (S^2,B)[dr]^/L_f      _(S^2,A)[dd]^/ (S^2,A)
       _f[dl]_q[dr]^p_U
    _(S^2,B)      _(S^2,A).

All vertical arrows and the left three diagonal arrows are unbranched covering maps.  The right diagonal arrows may not be surjective (see <cit.>).
Let _f,ι be the image of the deformation space _f,ι in _f.  Then we have

    _f,ι = (p_L, p_U).




 §.§ Stabilizer of deformation space
  
Fix a basepoint d_0 ∈_f,ι, and let m_0 ∈_f,ι be the image of d_0 under the map
ρ. Then we have an identification

    ℓ: π_1(_f,ι,m_0) → L_f,

defined by the path-lifting theorem for coverings.  That is, for each γ∈π_1(_f,ι,m_0), we lift γ
to a path γ' on _(S^2,B) based at d_0, and let ℓ(γ) be the 
(unique, since the covering is regular) mapping class that takes d_0 to the end point 
of γ'.   

  The restriction of ρ : _f →_f to 
_f,ι gives a covering map

    ρ_𝒟 : _f,ι→_f,ι,

and the image E_f,ι,d_0 of 

    π_1(_f,ι, m_0) →π_1(_f,ι,m_0) ℓ→ L_f

is the stabilizer of the connected component of _f,ι that contains d_0.


  We show that the projection ρ__f,ι satisfies the path-lifting theorem.
Let d_0 = (ϕ_0,ψ_0,F_0), and let
(i_t,j_t,F_t) be a path in _f,ι with m_0 = (j_0,i_0,F_0).   Let ξ and η be a representatives of the class
of ϕ_0 and ψ_0 so that the diagram commutes

    S^2 [d]_f [r]^η   ^1 [d]^F
    
    S^2 [r]^ξ   ^1,

η |_A = ψ_0 |_A and ξ |_B = ψ |_B.
Let p_t : (S^2,B) → (S^2,B_t) be any continuous family of homeomorphisms. Then

    (ξ∘ p_t |_B_t, η∘ p_t |_A_t, F_t) ∼ (i_t, j_t, F_t)

and (ξ∘ p_t, η∘ p_t, F_t) is a lift of (i_t,j_t,F_t) and lies in _f,ι.
 
Thus, as a restriction of
an unbranched covering map, ρ_𝒟 is itself is a covering map.  
The rest follows from basic covering space theory. 


The lifting map  f^♯ : L_f →(S^2,A) is uniquely defined
and satisfies the commutative diagram

    (S^2,A)[d]_f [r]^f^♯h    (S^2,A)[d]^f
    
    (S^2,B) [r]^h    (S^2,B).

For a homeomorphism h : (S^2,B) → (S^2,B), let h_A be the element of (S^2,A) defined by
ignoring the points of B ∖ A.  Then h_A is the isotopy class of h defined up to homeomorphisms 
isotopic to the identity rel A.  Define

    ι^♯ : (S^2,B)    →   (S^2,A)
    
    h   ↦    h_A.


Let S_f,ι⊂ L_f be the equalizer

    S_f,ι = (f^♯,ι^♯) ⊂ L_f.

By the identification of L_f with π_1(_f,ι,m_0),  S_f,ι can equivalently be defined as the
equalizer of the  homomorphisms

    (p_L)_*, (p_U)_* : π_1(_f,ι,m_0) →π_1(_(S^2,A),a_0)

where a_0 = p_L(m_0) = p_U(m_0), where m_0 ∈_f,ι  =  (p_L,p_U).

 The stabilizer in L_f of _f,ι equals S_f,ι, and 
S_f,ι acts transitively on the fibers of the covering

    _f,ι→_f,ι.

Thus the covering is regular and S_f,ι is the group of covering automorphisms.


  The deformation space _f,ι is connected if and only if _f,ι is connected and  S_f,ι = E_f,ι,d_0.


In the case when (f,ι) is associated to an element of _4(0)^*, E_f,ι,d_0 has infinite index in S_f,ι (<cit.>, Proposition 2.11).



 §.§ Augmented spaces
  
By a rational curve  we mean a nodal curve with the following properties


  * the irreducible components of  are isomorphic to ^1, and

  * the fundamental group of  is trivial.

A pre-stable rational curve (,) is a rational curve  together with a finite set  contained in the complement of the nodes of .
The set of nodes of  union the points of  form 
 the distinguished points of .
A stable rational curve is a pre-stable rational curve with the following additional property:


  * the number of distinguished points on each irreducible component of  is greater than or equal to 3.


For each component C of a pre-stable rational curve (,) there are three possibilities:


  * C is stable, i.e., it contains at least 3 distinguished points;

  * C is unstable, and contains two nodes; or

  * C is unstable, and contains one node and zero or one point in .

Let Σ^_(S^2,A) be the set of pre-stable rational curves (,) with a bijection
A →,
and let Σ_(S^2,A)⊂Σ^_(S^2,A) be the space of stable rational curves.
Define a map

    𝔰 : Σ^_(S^2,A)→Σ_(S^2,A)

sending (,) to the result of contracting components of  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 , 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  in C.  We call 𝔰 the stabilization map.

A marking of a pre-stable rational curve (,) ∈Σ^_(S^2,A) is a quotient map

    ϕ : (S^2,A) → (,),

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 →, and the components
of γ are in one-to-one correspondence with the nodes of .    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  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 ^_(S^2,A) and
 _(S^2,A), respectively.
 Post-composition by 𝔰 defines a surjection 
 
    ^_(S^2,A)→_(S^2,A).

The space _(S^2,A) is called the augmented Teichmüller space of (S^2,A).
There is a natural topology on _(S^2,A) such that points on
_(S^2,A)∖_(S^2,A) are the limits of sequences points on _(S^2,A) for which the length of  γ tends to zero (see <cit.> for more precise definitions).  



The mapping class group (S^2,A) extends to actions on ^_(S^2,A) and _(S^2,A).    For points in ^ the stabilizer contains a copy of the group of
Möbius transformations.
Given a point in _(S^2,A), the stabilitzer is the free abelian group of mapping classes generated by Dehn twists along
the components of the contracting curve.  


The action of (S^2,A) on _(S^2,A) defines a branched covering

    _(S^2,A)→_(S^2,A)

where _(S^2,A) is the space of inclusions

    A ↪

of A into a rational curve 
up to holomorphic automorphism of  that do not permute components (cf. <cit.>).

We now define  the augmented Teichmüller and moduli spaces of f.
A rational map F : (_U,) → (_L,) is pre-admissible if


  * F defines a surjective map from _U to _L of generically constant degree that maps nodes to nodes;

  * locally near each node of _U, F has generically constant degree; and

  * (_L,) is a stable rational curve and  (_U,) is pre-stable.

We say that F is admissible if in addition 


  * (_U,) is stable.

The augmented Teichmüller space _f for f  is the collection of holomorphic markings

    (S^2,A) [r]^ψ[d]_f    (_U,)[d]^F
    
    (S^2,B) [r]^ϕ    (_L,)

where the horizontal maps are markings in _(S^2,A) and _(S^2,B) respectively, and F is a pre-admissible covering.
Here, as in the definition of _f, we take (ϕ,ψ,F) up to the natural equivalences.

With this definition, the projection q : _f →_(S^2,B) defines an isomorphism.
Let p_U, p_L : _f →_(S^2,A) be defined by

    p_L (ϕ,ψ,F)    =   𝔰(ϕ)
    p_U(ϕ,ψ,F)     =   𝔰(ψ)


 The subgroup of liftables L_f ⊂(S^2,B) extends to an action on _f.


 Let ϕ : (S^2,B) → (_L,) be an element of _(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

    f^♯ h(f^-1(γ)) = f^-1(h(γ)).

Thus h ∈ L_f takes (ϕ,ψ,F) to (ϕ∘ h, ψ∘ f^♯ h, F) where ϕ∘ h and ψ∘ f^♯ h
contract the curves h(γ) and f^-1(h(γ)) respectively.


Let _f = _f/L_f.  Then the points of _f are defined by diagrams

    A[d]_f|_A@^(->[r]^i   _U[d]^F
    
    B@^(->[r]^j    _L

where (_L, ) is stable, (_U,) is pre-stable, and  the inclusions i,j 
are defined up to holomorphic automorphisms of _U and _L.  
We denote an element by (i,j,F).
Let p_U, p_L : _f →_(S^2,A) be defined by

    p_L ([ϕ,ψ,F])    =   𝔰∘ j
    p_U([ϕ,ψ,F])     =   𝔰∘ i

While the action of L_f on _f has no fixed points and the quotient map _f →_f is a covering,
the quotient map _f →_f can have branch points.

Summarizing, we have a commutative diagram of augmented spaces:

    _f[dl]_q[dr]^p_U[dd]^ρ
    _(S^2,B)[dd]_/ (S^2,B)[dr]^/L_f      _(S^2,A)[dd]^/ (S^2,A)
       _f[dl]_q[dr]^p_U
    _(S^2,B)      _(S^2,A),

The augmented deformation space is defined to be the equalizer

    _f,ι = (p_U,p_L)

and contains the deformation space _f,ι.  Let _f,ι = ρ(_f,ι).
Then we have

    _f,ι = (p_U,p_L),

 



 §.§ Stabilizer of augmented deformation space

In this section, we give a necessary and sufficient condition for _f to be connected.  Unlike in the case
for Corollary <ref>, _f →_f is not an unbranched covering.  To get around this
we use the notion of regular neighborhoods.
 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. 
 
 The stabilizer in L_f of _f,ι equals S_f,ι, that is, if g ∈ L_f is such that
g(α) = α for all α∈_f,ι, then g ∈ S_f,ι.


 The stabilizer in L_f of _f,ι must be contained in S_f,ι, since _f,ι⊂_f,ι.
Let h ∈ S_f,ι and let (ϕ,ψ, F) ∈_f,ι.  Then by definition f^♯ h = i^♯ h.  Thus,
we have a commutative diagram

    (S^2,A) [r]^f^♯ h[d]_f    (S^2,A) [d]^f[r]^ψ    (_U,)[d]^F
    
    (S^2,B) [r]^h    (S^2,B) [r]^ϕ    (_L,)

and we have

    f^* (h([ϕ]))  = [ψ∘ f^♯ h] =  [ψ∘ι^♯ h] =ι^*(h([ϕ])).

Thus h stabilizes _f,ι.  


  Conversely, one can ask whether if g ∈ L_f and g(α) = α for some α∈_f,ι, then does it follow that
g ∈ S_f,ι?  This is not true in general, since points in the boundary of _f,ι have extra automorphisms that need not be
in S_f,ι (see Remark <ref>).


  Suppose there is a  connected quasi-projective variety
X with _f,ι⊂ X ⊂_f,ι such that
X has a regular neighborhood N(X) ⊂_f with the properties


  * N(X) ∩_f is connected,
and 

  * the image
of the homomorphism induced by inclusion

    π_1(N(X) ∩_f,m_0) →π_1(_f,m_0)

contains S_f,ι.  

Then _f,ι is contained in a connected component of _f,ι.


  Let Y_0 ⊂_f be the connected component of the preimage of X in _f containing d_0.
Let 
U be the connected component of the preimage of N(X) in _f containing d_0.  Then 
since N(X) has a retract to X, the set U has a corresponding
retraction to a component of ρ^-1(X).  This component is necessarily Y_0, since
Y_0 ∩ U ≠∅. 
Since the image of π_1(N(X) ∩_f,m_0) in L_f = π_1(_f,m_0) contains S_f,ι, it follows that the action of S_f,ι on  _f preserves U and hence Y_0.
Thus, we have

    _f,ι⊂ Y_0 ⊂_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 <cit.> or <cit.>
for a review of elementary blowup theory for surfaces).




 §.§ Regular neighborhoods of algebraic curves on surfaces


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 (^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 

    N(q) = T(V_i) ∩ T(V_j)

is a neighborhood of q so that (N(q),V,q) is homeomorphic to 

    ({|x|<1}×{|y|<1}, {x=0}∪{y=0}, (0,0)).


For i=1,…,k, let

    V_i^c = V_i ∖⋃_q ∈ Q N(q)

and let T(V_i^c) be the tubular neighborhood given by 

    T(V_i^c) = T(V_i)  ∖⋃_q ∈ Q N(q).

Let T(V) = ⋃_i=1^k T(V_i^c), called the tubular neighborhood of V.

Let S(V_i^c) be the circle bundle over V_i^c contained in the boundary of T(V_i^c).  Then S(V_i^c) is an oriented 3-manifold
with torus boundary components corresponding to the intersections of V_i with other components of V.  In particular, if V_i is
isomorphic to ^1, then S(V_i^c) is the complement of thickened Hopf links in the 3-sphere.
The boundary manifold of V is given by

    S(V) = ⋃_i=1^k S(V_i^c).

This manifold and its embedding in X ∖ V is uniquely determined up to homeomorphisms of X that are isotopic to the identity rel V.

  The punctured tubular neighborhood T(V) ∖ V has a deformation retraction to S(V).


  For each i, T(V_i^c) ∖ V_i^c has a deformation retract to S(V_i^c) 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

    { |x| < 1}×{ |y| < 1}∖{x=0}∪{y=0}

has a deformation retract to {|x|=1}×{|y|=1}.  Such a retraction is defined by the map

    ((x,y),t) ↦ (x/1 + t(|x|-1), y/1+t(|y|-1) ).





By its construction, S(V) is naturally homeomorphic to a  graph manifold (see, for example, <cit.> for definitions) over the incidence graph 
Γ of the components of V;
this is the bipartite graph with vertices 

    {v_i  |  i=1,… k}

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_i^c) 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 <cit.> and a useful corollary.


  Let 𝔭 : X →^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 

    π_1(N(V) ∖ C) → (X ∖ C)

induced by inclusion is surjective.



  Let 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(^1 ∖ P) ∖ C.  Then

    𝔭^o: X^o →^1 ∖ P

is a fiber bundle, and the Zariski-Van Kampen theorem <cit.> <cit.> 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^-1β (γ), where
β ranges over automorphisms of π_1(V') determined by the action of π_1(^1 ∖ P) on
V'.   In particular, the homomorphism 

    π_1(V') →π_1(X ∖ C)

is surjective.   Let
N(V) =  (𝔭^o)^-1(U) where U is a small neighborhood of 𝔭 (V).
Then N(V) ∖ C contains a general fiber of 𝔭^o, and 
the claim follows.


  Let C ⊂^2 be an algebraic curve, and let V ⊂^2 be a line not contained in C.
Then we can include ^2 as a Zariski open subset of a smooth projective surface X, and find a sequence of blowups
σ : X→ X
such that 


  * σ is an isomorphism over ^2 ∖ C; and

  * 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

    π_1(N(V) ∩^2 ∖ C) →π_1 (^2 ∖ C)

is surjective.  



 Let 𝔭 : ^2 → be a linear projection so that V is a fiber.   Then
we can define completions ^2 ⊂^1 ×^1 and ⊂^1, so that 𝔭
extends to a projection

    𝔭: ^1 ×^1 →^1

so that the closure V of V in ^1 ×^1 is a fiber.
Let C be the union of the closure of C in ^1 ×^1, and the two lines in ^1 ×^1 ∖^2.

Let σ : ^1 ×^1→^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 <ref>.  Then
 the homomorphism 

    π_1(N(V) ∖C) →π_1(^1 ×^1∖C)

induced by inclusion is surjective.

Since σ is an isomorphism outside the preimage of C∩V,
we have a commutative diagram
 
    V[d] [r]    N(V) [d]
    V[r]    N(V)

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 ×^1 ∖C = ^2 ∖ C, so setting N(V) = N(V) ∩^2, it follows 
that N(V) is a regular neighborhood of V and 
the homomorphism

    π_1(N(V) ∖ C) →π_1(^2 ∖ C)

defined by inclusion is surjective.


Corollary <ref> is used in our discussion of connectivity of augmented deformation space  in Section <ref>.



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 <ref>).  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 ∈_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 _f,ι⊂_f,ι by looking at their images _f,ι and _f,ι in 
_f and _f. 




 §.§ Parameterization of moduli space

We begin by embedding _f in ^1 ×^1 as follows.
Consider an element of _f represented as a commutative diagram

    A [d]_f |_A@^(->[r]^i    ^1[d]^F
    
    B @^(->[r]^j    ^1.

By applying automorphisms of ^1 on the right side, we can assume that

    i(B) = {0,1,∞, y,z}   j(A) = {0,1,∞,x},

where 


  * ∞ and z are the critical values of f;

  * 0 is a critical point in A with f(0) = ∞; and

  * f(∞) = 1, f(x) = 0.  


The above data completely determines F : ^1 ×^1 as a rational function in the variable t:

    F(t) = (t-x)(t-r)/t^2     r = x + y - 1/x -1.

It follows that in this example z is determined by x and y:

    z = - (1 - 2x + x^2 - y)^2/4 x (x-1)(x+y-1).

We have the following (see also, <cit.>).

  There is an identification 

    _f  =   ^1 ×^1 ∖∪,

assigning (i,j,F) to (x,y),
 where

    = {x = 0}∪{y = 0}∪{x = 1}∪{y = 1}∪{x =  ∞}∪{y = ∞}

and  is the closure in ^1 ×^1 of the affine union of curves

    {1-2x + x^2 - y =0 }∪{x^2+y = 1}∪{ x + y = 1}∪{2xy+x^2-y-2x+1=0}







By this parameterization, the image  of _f in _f equals the diagonal

    = {(x,y) ∈^1 ×^1 ∖∪ |  x = y}.

Figure <ref> gives a picture[This figure was provided courtesy of Sarah Koch.] of the real part
of ∪ in the affine open subset ^2 ⊂^1 ×^1, and Figure <ref> gives a picture near (∞,∞).






     


 §.§ The quotient of augmented deformation space


Let _f be the quotient of _f by the action of L_f,  and let _f,ι be the image of _f,ι
in _f.  Our goal in this section is to concretely describe a subspace X ⊂_f,ι satisfying the properties in Proposition <ref>.

First we recall that the elements of _f,ι are the elements of _f that equalize the two maps

    p_L, p_U : _f →_(S^2,A).

Each stratum of _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 <ref>.
Here, the empty multi-curve corresponds to the principal stratum _f ⊂_f.

The corresponding stable curves are shown in Figure <ref>, with each ^1, homeomorphic to S^2, is drawn as a line.



For _(S^2,A) there are only two isomorphism types (shown in Figure <ref>).  The left picture depicts points 
belonging
to the main component
_(S^2,A)⊂_(S^2,A),  which is isomorphic to a thrice punctured sphere, while 
the right picture depicts one of the three single point boundary points.



The elements α∈_f that lie in positive dimensional strata must be of the form (1) or (2)
in Figure <ref>.  Those of type (1) lie in _f = ^1 ×^1 ∖∪
and map under both p_U and p_L  to elements of _(S^2,A) of type (I).
Those of type (2) divide into four subtypes: those that correspond to a partition of the form


  (2a) {a,b,z}∪{c,∞}; 

  (2b) {a,∞,z}∪{b,c};

  (2c) {a,b,c}∪{∞,z}; or

  (2d) {a,b,∞}∪{c,z}.


For types (2a) and (2b), p_L maps α to an element in _(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 : _U →_L must have two isomorphic
irreducible components in _U lying over the unramified component in _L upon which the distinguished points will be distributed.

From this we can reduce the types that can be in  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 _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 _(S^2,A) must give the partition
 
    {0,1}∪{x,∞}.

 Thus, the partition given in (2a) can only be
  
    {0,1,z}∪{y,∞}

 and for (2b) it can only be
 
    {∞, y, z}∪{0,1}.

 For type (2c), p_L(α) and p_U(α) must be equal, and F defines the isomorphism on stable curves.
Taking into account the combinatorics of f, this implies equality of the cross ratios:
 (0,∞;1,x) and (∞,1; x,0), which is false under the assumption that x ∉{0,1,∞}. 
 
 Let _1, _2 ⊂_f be the subsets corresponding to the partitions
  
    _1:  {∞, y,z}∪{0,1},

  and 
 
    _2:  {0,1,z}∪{y,∞}.

 Then each of these is isomorphic to _0,4×_(0,3).
 Furthermore, the closures of _1 and _2 in _f intersect at the point corresponding to the partition
 
    {0,1}∪{z}∪{y,∞}.


We have shown the following.

 
The pure 1-dimensional algebraic set ∪_1 ∪_2 ⊂_f is contained in _f,ι, and its
complement is a finite set of points (possibly empty).

 
  We leave the question of whether _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 ⊂
that contains , and whose complement in  is finite.

Let

    σ : ^1 ×^1→^1 ×^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 <ref> (compare Figure <ref>).



  The map σ has the following properties:


  * σ restricts to an isomorphism on ^1 ×^1∖σ^-1(Q); and

  * the total transform  and the proper transform ∪ meet in normal crossing
singularities.

  * inclusion induces a surjection on fundamental groups

    π_1(() ∩_f) →π_1(_f).




  Properties (1) and (2) follow from the definitions, and property (3) follows from Lemma <ref>.



Let 

    X = (∪ E_∞∪ E_q) ∖∪.

Then X is connected since  the punctures of  at intersections with ∪ occur
only in smooth points of .  Since σ is an isomorphism over , there is a natural inclusion
ν: ↪ X.

 The inclusion ↪ induced by _f,ι↪_f,ι factors  as
ξ∘ν for some embedding 

    ξ: X ↪.



   
Recall from Proposition <ref> the subset ∪_1 ∪_2 ⊂.
In what follows we define embeddings

    κ_∞:  E_∞∖∪   →   _1
    κ_q :  E_q ∖∪   →   _2

that together with the projection V→ 
extend to define ξ.

When α∈_f approaches a general point of ∪, it corresponds to 
two points in {0,1,∞,y,z} coming together.  For the lines (y=0, y=1 and y=∞, the pairs
{y,0}, {y,1} and {y,∞} approach each other,
while simultaneously {x,1}, {1,∞} and {1,0} approach each other.
Near the lines x = 0, x=1, x=∞
the pairs {x,0}) (and {0,∞}), {x,1} (and {0,y}) and {x,∞} (and {0,1}) approach each other.
For z approaching 0,1,∞ or y, we have

    {z,0}       1 - 2x + x^2 - y = 0; x ≠ 0, 1; x + y ≠ 1
    {z,1}       x^2 + y = 1; x ≠ 0; x+y ≠ 1
    {z,∞}       x = 0, x=1,   x+y=1
    {z,y}       2xy + x^2 - y - 2x =0


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 ^1 ×^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 ^1 ×^1 is isomorphic to the algebraic subset of ^2 ×^1 defined by

    {(x,y) × u ∈^2 ×^1 | x =   u y},

and

    E_∞ = {(x, y) × u ∈^2 ×^1 |  x = y = 0}.

With respect to the x, y coordinates, we have

    z    =    - x^4 y^2 (1 - 2x + x^2 - y)^2/4 x^4y^2x (x-1)(x+y-1)
       =    - (x^2y - 2xy + y - x^2)^2/4 xy(1 - x)(y + x -xy)


Using the identity x = u y, we have

    z    =   - (u^2 y^2 - 2 uy + 1 - u^2 y)^2/4 u y(1-uy)(1 + u (1 - y)).

To each (u, y), associate the point in _(S^2,B)
defined by

    ([u,∞ ; y, z] , *)

where the first component is the cross ratio of the points u,∞, y,z and the second
component is the unique triple of points in ^1 up to automorphism.   This defines a point in 
_(S^2,B).  

Since cross ratio is preserved under automorphisms of ^1 we have

    [u,∞: y, z] = [uy,∞, 1,z y].

As y approaches 0, the cross ratio approaches

    [0,∞; 1, -1/4 u(1+u)]

and is degenerate only when  u = 0 (x = ∞),  u = -1 (z = ∞), u = -1/2 (z = y), or u = ∞ (y = ∞).

We thus have a well-defined embedding:

    κ_∞ : E_∞∖{∪}   →   _1
    
    u    ↦    ([u, ∞; y,z], *).

Here u corresponds to a line through (∞,∞) in ^1 ×^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) ∈^2 ×^1 where y = v u.  
Then
x = u y = v and

    z
       =    -(1 -2x + x^2 - y)^2/4 x(x-1)(x+y-1)
       =    
     - (u(1 -2v + v^2) - v)^2/4 v (v-1)(v + uv - 1)

and as u goes to 0, 

    z = -v/4 (v-1)^2 = -v/4(1-v)^2.

Then we have

    [v,0;1,z] =  [1,0; v, v z] = [1,0; v, -1/4(1-v)^2].

The cross ratio is degenerate when v = ∞ (where
{y=∞} meets E_q), v = 1 (where E_∞ and E_q intersect), 
and where z = 0 and z = 1 (corresponding to two distinct values of v not equal to 1 or ∞).
The points correspond to v ∈ E_q ∖∪ except at the point
where v = 1.  

For this point, consider the 2-component multicurve γ on S^2 ∖ B
determined by two loops on the planes defined by
E_∞ and {y=∞} around (y, u) = (0,0).  
The corresponding point α_0 ∈_f corresponds to the partition

    {y,∞}∪{z}∪{0,1}.

Define

    κ_q : E_q ∖{∪}   →   _2
    
    v    ↦   {[ ([0,1; ∞, v], *)                 ;              α_0                  ] .


Then κ_∞ and κ_q extend to a morphism ξ : X →.



  Fix d_0 ∈_f,ι, and let _0 be the connected component of 
_f,ι that contains d_0.  Let _0 be the closure of _0 in _f,ι.  Then 
_0 maps to  under the projection from _f →_f.
Let _0 ⊂_f  be the connected component of the preimage of  that intersects _0.
Then, since  is closed in X and 
X is connected, we can find a connected component Y_0 of the preimage of X in _f that contains D_0,
and 
_0 is the closure
in Y_0 of _0.  That is, _0 = _0.

We claim that there is an element 

    g ∈ S_f,ι∩( π_1(N(X) ∩_f) →π_1(_f))
    ∖(π_1(V) →π_1(_f)).

Let v_0 be the image of d_0 in , and let γ be a close path starting at v_0 and passing along X to 
a point near an intersection of  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 _f to an closed 
path on V since Z and V do not intersect.
 Let g be the image of γ in π_1(_f) after pushing off the boundary of _f
into _f.

Since g ∉(π_1(V) →π_1(_f)),  g does not preserve
D_0, and hence g(_0) and _0 are disjoint.  These are the closures of
_0 and g(_0) in _f,ι and the claim follows.




 §.§ Connectivity of augmented deformation space


We finish this paper by giving a sufficient condition for _f,iota connected in the _4(0)^* case.

By Corollary <ref>, we know that V has a  regular neighborhood N in ^1 ×^1 so
that 

    π_1(N ∩_f) →π_1(_f)

is surjective. 

As a closure of _f, ^1 ×^1 is birationally equivalent to _f (but not isomorphic). By the
birational theory of 
complex projective surfaces, there is a minimal smooth surface Z with birational morphisms to _f and ^1 ×^1 

    Z[dl][dr]
    _f       ^1 ×^1.

Lifting U to Z and projecting to _f gives a regular neighborhood U' of ∪𝒦, where 𝒦 is a set
of boundary curves in _f.   Since _f is a smooth subset of both _f and ^1 ×^1 the
two projects of Z are isomorphisms over _f.  Thus

    U' ∩_f = U ∩_f,

 and hence

    π_1(U' ∩_f) →π_1(_f)

is surjective.  It follows that U' has a connected preimage in _f an that the preimage of V∪𝒦 in _f
is connected.

We have thus shown the following sufficient condition for _f,ι to be connected.

 If  is connected and is Zariski dense in ∪𝒦, then _f,ι is connected.





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