Connectedness of The Moduli Space of Artin-Schreier Curves of Fixed Genus
Huy Dang
math.NT math.AG






Connectedness of The Moduli Space of Artin-Schreier Curves of Fixed Genus 

    Huy Dang 

    
===========================================================================




We study the moduli space 𝒜𝒮_g
 of Artin-Schreier curves of genus g over an algebraically closed field k of positive characteristic p. The moduli space is partitioned into strata, which are irreducible. Each stratum parameterizes Artin-Schreier curves whose ramification divisors have the same coefficients. We construct deformations of these curves to study the relations between those strata. As an application, when p=3, we prove that  𝒜𝒮_g is connected for all g. When p>3, it turns out that 𝒜𝒮_g is connected for a sufficiently large value of g. In the course of our work, we answer a question of Pries and Zhu about how a combinatorial graph determines the geometry of 𝒜𝒮_g.
 



§ INTRODUCTION



Let k be an algebraically closed field of characteristic p>0. An Artin-Schreier k-curve is a smooth projective connected k-curve Y which is a ℤ/p-cover of the projective line ℙ^1_k. In <cit.>, the authors introduce the moduli space 𝒜𝒮_g of Artin-Schreier k-curves of genus g. They enumerate a family of irreducible strata of 𝒜𝒮_g by partitions of the integer d+2 where d:=2g/(p-1) such that no entries are congruent to 1 modulo p. From there, Pries and Zhu give the formula to calculate the dimension of a stratum from its corresponding partition (see <cit.>). Furthermore, they construct a deformation to show whether a stratum lies in the closure of another under certain conditions (see <cit.>). That technique, combine with the results of Maugeais in <cit.> which showed that each irreducible component of 𝒜𝒮_g has dimension 2g/(p-1)-1, allowed them to determine all the cases when the moduli space of such covers is irreducible (see <cit.>).

In this paper, based on the framework of <cit.>, we continue to study the relations between these irreducible strata of 𝒜𝒮_g with the aim of finding sufficient conditions for the moduli space to be connected. Corollary 1.2 of <cit.> implies that 𝒜𝒮_g is irreducible and thus connected when p=2. For p>2, the same corollary only tells us that 𝒜𝒮_g is connected for small values of g. We are able to prove that for p=3, the moduli space 𝒜𝒮_g is also connected for all g. When p>3, the following result, which implies that 𝒜𝒮_g is connected for g sufficiently large, is the main result of this paper.






When p=3, the moduli space 𝒜𝒮_g is connected for any g.

When p=5,  𝒜𝒮_g  is connected for any g ≥ 14.

 When p>5, 𝒜𝒮_g is connected if g ≥(p^3-2p^2+p-8)(p-1)/8.
 
 



When p=5, the only cases that the theorem does not cover are g=10 and g=12. The techniques we use in this paper are inadequate to study these cases. A more conceptual approach, which will be introduced in my thesis, will show that 𝒜𝒮_g is disconnected when g=10 or g=12.

 
 
To prove Theorem <ref>, we use an idea of Pries and Zhu in <cit.> about how the deformations of Artin-Schreier covers determine the geometry of the moduli space as follows. Suppose E_1 and E_2 are two partitions that satisfy the conditions in the previous paragraph. Then the stratum indexed by E_1, which is denoted by Γ_E_1, is in the closure of Γ_E_2 if and only if each k-point of Γ_E_1 can be deformed over k[[t]] to a k((t))-point of Γ_E_2 (see Proposition <ref>). These kind deformations, which actually change the number of branch points but do not change the genus, are only possible for wildly ramified cover. They were first studied by Mézard in her thesis <cit.> and were also used by Pries and Zhu in <cit.>. With the aim to prove Theorem <ref>, we will construct explicitly some deformations of this type in Theorem <ref>.


In <cit.>, the authors define a combinatorial invariant for 𝒜𝒮_g. They call it the graph G_d. Each vertex of G_d indexes a stratum of 𝒜𝒮_g of the same type as ones were mentioned previously. More details will be given in the following chapter. The existence of a chain of oriented edges connecting two vertices in G_d is necessary for the stratum corresponding to one vertex to lie in the closure of another (see <cit.>). When considering two vertices connected by a single edge, we able to show whether the necessary condition is also sufficient by comparing the dimensions of the corresponding strata (see Corollary <ref>). That answers Open Question 1 of <cit.>. 



 
 Here is an outline of this paper. In Section <ref>, we give a review on Artin-Schreier theory and some results of <cit.> about the relationship between irreducible strata of 𝒜𝒮_g and partitions of integers. Section <ref> introduces some deformation results and gives us information on how irreducible strata fit together in 𝒜𝒮_g by looking at their corresponding partitions (Theorem <ref>). Open Question 1 of <cit.> is also answered in this section. Connectedness of 𝒜𝒮_g is discussed in Section <ref>. In particular, Theorem <ref> will be proved in this section using Theorem <ref> and graph theory on the "tree" formed by partitions of d. The details of how the deformations are constructed are postponed to Section <ref>, which is the most novel part, but also the most technical part of this paper.
 
 
 
 
 
    I would like to thank Andrew Obus,  my advisor, for his guidance, support, and patience during the development of this paper. He also gave me the idea for the deformation in <ref>  (Proposition <ref>), which is the last piece of the puzzle to prove the connectedness result. I thank Rachel Pries and David Harbater for useful suggestions and proofreading an earlier version of this paper. I would also thank Jonathan Gerhard and Bennett Rennier for making the programs that help me to check my results. 

 


§ MODULI SPACE OF ARTIN-SCHREIER CURVES




 §.§ Artin-Schreier Theory





Let K be a field of characteristic p. If L is a separable extension of K of degree p, then Artin-Schreier theory says that L=K(α) where α is a root of the polynomial y^p-y=a for some element a ∈ K (detailed in <cit.>).

Let Y be an Artin-Schreier k-curve. Then there is a ℤ/p-cover ϕ: Y →ℙ_k^1 with an affine equation of the form y^p-y=f(x) for some non-constant rational function f(x)∈ k(x). A cover ϕ': Y' →ℙ_k^1 defined by an affine equation y^p-y=g(x) is isomorphic to ϕ if g(x)=f(x)+h(x)^p-h(x) for some h(x) ∈ K(x). At each ramification point, there is a filtration of the inertia group ℤ/p, called the filtration of higher ramification groups (see, e.g., <cit.>, IV). Suppose f(x) has r poles: {P_1,…,P_r} on ℙ^1_k. Let d_j be the order of the pole of f(x) at P_j. One may assume that f(x) has minimal form; i.e., the number d_j is prime to p by Artin-Schreier theory. It is an easy exercise to show that d_j is the ramification jump at P_j. Let e_j=d_j+1. Then e_j ≥ 2 and e_j ≢1  p. The ramification divisor of ϕ is D:=∑_j=1^r (p-1)e_j Q_j where Q_j is the ramification point above P_j (<cit.>, IV, Proposition 4). The p-rank of Y is an integer s such that the cardinality of Jac[p](k)=p^s.  Applying Riemann-Hurwitz formula (<cit.>, IV, Corollary 2.4) and Deuring-Shafarevich formula (<cit.>, Corollary 1.8), we have the following lemma. 



The genus of Y is g_Y=((∑_j=1^re_j)-2)(p-1)/2. The p-rank of Y is s_Y=(r-1)(p-1).




Throughout the paper, we define d by g=d(p-1)/2. So, we have the identities: d=(∑_j=1^re_j)-2 and ∑_j=1^re_j=d+2=2g/(p-1)+2.


The above lemma shows that all the Artin-Schreier k-curves with the same genus g have the same d. That is the essential difference between ℤ/p-curves and ℤ/q-curves where q≠ p is prime. For each ℤ/q-curve, a branch point contributes q-1 to the degree of its ramification divisor. Thus, every ℤ/q-cover of genus g must have the same number of branch points.



Lemma <ref> also implies that for p>2, the moduli space 𝒜𝒮_g is empty and thus is connected if g is not a multiple of (p-1)/2. Hence, throughout the paper we will assume g is a multiple of (p-1)/2 when p>2. 







 §.§ Partitions of Integers



Fix a prime p>0 and an integer d≥ 1. Let Ω_d be the set of partitions of d+2 into positive integers e_1,e_2, … with each e_j ≢1  p. Without loss of generality, suppose e_1 ≥ e_2 ≥…≥ e_n. Define a partial ordering ≺ on Ω_d: E≺E' if E' is a refinement of E. Using this partial ordering, one can construct a directed graph G_d. The vertices of the graph correspond to the partitions E in Ω_d. We say a partition E={e_1,…,e_n} of length n is in the n-th level of G_d, denoted E∈Ω_d,n-1 . There is an edge from E to E' if and only if E≺E' and there is no partition lying strictly between them. It is easy to see that there are only two types of edges. The first type has the form {e}→{e_1, e_2} and the second type has the form {e}→{e_1,e_2,e_3} where e_i ≡p+1/2 p. We say there is a path connecting E and E' if one can go from E to E' through some sequence of edges in G_d.

Below are some straightforward results about the set Ω_d and the graph C_d from <cit.> that will be used in <ref>.



The set Ω_d,0 is nonempty if and only if p ∤ (d+1). If p ∤ (d+1), then Ω_d,0 contains one partition {d+2} which is an initial vertex of G_d. If p | (d+1), then Ω_d,1 consists of ⌈(d+1)/(p-2)/2p ⌉ partitions, and every vertex of G_d is larger than one of these.





Let p≥ 3. A partition is maximal if and only if its entries are all equal to two or three. Every integer r+1 with (d-1)/3 ≤ r ≤ d/2 occurs exactly once as the length of a maximal partition. There are ⌊ d/2 ⌋-⌈(d-4)/3⌉ maximal partitions. There is a unique maximal partition if and only if d ∈{1,2,3,4}. 




Lemma 2.4 of <cit.>.


The next proposition, which shows that G_d is connected when k has small characteristic, will be used to show 𝒜𝒮_g is connected.



If k has characteristic 2 or 3, then G_d is connected for all d.





Case p=2: Since d+2=2g/(2-1)+2 is even for any g, the partition E={d+2} is the unique minimal partition of Ω_d by Lemma <ref>. Thus, G_d is connected.

Case p=3: 




  * If d+2 ≢1  3, all partitions E∈Ω_d satisfy {d+2}≺E.  Thus G_d is connected.


    
  * If d+2 ≡ 1  3, i.e. d+2=3n+1 for some n ≥ 1, all the minimum partitions are of the form E={3N+2,3M+2} where N+M=n-1 by Lemma <ref>. All the branches of the graph G_d starting from these E_i must have the same maximal partition {3,…,3,2,2} by Lemma <ref>. Thus, G_d is connected.






The above proposition is not true when k has characteristic p>3. For instance, when p=5 then Ω_4 has only two partitions: {3,3} and {2,4}. Since it is clear that {3,3}⊀{2,4} or vice versa, G_4 is disconnected.




 
 Let p=5 and d=7. Here is the graph G_7 for Ω_7.
 
 


[-latex ,auto ,node distance =1.5 cm and 2.5cm ,on grid ,
semithick ] 
(C)
{9};
(A) [below left=of C] {5,4};
(B) [below right =of C] {7,2};
(D) [below =of C] {3,3,3} ;
(E) [below =of A] {4,3,2} ;
(F) [below =of B] {5,2,2} ;
(G) [below =of E] {3,2,2,2} ;
(C) edge node[below =0.15 cm]  (A);
(C) edge node[below =0.15 cm]  (D);
(C) edge node  (B);
(B) edge node  (F);
(B) edge [bend left =15] node[below =0.15 cm]  (E);
(A) edge node  (E);
(A) edge [bend right =15] node[below =0.15 cm]  (F);
(E) edge node  (G);
(F) edge node  (G);














 §.§ Moduli of Artin-Schreier Curves and Partitions of Integers













Recall that 𝒜𝒮_g is the moduli space that parameterizes Artin-Schreier curves of genus g. For detail, see <cit.>. From the fact that Artin-Schreier curves of the same genus g have the same invariant d+2, Pries and Zhu show that 𝒜𝒮_g can be partitioned by locally closed strata corresponding to elements of Ω_d. In particular, the partition E={e_1,…,e_r}  of d+2 is associated with the stratum 𝒜𝒮_g,E (or Γ_E) which is the collection of all the points of 𝒜𝒮_g that represent Artin-Schreier curves with ramification divisors of the form ∑_j=1^r (p-1)e_j P_j, where P_j are distinct points of ℙ^1_k.







The moduli space 𝒜𝒮_g is partitioned by a set of irreducible strata that is in one-to-one correspondence with elements of Ω_d. If E={e_1, e_2, …, e_r}, then the dimension d_E over k of the irreducible stratum AS_g,E is d-1-∑_j=1^r⌊ (e_j-1)/p ⌋.




Corollary 3.11 of <cit.>.


As a result of the above proposition, one has a necessary condition for one stratum to be in the closure of another stratum by comparing their dimensions. 


For a ∈ℤ_>0, let a̅ be the integer so that a̅≡ a  p and 0 ≤a̅ < p.


  
  * If the edge from E_1 to E_2 is of the form {e} → {e_1,e_2} with 2 < e̅_2+e̅_2 ≤ p and e̅_1e̅_2 ≠ 0, then _k (Γ_E_1)=_k (Γ_E_2) and Γ_E_1 is not in the closure of Γ_E_2 in 𝒜𝒮_g.
  
  * In all other cases, _k (Γ_E_1)=_k (Γ_E_2)-1.





Lemma 4.5 of <cit.>.




Suppose we have two strata connected by an edge in G_d. We will see later in Corollary <ref> that comparing dimensions also gives a sufficient condition for one stratum lie in the closure of another.




The original statement of Lemma 4.5 of <cit.> does not have the condition that e_1 e_2 ≠ 0, which is a mistake. Since, for instance, if p=5, E_1={7} and E_2={5,2}, then _k (Γ_E_1)=3=_k (Γ_E_2)-1.



Finally, it is known when the moduli space 𝒜𝒮_g is irreducible.




The moduli space 𝒜𝒮_g is irreducible in exactly the following cases: 
  * p=2; or 
  * g=0 or g=(p-1)/2; or 
  * p=3 and g=2,3,5.



Corollary 1.2 of <cit.>




The above corollary implies that the moduli space 𝒜𝒮_g is connected when p=2. 






§ STRATA LYING IN CLOSURES OF THE OTHER STRATA



In this section, we introduce some techniques to study whether a stratum lies in the closure of certain other ones in 𝒜𝒮_g.  This involves deformations of wildly ramified curves with non-constant branch locus.



 §.§ Some Deformation Results





Let us first prove a general result for k-varieties.




Suppose A is an affine ring of dimension d over the field k and m is a maximal ideal of A. Suppose moreover that we have a chain of prime ideals (0) ⊊ I ⊊ m of A with d>(A/I)=d_1>0. Then there exists a k-algebra homomorphism


    ϕ: A → k[[t]],


where ϕ^-1((t))=m and ϕ^-1(0) ⊉ 
 I.




By Noether's Normalization Lemma (see <cit.>, 13, Theorem 13.3), A is isomorphic as a k-algebra to k[x_1, … ,x_d, y_1, …, y_l]/(p_1,…, p_n) where each p_i is irreducible and of the form


    p_i(X,y_i)= y_i^L_i+Q^i_k_i-1(X)y_i^L_i-1+ … +Q^i_0(X),


with Q^i_k(X)=∑_ν∈ℕ^dα_νX^ν∈ k[x_1,…,x_d]. In other words, A is integral over k[x_1,…,x_d]. With a good choice of x_i and y_j, one may assume that m=(x_1, … ,x_d, y_1, …, y_l), I∩ k[x_1,…,x_d]=(x_d_1+1,…, x_d) ≠ (0) and Q^j_0(X) has no constant term for all j. Define a map


    ϕ: A=k[x_1, … ,x_d, y_1, …, y_m]/(p_1,…, p_m) →k((T)),


by mapping each x_i to T and each y_j to a root of 


    p_j(ϕ(X),Y_j)=Y_j^L_i+Q^i_k_i-1(ϕ(X))Y_j^L_i-1+ … +Q^i_0(ϕ(X))


in k((T)). Since Q^i_0(X) has no constant term, Q^i_0(ϕ(X)) is a multiple of T, i.e. v_T (Q^i_0(ϕ(X))∈ℤ_+. Let L/k((T)) be the splitting field of the polynomials p_j(ϕ(X),Y_j). Suppose [L:k((T))]=s. Then there exists t ∈ L such that L=k((t)). Because k((T)) is complete with respect to the T-adic valuation, the valuation v_T extends uniquely to k((t)) (see <cit.>, II, 4, Theorem 4.8). Moreover, the extension v_t is given by the formula:


    v_t(α)=1sv_T(N_k((t))/k((T))(α)).


Hence, v_t(ϕ(y_j))=1/sv_T(Q^j_0(X)^(s/L_j))=1/L_jv_T(Q^j_0(X)) ∈1/sℤ_+ for each j=1,2,… m. Thus, the map ϕ restricts to a k-algebra homomorphism:


    ϕ: k[x_1, … ,x_d, y_1, …, y_m]/(p_1,…, p_m) → k[[t]],


where each of the generators of A is mapped to a multiple of a positive integer power of t. The proposition follows easily. 








Suppose U and V are two irreducible disjoint subsets of a k-variety X and 0 <  U= U <  V=:d. If U is contained in the closure of V then for each point u of U, there exists a k[[t]]-point of X whose generic point is in V and whose closed point is u.





Suppose A ⊊ (U∪ V) is an irreducible affine variety of dimension equal to d that containing u. Then there exists a chain of prime ideals (0) ⊊ I ⊊ m of A where U∩ A=𝒱(I) and u=𝒱(m) and A>(A/I)=:d_1>0 and (A/m)=0. By Lemma <ref>, there exists a k-algebra homomorphism: 


    θ̃: A → k[[t]],


where θ̃^-1(t)=m and θ̃^-1(0) ⊉ 
 I. Thus θ̃ induces a morphism of k-varieties:
 
 
    θ:  k[[t]] → A,

 
where θ(𝒱(t))=𝒱(m) and θ(𝒱(0)) ∉𝒱(I), that is θ(𝒱(0))∈ (V∩ A). Hence, θ is a k[[t]]-point whose generic point is in V and whose closed point is u.



The next fact about germs of Artin-Schreier curves will help us to reduce the problem of constructing deformations for an Artin-Schreier curve to the local case.



Suppose ϕ: Y →ℙ^1_k is an Artin-Schreier cover and P ∈ℙ^1_k is a branch point of ϕ. Then the localization of ϕ at P is determined by the ramification jump at P. 




See Lemma 2.1.2 of <cit.>.





The next proposition, which is about the relation between the irreducible strata of 𝒜𝒮_g and deformations of wildly ramified curves, will be used intensively in <ref>.



Suppose E_1={e_1,e_2,…,e_n}≺E_2={f_11,f_12,…,f_1m_1,f_21,f_22,…,f_nm_n}, with {e_i}≺{f_i1,…, f_im_i} for i=1,2,…,n. Then Γ_E_1 is in the closure of Γ_E_2 if and only if there exist n Artin-Schreier covers Φ_1, Φ_2, …, Φ_n over S= k[[t]], where the generic fiber of Φ_i yields a k((t))-point of Γ_{f_1i,…, f_im_i} and the special fiber yields a k-point of Γ_{e_i}.





The “if” conditions of the above proposition can be rephrased as follows. There are n Artin-Schreier covers Φ_i:Y_i → Z_i ≃ℙ^1_S, for i=1,2, …,n, whose special fiber is an Artin-Schreier curve over k, branched at a point b_i with ramification jump e_i-1, and whose generic fiber is an Artin-Schreier curve over k((t)), branched at m_i points that specialize to b_i and which have ramification jumps f_i1-1,…, f_im_i-1. Consider a point in Γ_E_1 that represents an Artin-Schreier cover ϕ: Y →ℙ^1_k, which is wildly ramified at y_1,…,y_n ∈ Y over b_1,…,b_n ∈ℙ^1_k, respectively,  with ramification jumps e_1-1,…,e_n-1, respectively. Let ϕ̂_j be the germ of ϕ at y_j. The localization of the special fiber of Φ_j at its branch point has ramification jump e_j-1 by the hypothesis. Since the germ of an Artin-Schreier cover at its branch point is uniquely determined by the ramification jump at this point by Proposition <ref>, the germ in the previous sentence is isomorphic to ϕ̂_j. Thus, Φ_j is a deformation of a curve such that the localization at its branch point is isomorphic to ϕ̂_j.  Using formal patching, see e.g., <cit.>, one can construct a deformation Φ: 𝒴→ℙ_S^1 of ϕ locally via the deformations Φ_j of ϕ̂_j. It follows from the construction that Φ is a ℤ/p-cover whose special fiber is isomorphic to ϕ and whose generic fiber can be represented by a k((t))-point of Γ_E_2. Whence, the point in Γ_E_1 that indexes ϕ is in the closure of Γ_E_2.

Conversely, if Γ_E_1 is in the closure of Γ_E_2, then Γ_E_1< Γ_E_2. Thus, by Proposition <ref>, for each k-point of Γ_E_1 that represents a cover ϕ: Y →ℙ^1_k, there exists an Artin-Schreier curve Φ: 𝒴→ℙ^1_S so that the generic fiber yields a k((t))-point of Γ_E_2 and the special fiber is isomorphic to ϕ. Suppose ϕ is wildly ramified at y_1,…,y_n ∈ Y over b_1,…,b_n ∈ℙ^1_k, respectively,  with ramification jumps e_1-1,…,e_n-1, respectively. We can get each of the local deformations Φ_i simply by restricting Φ to the spectrum of the complete local ring 𝒪̂_𝒴, y_i.
 



The above proposition implies that it suffices to study the deformations of germs or, in other words, to reduce to the case E_1 has only one entry.









The following corollary will be used in <ref> to prove a result about disconnectedness of 𝒜𝒮_g.





Let E_1 and E_2 be as in the preceding proposition. Suppose Γ_E_1 is not in the closure of Γ_E_2. Then Γ_E_1 is disjoint from the closure of Γ_E_2.





Suppose that there is a k-point of Γ_E_1, which corresponds to a cover ϕ: Y →ℙ^1_k, that is in the closure of Γ_E_2. Then, there is a deformation Φ over S= k[[t]] whose generic fiber yields a k((t))-point of Γ_E_2 and its special fiber is isomorphic to ϕ. Thus, for each e_i,  there exists a deformation over S from a k((t))-point of Γ_{f_1i,…, f_im_i} to a k-point of Γ_{e_i} formed by restricting Φ. Hence, by the preceding proposition, Γ_E_1 is in the closure of Γ_E_2.





Motivated by Proposition 4.1 of  <cit.> and with the aim to find sufficient conditions for the moduli space 𝒜𝒮_g to be connected, we construct deformations that will prove Theorem <ref>. 






 Let e ≢1 p be an integer and {e_1, e_2, …, e_n} be a partition of e such that e_i ≢1 p for all i. Fix a projective line ℙ^1_k[[t]] and a k-point b of ℙ^1_k[[t]]. Identify a projective line ℙ^1_k((t)) with the generic fiber of ℙ^1_k[[t]]. Then there exists a ℤ/p-Galois cover ψ_S over S= k[[t]] whose special fiber is an Artin-Schreier cover over k, branched only at b with ramification jump e-1, and whose generic fiber is isomorphic to an Artin-Schreier cover ψ_η over ℙ^1_k((t)) that is branched at n distinct points a_1, a_2, …, a_n which specialize to b and which have ramification jumps e_i-1 at a_i in the following situations:



  * n=2 and ⌊e-1/p⌋ > ⌊e_1-1/p⌋ + ⌊e_2-1/p⌋, or

  * n=3 and e_i ≡p+1/2 p for all i, or

  * n=3 and e_i=p-1 for some i, or

  * n=p-m+1 and e_i=m+1 for all i, or

  * n=4 and {e_1,e_2,e_3,e_4}={3,2,2,2} or {3,3,2,2} where p=5.













The proof will be provided in <ref>. 




Proposition 4.1 of <cit.> is a special case of Theorem <ref> part (a).





  §.§.§ The Oort-Sekiguchi-Suwa Component



Suppose Ψ is a ℤ/p-cover over R=𝒲(k)[ζ_p]. In <cit.>, Bertin and Mézard study the versal deformation ring R_Ψ whose spectrum is the formal deformation space of Ψ. They construct explicitly a family of infinitesimal deformations for Ψ that is parameterized by an irreducible component of the formal deformation space, called the Oort-Sekiguchi-Suwa component. It turns out that the dimension of the component is equal to the dimension of the versal deformation ring. In particular, if Ψ has ramification jump m=pq-l, where l ∈ [1,p[, we have:


    _Krull R_Ψ= 
          q     l ≠ 1 
    
          q+1     l=1


For more details, see Theorem 5.3.3 of <cit.>. Most of the deformations that we will construct to prove Theorem <ref> are explicit examples of ones that do not lie in the Oort-Sekiguchi-Suwa component. In particular, only the local deformation ψ_S in part (a) of Theorem <ref>, under the additional condition e_1=p and e_2 ≤ p is an element of the component.

In the rest of this subsection, we shall realize the characteristic p fibers of the deformations in the Oort-Sekiguchi-Suwa component as special cases of ones that are used by Pries and Zhu in the proof of Proposition 4.1 of <cit.>. Consider an Artin-Schreier cover ϕ over k defined by an affine equation:


    y^p-y=t^-m,


where m=pq-l, l ∈ [1,p[. One can think of ϕ as a k-point of Γ_{m+1}. We will study the case l=1 and l  1 separately.


Case l=1. Let ζ=yt^q, then (<ref>) can be written as:


    ζ^p-t^(p-1)qζ=t.


Consider the Artin-Schreier cover Φ over S= k[[x_1, …, x_q]] given by the following affine equation:


    ζ^p-(t^q+x_1t^q-1+…+x_q)^p-1ζ=t.


The special fiber of Φ is clearly birationally equivalent to ϕ. On the generic fiber of Φ, we define Z=ζ/(t^q+x_1t^q-1+…+x_q). Equation (<ref>) can be written in familiar form:


    Z^p-Z=t(t^q+x_1t^q-1+…+x_q)^p=:F(t).


Let us study a(t)=t^q+x_1t^q-1+…+x_q. One can show that the polynomial is separable over k[[t]][[x_1,…,x_q]]. Suppose u_0,u_1, …, u_q are roots of x^q+x_1t^q-1+…+x_q over a fixed algebraic closure of k[[t]][[x_1,…,x_q]]. Then the right hand side of the above equation is:


    F(t)=t∏_i=1^q(t-u_i)^p.


Thus, 


    ((t-u_j)^pF(t))'=1∏_i=1 
     i  j^q(t-u_i)^p


is clearly not equal to 0 or undefined when we plug in t=u_j Hence, the minimal form of Artin-Schreier equation (<ref>) has precisely q distinct poles of order p-1. It follows from Lemma IV.2.3 of <cit.> that Φ is a deformation from a point of Γ_{pq} to a point of Γ_E_2 where E_2={p,p,…,p}_q.

Case l  1. Again, let ζ=yt^q. This time, equation (<ref>) is written as:


    ζ^p-t^(p-1)qζ=t^l.


Consider the Artin-Schreier cover Φ over S=k[[x_1, …, x_q-1]] defined by the equation:


    ζ^p-(t^q+x_1t^q-1+…+x_q-1t)^p-1ζ=t^l.


Again, the special fiber is birationally equivalent to ϕ. On the generic fiber, just like when l=1, define Z=ζ/(t^q+x_1t^q-1+…+x_q-1t). Equation (<ref>) becomes:


    Z^p-Z=t^l(t^q+x_1t^q-1+…+x_q-1t)^p.


By a similar argument as in the case l=1, the minimal form of Artin-Schrier equation (<ref>) has precisely a pole at 0 of order p-l and q-1 other distinct poles of order p-1. Thus, again by Lemma IV.2.3 of <cit.>, Φ is a deformation from a point of Γ_{m+1} to a point of Γ_E_2 where E_2={p,p,…,p_q-1,p-l+1}.


Let us compare those deformations above with one in <cit.>. Pries and Zhu's deformations in Proposition 4.1 of <cit.> are defined by equations of the form:


    y^p-y=1x^e_2-1(x-t)^e_1,




where p | e_1, say e_1=pn_1. One can check the minimal form of the Artin-Schreier equation (<ref>) has a pole at 0 of order e_2-1 and at t of order e_1-1. Thus, this is a deformation from a point of Γ_{e} to a point of Γ_{pn_1,e_2}. One can get all elements of the Oort-Sekiguchi-Suwa component by iterating Pries and Zhu's deformation.


When p=2, if E={e_1,e_2,…,e_n}∈Ω_d then e_i must all be even, or else e_i would be congruent to 1 modulo p. Thus, all deformations are formed by repeated applications of equation (<ref>).


 


 §.§ Some Closure Results



In this section, we show that the combinatorial data in the graph G_d gives partial information about how irreducible strata of 𝒜𝒮_g relate. In fact, we will see in <ref> that the graph G_d gives complete information about this question when p=2 or p=3.

Let us fix a prime p and construct a directed graph C_d. The vertices of the graph correspond to the partitions E in Ω_d. There is an edge from E to E' if and only if E≺E', and Γ_E lies in the closure Γ_E'. We will show in <ref> that for p=2 or p=3, the graph G_d is a subgraph of C_d.
 
In general topology, if one irreducible subset of a space lies in the closure of another, then they are contained in the same connected component of that space. Thus, if C_d is connected then so is 𝒜𝒮_g. From now on, we will study the connectedness of C_d.






Suppose E_1 ≺E_2. Then there is an edge from the former to the latter in C_d if each entry of E_1 splits into entries of E_2 of the form:

    
    
  * {e}→{e_1,e_2} where  ⌊e-1/p⌋ > ⌊e_1-1/p⌋ + ⌊e_2-1/p⌋.
    
  * {e}→{e_1,e_2,e_3} where e_i ≡p+1/2 p for i=1,2,3.
    
  * {e}→{p-1, e_1,e_2}. 
    
  * {(n+1)(p-n+1)}→{n+1, n+1, …,n+1}_p-n+1 where 1≤ n ≤ p-1.
    
  * {9}→{3,2,2,2} or {10}→{3,3,2,2} where p=5.
    






It suffices to show that if E_1 and E_2 satisfy the conditions listed above then Γ_E_1 is in the closure of Γ_E_2. By Proposition <ref>, we can reduce to the case E_1={e}.

(a) Let ϕ_o: Y_o →ℙ_k ^1 be an Artin-Schreier cover that corresponds to a k-point of Γ_E_1. The element e in the partition Γ_E⃗_1 indicates the ramification jump of a branch point b ∈ℙ_k^1 so that the ramification jump of ϕ_o above b is e-1.

Let S=(k[[t]]). By Theorem <ref> (a), there exists an Artin-Schreier cover ϕ_S over S whose special fiber is isomorphic to ϕ_o and whose generic fiber is branched at two points that specialize to b and that have ramification jumps e_1-1 and e_2-1. Furthermore, the ramification divisor is otherwise constant. Thus the generic fiber ϕ_S has partition E⃗_2. Thus Γ_E⃗_1 is in the closure of Γ_E⃗_2 in 𝒜𝒮_g.

Parts (b),(c),(d),(e), are proved using similar arguments as in part (a) and Theorem <ref> parts (b),(c),(d),(e).



Item (e) of the above lemma is used to lower the bound for the genus g so that the moduli space 𝒜𝒮_g is connected when the base field has characteristic 5.


Combining the result above with the discussion in Section <ref> about the two types of edges in the directed graph G_d, we have the complete answer for Open Question 1 of <cit.>: the necessary and sufficient conditions on the edge {e}→{e_1,e_2} or the edge {e}→{e_1,e_2,e_3} in G_d for Γ_E⃗_1 to be in the closure of Γ_E⃗_2 in 𝒜𝒮_g.







  * If E⃗_1 ≺E⃗_2 with an edge of the form {e}→{ e_1, e_2} in G_d from E⃗_1 to E⃗_2, then Γ_E⃗_1 is in the closure of Γ_E⃗_2 if and only if ⌊e-1/p⌋ > ⌊e_1-1/p⌋ + ⌊e_2-1/p⌋.


  * If E⃗_1 ≺E⃗_2 with an edge of the form {e}→{ e_1, e_2, e_3} in G_d from E⃗_1 to E⃗_2, then Γ_E⃗_1 is in the closure of Γ_E⃗_2. 


  * Suppose E⃗_1 ≺E⃗_2 with an edge between them in G_d. Then Γ_E⃗_1 is in the closure of Γ_E⃗_2 if and only if _k(Γ_E⃗_1)< _k(Γ_E⃗_2).







For edge of the form {e}→{ e_1, e_2}:. The “if” direction follows from part (a) of Theorem <ref>. Conversely, if ⌊e-1/p⌋ = ⌊e_1-1/p⌋ + ⌊e_2-1/p⌋, then by theorem 3.10 of <cit.>, dim_k (Γ_E_1)=dim_k (Γ_E_2) and Γ_E_1 is not in the closure of Γ_E_2 in 𝒜𝒮_g.

For edge of the form {e}→{ e_1, e_2, e_3}: In order to have an edge of type {e}→{e_1,e_2,e_3} in G_d, we must have e_1+e_2 ≡ 1  p, e_1+e_3 ≡ 1  p and e_2+e_3 ≡ 1  p. Thus, e_1 ≡ e_2 ≡ e_3 ≡p+1/2 p. This is then an immediate result of part (b) of Theorem <ref>.

Since an edge in G_d is either of a type in part (a) or type in part (b), part (c) easily follows.









 

 



§ CONNECTEDNESS



In this section, we will prove Theorem <ref>. Let's take a look at one example.



Let p=5 and g=14. Then d=7. Using Lemma <ref>, we can draw a subgraph of C_7.


 

[-latex ,auto ,node distance =1.5 cm and 2.5cm ,on grid ,
semithick] 
(C)
{9};
(A) [below left=of C] {5,4};
(B) [below right =of C] {7,2};
(D) [below =of C] {3,3,3} ;
(E) [below =of A] {4,3,2} ;
(F) [below =of B] {5,2,2} ;
(G) [below =of E] {3,2,2,2} ;
(C) edge [bend right =15] node[below =0.15 cm]   (A);
(C) edge node[below =0.15 cm]   (D);
(B) edge [bend left =15] node[below =0.15 cm]   (E);
(B) edge node[below =0.15 cm]   (F);
(C) edge [bend right =90] node[below =0.15 cm]   (G);
(C) edge [bend left =5] node[below =0.15 cm]   (E);





































Since the subgraph is connected, the graph C_7 is connected. Thus, follow the discussion at the beginning of <ref>, 𝒜𝒮_14 is also connected.








Suppose E_1={e,e_3,…,e_n}≺E_2={e_1,e_2,e_3,…,e_n}  with an edge in G_d of the form {e} to {e_1,e_2} such that e_1 >p and Γ_E_1 is not in the closure of Γ_E_2. Then Γ_E_1 and Γ_E_2 are both in the closure of Γ_E_3, where  E_3={p-1,e_1-p+1,e_2,e_3,… e_n}.







Suppose e_i-1=pr_i+s_i where 1 ≤ s_i ≤ p-1 and r_1 ≥ 1. Then e-1=p(r_1+r_2)+s_1+s_2+1. By Corollary <ref>, Γ_E_1 is not in the closure of Γ_E_2 if and only if ⌊e-1/p⌋ = ⌊e_1-1/p⌋ + ⌊e_2-1/p⌋, or 3 ≤ s_1+s_2+1 ≤ p-1. Thus, e_1-p=p(r_1-1)+s_1+1, and s_1+1 <p-1. It follows that ⌊e_1-1/p⌋> ⌊p-1/p⌋ + ⌊e_1-p+1-1/p⌋. Whence, Γ_E_2 is in the closure of Γ_E_3 by Lemma <ref> (a). Also, by Lemma <ref> (c), Γ_E_1 is in the closure of Γ_E_3.








The above proposition gives us a strategy to study the connectedness of C_d: even though there is no edge on C_d directly connecting two vertices, we can still go from one vertex to another through a third one. The following corollary will show that for most choices of the genus g, we can connect all the vertices of C_d corresponding to partitions where at least one entry has a value greater than or equal to p.





Fix a base field of characteristic p>0. 


    
  * If d+2 ≢1  p, then all vertices of C_d that correspond to partitions with at least one entry having a value greater than or equal to p lie in the same connected component.
    
  * If d+2 ≡ 1  p, then all vertices of C_d that correspond to partitions with at least one entry having a value greater than or equal to p are in the same connected component with a partition on the second level.
 




Case d+2 ≢1  p. By Lemma <ref>, we have that E≻{d+2} for all E∈Ω_d. Thus, it suffices to show that all partitions having at least one entry greater than or equal to p can be connected with {d+2} in C_d.

Suppose E={e_1,…,e_n} with e_1 > p. Then there is a “path” from {d+2} to {e_1,…,e_n} in G_d such that each partition in the path has at least one entry greater than p. By Proposition <ref>, all the partitions in the path are in the same connected component in C_d. In other words, one can go from {d+2} to E in C_d.

If E={p,e_2, …,e_n} then ⌊p+e_2-1/p⌋=1+⌊e_2-1/p⌋>⌊p-1/p⌋ + ⌊e_2-1/p⌋=⌊e_2-1/p⌋. Thus, there is an edge from E'={p+e_2, …, e_n} to E in C_d by Lemma <ref>(a). By the argument in the previous paragraph, E' is in the same component with {d+2}, then so is E.

Case d+2 ≡ 1  p. Consider a partition E={e_1,…,e_n}∈Ω_d. Suppose, without loss of generality, that e_1 >p and e_n ≢0 p. Then, the number e'=∑_i=1^n-1e_i is greater than p and not congruent to 1 modulo p. Thus, E={e_1,…,e_n}≺E'={e',e_n}, and by the previous discussion, {e'} and {e_1, …, e_n-1} are linked in C_e'-2. Thus, E' and E lie in the same connected component in C_d.



Corollary <ref> implies that the problem of determining whether 𝒜𝒮_g with d+2 ≢1  p is connected can be reduced to the study of whether all strata indexed by partitions with all entries less than p contain in their closure at least one stratum enumerated by a partition with at least one entry greater than or equal to p.






 §.§ Case p=3





The strata E_1, E_2 satisfy E_1 ≺E_2 if and only if Γ_E_1 is in the closure of Γ_E_2.




“If” follows from Lemma 4.3 of <cit.>.

Conversly, suppose E_1 and E_2 are connected by an edge of the form {e}→{e_1,e_2}. Suppose moreover that e_2-1=3a+u and e_1-1=3b+v with 1 ≤ u,v <3. Then e-1=e_1+e_2-1=3(a+b)+u+v+1. In order for 3 ∤ (e-1), we need u+v+1  3. Moreover, since u,v ≥ 1, we must have u+v+1>3, thus ⌊e-1/3⌋ > ⌊e_1-1/3⌋ + ⌊e_2-1/3⌋. Hence, by Lemma <ref> (a),  Γ_E_1 is in the closure of Γ_E_2. If the edge connecting E_1 and E_2 is of the form {e}→{e_1,e_2,e_3}, we get the same result by Lemma <ref> (b). Thus, one can conclude that for any E_1 ≺E_2, Γ_E_1 is in the closure of Γ_E_2.













When the base field has characteristic 3, the moduli space 𝒜𝒮_g is connected for any value of g.






Just like the case p=2, Proposition <ref> implies that the graph G_d is a subgraph of C_d. Hence, in order to show that 𝒜𝒮_g is connected, it suffices to prove that G_d is connected, which follows from Proposition <ref>.





 §.§ Case p=5



 Given p=5, the graph C_d is connected when d ≥ 7. In particular, 𝒜𝒮_g is connected when g ≥ 14.







When g ≥ 14, then d+2 ≥ 9 so d ≥ 7.





Case d+2 ≢1  5: It suffices to show that every partition of Ω_d lies in the same connected component with {d+2} in C_d.




    

  * If E={e_1,…,e_n} has at least one entry greater than or equal to 5, E is in the same component with {d+2} by Proposition <ref>.



  * If E={e_1,…,e_n} has no entry greater than 5, then e_i can only be 4,3 or 2. 

Suppose there exists an entry equal to 4, say e_1=4. Since d+2 ≥ 9, there exists at least one more entry, say e_2. Let e'=e_1+e_2. If e_2=3 or e_2=4, then e'≢1  5 and ⌊e'-1/5⌋ = ⌊e_1+e_2-1/5⌋ >  ⌊4+2-1/5⌋ =1 and ⌊e_1-1/5⌋ +  ⌊e_2-1/5⌋ =0 . Thus, by Lemma <ref>(a), Γ_{e'} is in the closure of Γ_{e_1,e_2}, that means Γ_{e_1,…,e_n} is in the closure of Γ_{e',…,e_n} which is in the same component with Γ_{d+2}. If e_i=2 for all i ≥ 2, then n > 3. Thus, there is an edge from {8,e_4,…,e_n} to {4,2,2,e_4,…,e_n} in C_d by Lemma <ref>(c), and {8,e_4,…,e_n} is in the same component with {d+2}.


Suppose all entries are equal to 2 or 3. If there are more than three entries equal to 3 or more than 5 entries equal to 2, then, by Lemma <ref>(b) and (d), there is an edge in C_d that connects E with a partition E', which is different from E by edges in G_d of the form {9}→{3,3,3} or {10}→{2,2,2,2,2}. So, E is in the same component with  {d+2}. Otherwise, since d+2 ≥ 9, we are left with the only possible partitions: {3,2,2,2} or {3,3,2,2}. These particular cases are solved by Lemma <ref> (e) . 






Case d+2 ≡ 1  5: All the minimal deformations on the second level are of the form {5a_1+2,5a_2+4} or {5b_1+3,5b_2+3} where ∑_i a_i=∑_j b_j. It follows that:


    {5b_1+3,5b_2+3}≺{3,…,3_b_1+b_2+2,2,…,2_b_1+b_2} and {5a_1+2,5a_2+4}≺{3,…,3_a_1+a_2,2,…,2_a_1+a_2+3}.


Thus, by an argument similar to the case d+2 ≢1  5, all partitions on the branches started by ones of the form  {5a_1+2,5a_2+4} (or {5b_1+3,5b_2+3}) are in the same components in C_d.

In order to show C_d is connected, it now suffices to show that {5a_1+2,5a_2+4} and {5b_1+3,5b_2+3} are in the same component in G_d for some values of a_1,a_2,b_1 and b_2. Again, by the condition that d+2 ≥ 9, the smallest d+2 that can be congruent to 1 mod 5 is 11. Thus, there exists in the second level a partition of the form {5a_1+2,5a_2+4} where a_2 ≥ 1. Whence {5a_1+2,5a_2+4}≺{5a_1+2,5(a_2-1),9}≺{3,…,3,2,…,2}, where the number of entries of the latter partition equal to 3 is a_1+a_2-1+3=a_1+a_2+2=b_1+b_2+2 and the number of entries equal to 2 is a_1+1+a_2-1=a_1+a_2=b_1+b_2. Thus, all partitions are in the same component in C_d.












 §.§ Case p>5



Given p>5, if d≥p(p-1)^2/4 or d ≤ 1 then C_d is connected. In particular, the moduli space 𝒜𝒮_g is connected when g≥(p^3-2p^2+p-8)(p-1)/8 or g ≤p-1/2.



 Case d+2 ≢1  p: By Proposition <ref>, all partitions of Ω_d with at least one entry greater than or equal to p lie in the same connected component of C_d. Thus, it suffices to show that when g ≥(p^3-2p^2+p-8)(p-1)/8 i.e d ≥p(p-1)^2/4, every partition of the form {e_1,e_2,…,e_n} such that e_i <p for all i can be connected with some partition having an entry not less than p. 

Consider a partition of d+2 with all entries smaller than or equal to p+1/2. By the Pigeonhole Principle, we can guarantee to have p entries with the same value n between 1 and p-1/2 when the partition has at least p(p-1)/2 entries. Thus, when d ≥p(p-1)^2/4, the partition we consider is in the same component with E={(n+1)p-n^2+1,e_2,…,e_m} for some n between 1 and p-1/2 by Lemma <ref>(d). The first entry of E is greater than p. Hence, E and {d+2} are linked.

Consider a partition of d+2, called E={e_1,e_2, …,e_n}, where one entry has value m such that p+1/2<m<p. When ∑_i=1^n e_i>p(p-1)^2/4, we must have the sum of all entries except the one with value m greater than p(p-1)^2/4-m. If there is another entry which has value greater than p-m+1, call the value n, then the corresponding stratum lies in the same connected component with one indexed by {m+n, …}≺{m,n, …}=E by Theorem <ref>(a). Otherwise, all other entries are smaller than or equal to p-m+1. Again by the Pigeonhole Principle, we can guarantee to have p entries with the same values n such that 1≤ n≤ p-m+1 when the partitions have at least p(p-m+1)+1 entries (“+1” coming from the entry with value m). Thus, if p(p-1)^2/4-m< p(p-m+1)^2, by a similar argument as in the previous paragraph,  E and {d+2} are in the same connected component. In order to prove the above claim, we study the function:


    F(m):=p(p-m+1)^2-p(p-1)^24+m.








One can show that F(m) is strictly increasing on the closed interval [p+1/2, p-1]. Moreover, the value of F(m) at (p+1)/2 is:  F(p+1/2)=p+1/2>0. Hence, F(m) is positive on the whole interval [p+1/2, p-1] and we are done.



Case d+2 ≡ 1  p: All deformations on the second level are of the form {a+pe_1,p+1-a+pe_2} where 2≤ a ≤ p-1. Suppose d+2 is odd. Then one of the entries of each partition in the second level is even and the another is odd. Thus, we have {a+pe_1,p+1-a+pe_2}≺{2,2,…,2,3} for all possible a, e_1, e_2. Also, the number of entries equal to 2 in the latter partition is greater than p and one of the entries of the former partition is greater than 2p by the assumption that d+2 ≥p(p-1)^2/4. We have a chain of partitions {a+pe_1,p+1-a+pe_2}≺{2p,e+(e_1-2)p,p+1-a+pe_2} (or {2p,e+e_1 p,1-a+p(e_2-1)}) ≺{2p,2,…,2,3}≺{2,2,…,2,3} where the first three are in the same component of C_d by Corollary <ref>. Applying Lemma <ref>(d) for n=1, we know that there is a path in C_d connecting {2p,2,…,2,3} and {2,2,…,2,3}. Furthermore, {a+pe_1,p+1-a+pe_2} and {2p,2,…,2,3} are linked by Corollary <ref>. Hence, the partition {a+pe_1,p+1-a+pe_2} lies in the same connected component with {2,2,…,2,3}. Similarly, if d+2 is even, {a+pe_1,p+1-a+pe_2} can be linked with {2,2,…,2,3,3} in C_d and the proof follows as in the odd case.


When d ≤ 1, i.e. when d=1 or d=0, then Ω_d has exactly one partition: either {3} or {2}. Thus, the graph C_d is connected.






 §.§ Disconnectedness of AS_g





Suppose p ≥ 5. Then 𝒜𝒮_g is disconnected when p-1/2 < g ≤(p-1)^2/2.





If g ≤(p-1)^2/2, then d+2≤ p+1. This implies that all elements of Ω_d have entries with values less than p. By Corollary 3.16 of <cit.>, each irreducible stratum of 𝒜𝒮_g has dimension equal to d-1. Thus, no closure of one stratum contains another. Hence, by Corollary <ref>, each stratum is a connected component of 𝒜𝒮_g. Whence, if g satisfies the second inequality and Ω_d has more than one partition in Ω_d then 𝒜𝒮_g is disconnected. By Corollary <ref>, 𝒜𝒮_g is irreducible if and only if g=0 or g=p-1/2, which implies the first inequality.





The only obstructions for a deformation that have been known thus far were the dimensions of the strata and the “≺” relation of the partitions which says E_1 ≺E_2 if E_2 is a refinement of E_1. More precisely, there is a deformation from  Γ_E_1 to Γ_E_2 only if _k(Γ_E_1)< _k(Γ_E_2) and E_1 ≺E_2. Our new approach, which was mentioned in Remark <ref>, gives us another obstruction and improves Proposition <ref>. In particular, we are able to show that for p>5, 𝒜𝒮_g disconnected if (p-1)/2<g ≤ 2(p-1).


 
 
 
 

§ PROOF OF TECHNICAL RESULTS

 
 
 
 In this section, we give the proof of Theorem <ref>. Each subsection will prove one item of this theorem.
 
By the discussion at the beginning of <ref>, it suffices to study relations between two partitions E_1={e}≺E_2={e_1,e_2, …, e_n}.  





 §.§ Edge of Type {e}→{e_1,e_2}


Throughout this subsection, suppose E_1={e}≺E_2={e_1,e_2}. Then there is an edge in G_d from E_1 to E_2. Given e=e_1+e_2 such that p ∤ (e-1)(e_1-1)(e_2-1). Without loss of generality, one can assume that e_1 ≥ e_2. Let pw be the smallest multiple of p that is greater than or equal to e_1. Before going into the technical details, let us take a look at a concrete example.


Let p=5, g=10 and consider E_1={7} and E_2={4,3}. Then there is a deformation over S= k[[t]] given by the normalization of ℙ^1_S over the extension of its fraction field that is defined by the following affine equation:


    y^5-y=-2x+t(-2)x^5(x-t)^2=H(x,t).


The special fiber has the affine equation:


    y^5-y=1x^6,


which represents a k-point of Γ_E_1.

On the generic fiber, when t ≠ 0, the partial fraction decomposition of H(x,t) is of the form:


    -12tx^5+12t^3x^3+1t^4x^2+32t^5x+12t^4(x-t)^2-32t^5(x-t)


One can obtain a ℤ/p-curve that is isomorphic to the generic fiber by adding 1/2tx^5-1/√(2t)x to H(x). The curve is thus defined by the affine equation:


    y^5-y=-1√(2t)x+12t^3x^3+1t^4x^2+32t^5x+12t^4(x-t)^2-32t^5(x-t).


Hence, the generic fiber is branched at two points x=0 and x=t, which have ramification jumps 3 and 2, respectively. Thus, it represents a k((t))-point of Γ_E_2. 




The previous example also illustrates our main approach in the whole chapter. We will look for a rational function H(x,t) ∈(k[x,t])  such that H(x,0)=c/x^e-1 and has a desired partial fraction decomposition in terms of x. 



 




Suppose v ≤ u. Then:



    ∑_k=0^v u  k(-1)^k =(-1)^vv!∏_l=1^v (u-l).






Induction on v.

For v=1:  


    ∑_k=0^1 u  k(-1)^k =u  0 -u  1=1-u=(-1)^11!∏_l=1^1 (u-l).


Suppose the proposition is true for v=n. Then, for v=n+1:


    ∑_k=0^n+1u  k(-1)^k     =∑_k=0^n u  k(-1)^k + (-1)^n+1(n+1)!∏_m=0^n (u-m) 
       =(-1)^nn!∏_l=1^n (u-l)+ (-1)^n+1(n+1)!∏_m=0^n (u-m)  (by induction) 
        =(-1)^n+1(n+1)!∏_l=1^n+1 (u-l) .









Suppose ⌊e-1/p⌋ > ⌊e_1-1/p⌋ + ⌊e_2-1/p⌋. Then p ∤e_2-1pw-e_1,  p ∤e_2-1pw-e_1+1 and p ∤∑_k=0^pw-e_1e_2-1 k (-1)^e_2-1-k=:Z.






Suppose e_2-1=pa+u, and e_1-1=pb+v such that  1 ≤ u,v <p and either a ≤ b or u ≤ v if a=b. Thus, e-1=e_1+e_2-1=p(a+b)+u+v+1, and pw=p(b+1). Because ⌊e-1/p⌋ > ⌊e_1-1/p⌋ + ⌊e_2-1/p⌋, we have p+1 ≤ u+v+1 ≤ 2p-1.

Now consider:


    e_2-1pw-e_1=(e_2-1)!(pw-e_1)!(e_1+e_2-1-pw)!=1(pw-e_1)! (e_1+e_2-pw) … (e_2-1).


Because e_1+e_2-pw=pa+u+v-p+2 ≥ pa+2, we have p ∤   (e_1+e_2-pw) … (e_2-1). Thus, p ∤e_2-1pw-e_1.



Similarly,


    e_2-1pw-e_1+1=(e_2-1)!(pw-e_1+1)!(e_1+e_2-2-pw)!=1(pw-e_1+1)! (e_1+e_2-1-pw) … (e_2-1).

 
Because  e_1+e_2-1-pw=pa+u+v-p+1 ≥ pa+1, we have p ∤   (e_1+e_2-1-pw) … (e_2-1). Thus, p ∤e_2-1pw-e_1+1.



Now we want to show p ∤ Z=∑_k=0^pw-e_1(e_2-k)… (e_2-1)1· 2 … k (-1)^e_2-1-k. Taking reduction modulo p, it suffices to prove that:


  

    p ∤∑_k=0^p-v-1u  k (-1)^k  for  1 ≤ u,v <p , u+v ≥ p.






Apply Lemma <ref>, ∑_k=0^p-v-1u  k (-1)^k=(-1)^p-v-1(p-v-1)!∏_l=1^p-v-1 (u-l). Since p ∤ (u-l) for any 1 ≤ l ≤ (p-v-1), we conclude that p∤∑_k=0^p-v-1u  k (-1)^k.












Suppose ⌊e-1/p⌋ > ⌊e_1-1/p⌋ + ⌊e_2-1/p⌋. Then there exists a rational function G(x,t) ∈(k[x,t]) that has partial fraction decomposition in k(t)(x) of the form:



    G(x,t)= ∑_i=1^e_1-1a_ix^i + a_pwx^pw + ∑_j=1^e_2-1b_j(x-t)^j




such that a_pw, a_e_1-1, b_e_2-1 0; a_i,b_j are elements of k(t); and G(x,0) ∈(k[x]) has exactly one pole of order e-1 at x=0.










If G(x,t) is the rational function we are looking for, it should have the fraction form:



    G(x,t)= ∑_i=1^e_1-1 a_i x^pw-i (x-t)^e_2-1 + a_pw(x-t)^e_2-1+ ∑_j=1^e_2-1 b_j x^pw (x-t)^e_2-1-jx^pw(x-t)^e_2-1=N(x,t)D(x,t).






We want the numerator of G(x,t), called N(x,t), to lie in k[t][x] and N(x,0) to be a k^×- multiple of x^pw-e_1.

In order to construct such a function, it suffices to find values for a_pw, a_1, …,a_e_1-1,b_1, …, b_e_2-1 in k(t) that satisfy the following conditions:


    
  * Each term of N(x,t) having power of x less than pw-e_1 is a multiple of t.
    
  * The term with power of x equal to pw-e_1 is not a multiple of t.
    
  * All the terms of N(x,t) having x-degree greater than pw-e_1 are equal to 0.
    
  * a_pw, a_e_1-1 and b_e_2-1 are not equal to 0.




Observe that all the terms of b_j x^pw (x-t)^e_2-1-j have x-degree at least pw, which is strictly greater than pw-e_1. The terms of a_i x^pw-i (x-t)^e_2-1 have x-degree at least pw-e_1+1, which is also greater pw-e_1. Thus, all the terms of N(x,t) with x-degree less than or equal to pw-e_1 lie in a_pw (x-t)^e_2-1. Note that e_2-1 > pw-e_1 follows from the assumption that  ⌊e-1/p⌋ > ⌊e_1-1/p⌋ + ⌊e_2-1/p⌋. 







The term of N(x,t) with x-degree n, for pw-e_1+1 ≤ n ≤ pw+e_2-2, has coefficient of the form:






    ∑_pw-i+r=n 
     1 ≤ i ≤ e_1-1 
     0 ≤ r ≤ e_2-1  a_i e_2-1r (-t)^e_2-1-r+ ∑_pw+l=n 
     0 ≤ l ≤ e_2-1-j 
     1 ≤ j ≤ e_2-1  b_j e_2-1-jl (-t)^e_2-1-j-r+a_pwe_2-1 n(-t)^e_2-1-n



We want all these terms to be equal to 0, i.e., a_1, …, a_e_1-1,b_1, …, b_e_2-1 are the solutions of the linear system:


    ( [ A_ij ]) ×( [     b_1;     b_2;       ⋮; b_e_2-2; b_e_2-1;     a_1;     a_2;       ⋮; a_e_1-2; a_e_1-1 ]) = ( [                                       0;                                       0;                                       ⋮;                                       0;                                   -a_pw;                               -a_pw(-t);                                       ⋮;                                       ⋮; -a_pwe_2-1pw-e_1+2(-t)^e_2-1-(pw-e_1+2); -a_pwe_2-1pw-e_1+1(-t)^e_2-1-(pw-e_1+1) ])
 

Where A_ij is a size (e_1+e_2-2) square matrix of the form: 






    [
    [           1           0           …           0           0           1           0           …           0           0; (e_2-2)(-t)           1           …           …           …           …           …           …           …           …;           …           …           …           …           …           …           …           …           …           …;           …           …           …           1           0           …           …           …           …           …;           …           …           …           …           1           …           …           …           …           …;           0           0           …           0           0  (-t)^e_2-1           …           …           …           …;           0           0           …           0           0           0  (-t)^e_2-1           …           …           …;           …           …           …           …           …           …           …           …           …           …;           0           0           …           0           0           0           0           …  (-t)^e_2-1           …;           0           0           …           0           0           0           0           …           0  (-t)^e_2-1 ]]



The entry a_ij of the matrix is the multiple of b_j if 1 ≤ j ≤ e_2-1 or the multiple of a_j-e_2+1 if e_2 ≤ j ≤ e_1+e_2-2 in the coefficient of the term with x-degree pw+e_2-1-i. The right hand side of the linear system is a (e_1+e_2-2) × 1 matrix where the i-th row is the coefficient of x-degree pw+e_2-1-i term of the polynomial -a_pw(x-t)^e_2-1. One can easily see that A_ij is a block triangular matrix. The size of the blocks, clockwise from top, are (e_2-1)× (e_2-1), (e_1-1)× (e_1-1), (e_2-1)× (e_2-1), and (e_1-1)× (e_1-1). Thus, it is invertible and we can always find the desired values for a_pw, a_1, …,a_e_1-1,b_1, …, b_e_2-1.

Now consider:


    a_pw (x-t)^e_2-1= ∑_r=0^e_2-1 a_pwe_2-1 r x^r (-t)^e_2-1-r.


Pick a_pw=1t^e_1+e_2-1-pw. Then  G(x,t) has the form:


    G(x,t)= ∑_r=0^pw-e_1e_2-1 rx^r (-1)^e_2-1-r t^pw-e_1-rx^pw(x-t)^e_2-1.



With that choice of a_pw, conditions 1, 2, 3 and the part of condition 4 that a_pw≠ 0 are satisfied.

 It now suffices to show that a_e_1-1≠ 0 and b_e_2-1≠ 0. Consider the coefficient of the term of N(x,t) with x-degree pw-e_1+1, which is equal to 0 by the previous construction:


    a_e_1-1 (-t)^e_2-1 + a_pwe_2-1pw-e_1+1(-t)^e_2-1-(pw-e_1+1)=a_e_1-1 (-t)^e_2-1 +e_2-1pw-e_1+1 (-1)^e-1-pw1t.


By Lemma <ref>, p ∤e_2-1pw-e_1+1. So, in order for the  pw-e_1+1^th coefficient to be 0, we must have a_e_1-1 0. In particular, we have a_e_1-1=e_2-1pw-e_1+1 (-1)^e_1-e_2+1-pw t^-e_2.





Now, we want to prove that b_e_2-1 0. Consider:


    H(x,t)=G(x,t)(x-t)^e_2-1=∑_r=0^pw-e_1e_2-1 rx^r (-1)^e_2-1-r t^pw-e_1-rx^pw.


The coefficient of the leading term of the expansion of H(x,t) around x=t coincides with the coefficient of the leading term of the expansion of G(x,t) around x=t, i.e., b_e_2-1. Note that:


    H(t,t)= ∑_r=0^pw-e_1e_2-1 r  (-1)^e_2-1-rt^e_1=b_e_2-1.

By Lemma <ref>, p ∤∑_r=0^pw-e_1e_2-1 r (-1)^e_2-1-r. So b_e_2-1 0.







The following proposition will prove part (a) of Theorem <ref>.





Suppose ⌊e-1/p⌋ > ⌊e_1-1/p⌋ + ⌊e_2-1/p⌋ . Suppose moreover that ψ_0 is an Artin-Schreier cover over k, branched at a point b with ramification jump e_1+e_2-1. Then there exists an Artin-Schreier cover ψ_S over S= k[[t]] whose special fiber is isomorphic to ψ_0, whose generic fiber is branched at two points that specialize to b and which have ramification jumps e_1-1 and e_2-1, and which is unramified everywhere else.




We use notation from Lemma <ref>. 

By Proposition <ref> and <ref>, it suffices to construct a deformation for the germ of ψ_0 at b.

Let us study the Artin-Schreier cover Ψ over S= k[[t]] given by the normalization of ℙ^1_S over the extension of its fraction field that is defined by the following affine equation:



    y^p-y =∑_r=0^pw-e_1e_2-1 rx^r (-1)^e_2-1-r t^pw-e_1-re_2-1pw-e_1(-1)^e_1+e_2-1-pwx^pw(x-t)^e_2-1:=F(x,t)


where the right-hand-side of (<ref>) is just a scalar multiple of (<ref>). We will show that Ψ is locally a deformation of ψ_0 that satisfies the conditions we seek.




On the special fiber of Ψ, when t=0, we have:


    F(x,0)=e_2-1pw-e_1(-1)^e_1+e_2-1-pwe_2-1pw-e_1(-1)^e_1+e_2-1-pwx^e-1=1x^e-1.


Thus, the normalization of the special fiber Ψ_s of Ψ has ramification jump e-1 at its branch point. Hence, Ψ_s is birationally equivalent to ψ_0.

On the generic fiber, when t  0, the cover Ψ_η is branched above x=0 and x=t. The order of the pole of F(x,t) at x=t is e_2-1 by the construction from Proposition <ref>. Since, e_2-1 is prime to p by hypothesis, the ramification jump above x=t is e_2-1. 

The expansion of F(x,t) around x=0 is of the form:


    a_pwx^pw+∑_i=1^e_1-1a_ix^i +  terms with non negative x-degree,


Where a_pw=1e_2-1pw-e_1(-1)^e_1+e_2-1-pwt^e_1+e_2-1-pw. After a finite inseparable extension of k((t)) with equation t_1 ^p =t, the leading term of F(x,t) is a p-th power. The second term is a_e_1-1=e_2-1pw-e_1+1 (-1)^e_1-e_2+1-pw t^-e_2, which is non-zero and thus becomes the leading term of the affine equation in standard form for Ψ_η. Thus, the ramification jump above x=0 is e_1-1.

Hence, the cover Ψ_η is branched at two points that specialize to 0 which have ramification jumps e_1-1 and e_2-1. By Lemma <ref> and Lemma IV.2.3 of <cit.>, Ψ is a deformation of ψ_0 over S.






Suppose p | e_1 e_2. Without loss of generality, we may assume p | e_1. Then, we can use pw=e_1 and equation (<ref>) becomes:


    y^p-y=1x^e_1(x-t)^e_2-1.


Which is precisely equation (<ref>) after a change of variables.








 
 
 


 §.§ Edge of Type {e}→{e_1,e_2,e_3}


From the discussion in the beginning of section <ref>, we know that there is an edge of type {e}→{e_1,e_2,e_3} in G_d if and only if e_i ≡p+1/2 p. Let pw=e_1+p-1/2 be the smallest multiple of p greater than or equal to e_1.

The proof of the below proposition is easy, and we will leave it for the reader.



Suppose i<p. Then a  i≡a i p, for 0 ≤a <p, a ≡a p.


















 The polynomial ∑_i=0^p-1/2p-1/2i^2  x^p-1/2-i is separable over k[x] where k has characteristic p.




Theorem 4.1 in chapter V of <cit.>.








Suppose ψ_0 is an Artin-Schreier cover over k, branched at a point b with ramification jump e_1+e_2+e_3-1. Then there exists an Artin-Schreier cover Ψ over S= k[[t]] whose special fiber is isomorphic to ψ_0, whose generic fiber is branched at three points that specialize to b and which have ramification jumps e_1-1, e_2-1 and e_3-1, where e_i ≡p+1/2 p, and whose ramification divisor is otherwise constant.







Let us study the Artin-Schreier cover Ψ over S= k[[t]] given by the normalization of ℙ^1_S over the extension of its fraction field that is defined by the following affine equation:


    y^p-y =∑_n=0^p-3/2( ∑_i+j=np-1/2 ip-1/2 j r^e_3-1-j(-1)^p-3/2-n)t^p-3/2-nx^n-tx^p-1/2(∑_i+j=p-3/2p-1/2 ip-1/2 j r_0^e_3-1-j)x^pw(x-t)^e_2-1(x-tr)^e_3-1=:F(x,t),





where r is a solution of:


    ∑_i+j=p-1/2p-1/2ip-1/2j r^e_3-1-j =t^2.
 



over a fixed algebraic closure of k((t)) such that its value when t=0, called r_0, satisfies ∑_i+j=p-3/2p-1/2ip-1/2j r_0^e_3-1-j 0. We will show that Ψ is locally a deformation of ψ_0 that satisfies the conditions we seek.







Since the left-hand-side of (<ref>) is congruent to ∑_i=0^p-1/2p-1/2i^2  r^e_3-1-i modulo p by Lemma <ref>, showing that there exists r ∈k((t)) which satisfies the above conditions is equivalent to showing that there are some roots of the polynomial ∑_i=0^p-1/2p-1/2i^2  r^p-1/2-i that are not roots of the polynomial ∑_i+j=p-3/2p-1/2ip-1/2j r^e_3-1-j=r^e_3-1-p-3/2(∑_i+j=p-3/2p-1/2ip-1/2j r^p-3/2-j). Since the former polynomial has degree higher than ∑_i+j=p-3/2p-1/2ip-1/2j r^p-3/2-j, it suffices to show the former one is separable, which follows from Lemma <ref>.


On the special fiber, when t=0, then:


    F(x,0)=x^p-3/2x^pw(x-0)^e_2-1(x-0)^e_3-1= 1x^e-1.


Thus, the normalization of the special fiber Ψ_s has ramification jump e-1 at its only branch point. Hence, the cover Ψ_s is birationally equivalent to ψ_0.

On the generic fiber, when t  0, then Ψ_η is branched above x=0, x=t and x=tr. Consider:


    H(x,t)=F(x,t)(x-t)^e_2-1=∑_n=0^p-3/2∑_i+j=np-1/2 ip-1/2 j r^e_3-1-j(-1)^p-3/2-nt^p-3/2-nx^n-tx^p-1/2(∑_i+j=p-3/2p-1/2 ip-1/2 j r_0^e_3-1-j)x^pw(x-tr)^e_3-1.


Then:


    H(t,t)=∑_n=0^p-3/2∑_i+j=np-1/2 ip-1/2 j r^e_3-1-j(-1)^p-3/2-nt^p-3/2-t^p+1/2(∑_i+j=p-3/2p-1/2 ip-1/2 j r_0^e_3-1-j)t^pw(t-tr)^e_3-1 0.


Because H(x,t) does not have a zero at x=t, the order of the pole of F(x,t) at x=t is e_2-1. Moreover, e_2-1 prime to p by hypothesis. Thus, the ramification jump above x=t is e_2-1.

Similarly, consider:


    K(x,t)=F(x,t)(x-tr)^e_3-1=∑_n=0^p-3/2∑_i+j=np-1/2 ip-1/2 j r^e_3-1-j(-1)^p-3/2-nt^p-3/2-nx^n-tx^p-1/2(∑_i+j=p-3/2p-1/2 ip-1/2 j r_0^e_3-1-j)x^pw(x-t)^e_2-1.


Then:


    K(tr,t)=∑_n=0^p-3/2∑_i+j=np-1/2 ip-1/2 j r^e_3-1-j(-1)^p-3/2-nt^p-3/2r^n-t^p+1/2r^p-1/2(∑_i+j=p-3/2p-1/2 ip-1/2 j r_0^e_3-1-j)(tr)^pw(tr-t)^e_2-1 0.


By the same argument as above, the ramification jump above x=tr is e_3-1.

Finally, let us consider the expansion of F(x,t) around x=0:


    ∑_i=1^pwa_ix^i +  terms with non negative x-degree,






where a_pw=1/(∑_i+j=p-3/2p-1/2 ip-1/2 j r_0^e_3-1-j)(-t)^e_2+e_3-2-p-3/2. In order for a_pw/x^pw to have the same denominator as F(x,t), we multiply the top and the bottom of the fraction by (x-t)^e_2-1(x-tr)^e_3-1. It would have the form:




    (x-t)^e_2-1(x-tr)^e_3-1(-t)^-(e_2+e_3-2-p-3/2)/(∑_i+j=p-3/2p-1/2 ip-1/2 j r_0^e_3-1-j)(x-t)^e_2-1(x-tr)^e_3-1.


One can check that the coefficients of the terms of the numerator of the above fraction with x-degree 0,1, …, p-1/2 are precisely the corresponding ones of the numerator of F(x,t). Thus, we have a_pw-1=…=a_pw-p-1/2=0. Moreover, we must have a_pw-p+1/2=a_e_1-1 different from 0, or else we can not cancel out the term with x-degree p+1/2 of the numerator of (<ref>). After a finite inseparable extension of k((t)) with equation t_1 ^p =t, the leading term a_pw/x^pw of the expansion of F(x,t) around x=0 is a p-th power. The second term is a_e_1-1/x^e_1-1, which is non-zero and thus becomes the leading term of the affine equation in standard form for Ψ_η. Thus, the ramification jump above x=0 is e_1-1.


Hence, the cover Ψ_η is branched at three points that specialize to 0 which have ramification jumps e_1-1 ,e_2-1 and e_3-1. By Lemma <ref> and Lemma IV.2.3 of <cit.>, Ψ is a deformation of ψ_0 over S.




Theorem <ref> (b) is proved.





 §.§ Edge of Type {e}→{p-1,e_1,e_2}
 
 


Suppose ψ_0 is an Artin-Schreier cover over k, branched at a point b with ramification jump e-1=p+e_1+e_2-2 where e_1, e_2, e_1+e_2-1 ≢1  p. Then there exists an Artin-Schreier cover Ψ over S= k[[t]] whose special fiber is isomorphic to ψ_0, whose generic fiber is branched at three points which specialize to b and which have ramification jumps p-2, e_1-1 and e_2-1, and whose ramification divisor is otherwise constant.






We define an Artin-Schreier cover Ψ over S= k[[t]] given by the normalization of ℙ^1_S over the extension of its fraction field that is defined by the following affine equation:


    y^p-y=r-tx1-e_2/e_1-1x^p(x-t)^e_1-1(x-tr)^e_2-1=:F(x,t)


Where r=t^2-e_2+1/e_1-1. Going through a process similar to one in the proof of Proposition <ref>, one can show that Ψ is a deformation satisfies the proposition. In particular, its special fiber is an Artin-Schreier cover over k, branched at 0 with ramification jump e-1. Its generic fiber is an Artin-Schreier cover over k((t)) branched at 0, t, and tr, which all specialize to 0 and which have ramification jumps p-2, e_1-1, and e_2-1 respectively.



Part (c) of Theorem <ref> is proved.





 §.§ Edge of Type {(n+1)(p-n+1)}→{n+1, n+1, …,n+1}


 
 



Suppose ψ_0 is an Artin-Schreier cover over k, branched at a point b with ramification jump (n+1)p-n^2 where p ∤ n[(n+1)p-n^2],  then there exists an Artin-Schreier cover Ψ over S= k[[t]] whose special fiber is isomorphic to ψ_0, whose generic fiber is branched at p-n+1 points that specialize to b and which all have ramification jumps n, and whose ramification divisor is otherwise constant.






Let us study the Artin-Schreier cover Ψ over S= k[[t]] given by the normalization of ℙ^1_S over the extension of its fraction field that is defined by the following affine equation:


    y^p-y=1x^p(x^p-n-t^p-n)^n=:F(x,t).


We will show that Ψ is locally a deformation of ψ_0 that satisfies the conditions we seek.

On the special fiber, when t=0, we have:


    F(x,0)=1x^p+(p-n)n=1x^(n+1)p-n^2.


Thus, the special fiber Ψ_s has ramification jump e-1 at its only branch point. Hence, the cover Ψ_s is birationally equivalent to ψ_0.

On the generic fiber, where t  0, the derivative of x^p-n-t^p-n (with respect to x) is (p-n)x^p-n-1, which differ from 0 and only has x=0 as root. Thus x^p-n-t^p-n is relatively prime to its derivative i.e. x^p-n-t^p-n has p-n distinct roots. Whence F(x,t) on generic fiber has a pole at x=0, and p-n distinct poles that are roots of  x^p-n-t^p-n. By a similar argument as before, F(x,t) has poles of order exactly n at those later points.




Let us consider the Taylor expansion of F(x,t) around x=0:


    (-1)^nt^n(p-n)x^p(1+n(xt)^p-n+⋯)


After a finite inseparable extension of k((t)) with equation t_1^p=t, the leading term of expansion of F(x,t) around 0 is a p-th power. The second term is n (-1)^n/t^p(n+1)-n^2-nx^n which is nonzero and thus becomes the leading term of the affine equation in standard form. Thus, the ramification jump above x=0 is n.

Thus, the cover Ψ_η is branched at n+1 points that specialize to 0 which have ramification jumps all equal to n. By Lemma <ref> and Lemma IV.2.3 of <cit.>, Ψ is a deformation of ψ_0 over S.




 
 That proves part (d) of Theorem <ref>.



 

 §.§ Deformations in Characteristic 5

 
 In this section, the base field k has characteristic 5.
 
 


Suppose ψ_0 is an Artin-Schreier cover over k, branched at a point b with ramification jump 8. Then there exists an Artin-Schreier cover ψ_S over S= k[[t]] whose special fiber is isomorphic to ψ_0, whose generic fiber is branched at four points that specialize to b and which have ramification jumps 2,1,1 and 1, and whose ramification divisor is otherwise constant.







Let us study the Artin-Schreier cover Ψ over S= k[[t]] given by the normalization of ℙ^1_S over the extension of its fraction field that is defined by the following affine equation:


    y^5-y=-rs+tx-tx^2/-r_0s_0x^5(x-t)(x-tr)(x-ts)=:F(x,t).


Where r, s are solutions in k((t)) of the equation:


    X^2-(t^3-1)X+t^2-t^3+1=0.


And r_0, s_0 ∈ k are the values of r and s, respectively, when we plug in t=0. We will show that Ψ is locally a deformation of ψ_0 that satisfies the conditions we seek.

On the special fiber, when t=0, r_0, s_0 are solutions in k of the equation X^2+X+1=0 which does not have 0 as a root. Thus r_0, s_0 ≠ 0. Also, X^2+X+1 is separable on k(X). Hence, r_0 ≠ s_0. It follows that r ≠ s, and it is also clear that r,s ≠ 1. Moreover, the special fiber of Ψ is defined by:


    y^5-y=F(x,0)=-r_0s_0-r_0s_0x^8=1x^8.


Thus, the normalization of the special fiber Ψ_s has ramification jump 8 at its only branch point. Hence, the cover Ψ_s is birationally equivalent to ψ_0.

On the generic fiber, Ψ_η is branched above x=0, x=t, x=tr and x=ts. The order of the poles of F(x,t) at x=t, x=tr and x=ts is 1. 

The expansion of F(x,t) around x=0 is


    a_5x^5 +∑_i=1^4 a_ix^i+ terms with non negative x-degree,




where a_5=-1/r_0s_0t^3. In order for a_5/x^5 to have the same denominator as F(x,t), we multiply the top and the bottom of the fraction by (x-t)(x-t· r)(x-t · s)/t^3. It would have the form:




    (x-t)(x-tr)(x-ts)t^-3/-r_0s_0x^5(x-t)(x-tr)(x-ts).


One can check that the coefficients of the terms of the numerator of the above fraction with x-degree 0,1,2 are precisely the corresponding ones of the numerator of F(x,t). Thus, we have a_4=a_3=0. Moreover, we must have a_2 different from 0, or else we can not cancel out the term with x-degree 3 of the numerator of (<ref>). After a finite inseparable extension of k((t)) with equation t_1 ^5 =t, the leading term a_5/x^5 of the expansion of F(x,t) around x=0 is a 5-th power. The second term is a_2/x^2, which is non-zero and thus becomes the leading term of the affine equation in standard form for Ψ_η. Thus, the ramification jump above x=0 is 2.



Thus, the cover Ψ_η is branched at four points that specialize to 0 which have ramification jumps 2 ,1, 1 and 1. By Lemma <ref> and Lemma IV.2.3 of <cit.>, Ψ is a deformation of ψ_0 over S.



 


Suppose ψ_0 is an Artin-Schreier cover over k, branched at a point b with ramification jump 9. Then there exists an Artin-Schreier cover ψ_S over S= k[[t]] whose special fiber is isomorphic to ψ_0, whose generic fiber is branched at four points that specialize to b and which have ramification jumps 2,2,1 and 1, and whose ramification divisor is otherwise constant.





Let us study the Artin-Schreier cover Ψ over S= k[[t]] given by the normalization of ℙ^1_S over the extension of its fraction field that is defined by the following affine equation:


    y^5-y=rs-tx+tx^2/r_0s_0x^5(x-t)^2(x-tr)(x-ts)=:F(x,t).


Where r, s are solutions in k((t)) of the equation:


    X^2-2(2t^3-t^2-2)X-2(t^3-2t^2-1)=0.


And r_0, s_0 are the values of r and s, respectively, when t=0. We will show that Ψ is locally a deformation of ψ_0 that satisfies the conditions we seek.

On the special fiber, when t=0, r_0, s_0 are solutions in k of the equation X^2+4X+2=0 which does not have 0 as a root. Thus r_0, s_0 ≠ 0. Also, X^2+4X+2 is separable on k(X). Hence, r_0 ≠ s_0. It follows that r ≠ s ≠ 1.


    F(x,0)=r_0s_0r_0s_0x^9=1x^9.


Thus, the normalization of the special fiber Ψ_s has ramification jump 9 at its only branch point. Hence, the cover Ψ_s is birationally equivalent to ψ_0.

On the generic fiber, when t ≠ 0, then Ψ_η is branched above x=0, x=t, x=tr and x=ts. 
The order of the poles of F(x,t) at x=t, x=tr and x=ts is 1. Expansion of F(x,t) around 0:


    a_5x^5 +∑_i=1^4 a_ix^i+ terms with non negative x-degree,




where a_5=1/r_0s_0t^4. In order for a_5/x^5 to have the same denominator as F(x,t), we multiply the top and the bottom of the fraction by (x-t)^2(x-t· r)(x-t · s)/t^4. It would have the form:




    (x-t)^2(x-tr)(x-ts)t^-4/r_0s_0x^5(x-t)^2(x-tr)(x-ts).


One can check that the coefficients of the terms of the numerator of the above fraction with x-degree 0,1,2 are precisely the corresponding ones of the numerator of F(x,t). Thus, we have a_4=a_3=0. Moreover, we must have a_2 different from 0, or else we can not cancel out the term with x-degree 3 of the numerator of (<ref>). After a finite inseparable extension of k((t)) with equation t_1 ^5 =t, the leading term a_5/x^5 of the expansion of F(x,t) around x=0 is a 5-th power. The second term is a_2/x^2, which is non-zero and thus becomes the leading term of the affine equation in standard form for Ψ_η. Thus, the ramification jump above x=0 is 2.

Thus, the cover Ψ_η is branched at four points that specialize to 0 which have ramification jumps 2 ,2, 1 and 1. By Lemma <ref> and Lemma IV.2.3 of <cit.>, Ψ is a deformation of ψ_0 over S.




 We have proved part (e) and completed the proof of Theorem <ref>.
 
 

 
 
 

alpha




