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Deformations of Artin-Schreier covers Huy Dang University of Virginia November 5, 2019 Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 1 / 23

What is an Artin-Schreier curve? π → P 1 A Z /p -cover Y − k (characteristic of k is p > 0 ) y p − y = f ( x ) f ( x ) ∈ k ( x ) , unique up to adding an element of the form a p − a , a ∈ k ( x ) . Example 1 y 5 − y = 1 1 x 5 − ( x − 1) 2 Isomorphic to � − 1 � 5 � − 1 � y 5 − y = 1 1 = 1 1 x 5 − ( x − 1) 2 + − x − ( x − 1) 2 x x Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 2 / 23

Ramification 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 , p ∤ d j . Let e j := d j + 1 , the conductor of P j . Lemma 2 r � � � The genus of Y is g = e j − 2 ( p − 1) / 2 . j =1 � �� � d We say Y has branching datum [ e 1 , e 2 , . . . , e r ] ⊤ . Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 3 / 23

Deformation of Artin-Schreier covers φ → P 1 C − k is a G -Galois cover. A is a Noetherian complete k -algebra with residue field k . C C P 1 P 1 (1) k A Spec k Spec A, Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 4 / 23

Equal characteristic deformation → P 1 Suppose φ : Y − k is defined by y 5 − y = 1 x 6 → P 1 and A = k [[ t ]] . τ : Y − k [[ t ]] is defined by − 2 x + t 5 y 5 − y = ( − 2) x 5 ( x − t 5 ) 2 ⇒ The special fiber is birational equivalent to y 5 − y = 1 x 6 . ⇒ The generic fiber (where t � = 0 ) branches at two points x = 0 and x = t 5 , with conductors 4 and 3 , respectively. ⇒ The generic fiber and the special fiber have the same genus. → [4 , 3] ⊤ . ⇒ τ is a flat deformation of type [7] − Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 5 / 23

Mixed characteristic deformation → P 1 φ : Y − k is a Z / 2 -cover given by y 2 − y = 1 x 3 . It deforms to (lifts to) characteristic 0 by Z 2 = 1 + 4 X 3 . (2) If Z = 1 − 2 Y , 4 Y 2 − 4 Y + 1 = 1 + 4 X 3 (3) 1 Y 2 − Y = X 3 The generic fiber is a Kummer cover that branches at 4 points 0 , − 2 2 / 3 , − 2 2 / 3 ξ 3 , and − 2 2 / 3 ξ 2 3 . Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 6 / 23

Motivation C curve over k , char ( k ) = p Grothendieck = = = = = = = ⇒ C lifts to a curve C /W ( k ) (char W ( k ) = 0 ). 1 ( C ) � p = π et π et 1 ( C η ) � p Question 2.1 (Lifting problem for covers) Can we lift a G -Galois cover in characteristic p to characteristic 0 ? Obus, Wewers, and Pop show that every cyclic cover lifts to char 0 . Obus and Wewers first show that every cyclic cover with no essential ramification lifts. Pop completes the proof by showing that every cyclic cover can be deformed to one with no essential, hence lifts. Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 7 / 23

Local equicharacteristic deformation problem Suppose p = 5 . Can we deform a curve with branching datum to [7 , 4] ⊤ over R := k [[ t ]] to one with branching datum [4 , 3 , 2 , 2] ⊤ By a local-global principle, it is enough to answer the “local deformation problem”. Question 2.2 (Local equicharacteristic deformation problem) Suppose φ is a Z /p -extension k [[ z ]] /k [[ x ]] given by y p − y = 1 x e − 1 , and { e 1 , . . . , e r } is a partition of e . Does there exists a Z /p -deformation R [[ Z ]] /R [[ X ]] of φ over R := k [[ t ]] whose generic fiber has branching datum [ e 1 , . . . , e r ] ⊤ . R [[ X ]] can be thought of as an open unit disc! Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 8 / 23

Disc and annuli R complete discrete valuation ring. Open unit disc ↔ R [[ X ]] . � � � i a i X i | lim i →∞ | a i | = 0 R { X } := . Closed unit disc ↔ R { X } . Boundary of a disc ↔ R [[ X − 1 ]] { X } . Disc of radius r ↔ R [[ a − 1 X ]] where a ∈ K, | a | = r . Open annulus of thickness ǫ ↔ R [[ X, U ]] / ( XU − a ) , where v ( a ) = ǫ . Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 9 / 23

Skeleton of a Hurwitz tree − 2 x + t 5 y 5 − y = (4) ( − 2) x 5 ( x − t 5 ) 2 (4) branches at x = 0 and x = t 5 . Since, v t ( t 5 − 0) = 5 · 1 . I O L 3 i i i Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 10 / 23

Another example of a skeleton If a cover branches at 0 , t 5 , and t 10 t o t Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 11 / 23

Depth and differential conductors iii y Definition 3 1 the depth conductor δ ∈ Q ≥ 0 , is 0 if separable reduction. 2 When δ > 0 , the differential conductor ω ∈ Ω 1 K , exact. p . 3 When δ = 0 , the degeneration f ∈ K \ K Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 12 / 23

Hurwitz tree tf 3 O 4 f o 1 Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 13 / 23

Boundary conductor d y iii Boundary conductor bsw( x ) = − ord x ( ω ) − 1 , bsw( ∞ ) = ord ∞ ( ω ) + 1 . Give the instantaneous rate of change of the depth conductor on the direction corresponds to the boundary. Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 14 / 23

Flat deformations give rise to good trees → [ e 1 , . . . , e r ] ⊤ = Suppose there exists a flat deformation of type [ e ] − ⇒ → [ e 1 , . . . , e r ] ⊤ there exists a good Hurwitz tree of type [ e ] − → [3 , 2] ⊤ (when p = 5 ). = ⇒ there is no deformation of type [5] − ⇐ ? YES! Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 15 / 23

Z /p -action on a boundary Theorem 4 A Z /p -cover of a boundary Spec R [[ X − 1 ]] { X } is determined by its depth and its boundary Swan conductors. Remark It is not true for Z /p n -covers ( n > 1 ). Strategy: Construct desired covers for subdiscs and annuli, then glue them together along their boundaries. Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 16 / 23

Realize a vertex 4 Eitan Spec R { X, ( X − 0) − 1 , ( X − 1) − 1 } dx dx As x 4 ( x − 1) 3 is exact, x 4 ( x − 1) 3 = dg for some g ∈ k ( x ) . For instance, 2 g = x 3 ( x − 1) 2 . 2 y 5 − y = t 5 · 6 X 3 ( X − 1) 2 Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 17 / 23

Realize an edge 1 lost I Spec R [[ X 1 , X 0 ]] / ( X 1 X 0 − t 5 · 1 ) . y 5 − y = X 6 1 1 t 5 · 6 = X 6 0 Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 18 / 23

Lego time Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 19 / 23

Deformation is determined by exact differential forms Theorem 5 (D.) Suppose { e 1 , e 2 , . . . , e m } is a partition of { e } . Then there is a deformation → [ e 1 , . . . , e m ] ⊤ if and only if there exists an exact Hurwitz tree of type [ e ] − → [ e 1 , . . . , e m ] ⊤ . of type [ e ] − Question 5.1 Let k be an algebraically closed field of characteristic p > 0 . Suppose 1 < e i < p for i = 1 , 2 , . . . , n are integers ( n ≥ 2 ). What are the conditions on the e i ’s so that the rational function 1 � n i =1 ( x − P i ) e i is a derivative of some rational function in k ( x ) for some P i ’s in k pairwise distinct? Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 20 / 23

Applications 1 Study the geometry of AS g , the moduli space Artin-Schreier covers of fixed genus g . AS g is connected when g is large. 2 Study the deformation of all wildly ramified covers Z /p n , D p n , . . . 3 The lifting problem Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 21 / 23

Equal characteristic deformation of Z /p n -covers � � 1 1 ℘ ( Y 1 , Y 2 ) = x 2 ( x − t 4 ) , . (5) x 3 ( x − t 4 ) 2 ( x − t 2 ) 2 t 4 : 2 t 4 : 3 0 : 2 0 : 3 � � dx � � 12 , dx 24 , x 2 ( x − 1) 2 x 3 ( x − 1) 3 e 1 e 1 ǫ e 1 = 1 � � 6 , dx � � dx 14 , x 4 t 2 : 2 x 6 ( x − 1) 2 e 0 e 0 ǫ e 0 = 1 � 1 � � � �� 0 , 1 x 3 , 1 0 , x 3 x 7 Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 22 / 23

Mixed characteristic Hurwitz tree l H 01 Hurwitz tree of a lift of y 2 − y = 1 x 5 . Huy Dang (University of Virginia) Deformations of Artin-Schreier covers November 5, 2019 23 / 23