On wormholes in the moduli space of surfaces
aa r X i v : . [ m a t h . AG ] F e b ON WORMHOLES IN THE MODULI SPACE OFSURFACES
GIANCARLO URZ ´UA AND NICOL ´AS VILCHES
Abstract.
We study a certain wormholing phenomenon that takesplace in the Koll´ar–Shepherd-Barron–Alexeev (KSBA) compacti-fication of the moduli space of surfaces of general type. It occursbecause of the appearance of particular extremal P-resolutions insurfaces on the KBSA boundary. We state a general wormholeconjecture, and we prove it for a wide range of cases. At the end,we discuss some topological properties and open questions.
Contents
1. Introduction 12. A review on continued fractions and extremal P-resolutions 42.1. Continued fractions 42.2. Zero continued fractions 62.3. Cyclic quotient singularities 72.4. Extremal P-resolutions and wormhole singularities 83. General set-up and the wormhole conjecture 133.1. W-surfaces and their MMP 143.2. Wormholes 164. Proof of Theorem 1.2 195. Proof of Theorem 1.3 266. Open questions 316.1. Topological type of surfaces in a wormhole 316.2. What is left to prove the conjecture 34References 361.
Introduction
Since the breakthrough construction of simply connected Campedellisurfaces by Y. Lee and J. Park in [LP07], there have been several resultson various aspects of one parameter Q -Gorenstein degenerations of Date : February 4, 2021. urfaces, see e.g. [PPS09a], [PPS09b], [LN13], [HTU13], [U16a], [U16b],[RTU17], [LN18], [PPSU18], [CU18], [EU18]. One of those aspects hasbeen the study of KSBA surfaces with only Wahl singularities whichadmit Q -Gorenstein smoothings into surfaces of general type. Thesesmoothings could be seen as punctured disks D × on the moduli spaceof surfaces of general type M K ,χ , which are completed in the KSBAcompactification M K ,χ with a normal projective surface X with onlyWahl singularities and K X ample. (Here of course K X = K and χ ( O X ) = χ .) In this way, we have a Q -Gorenstein smoothing( X ⊂ X ) → (0 ∈ D ) , where D = D × . Nowadays there are many examples of such situationsin the literature, most of them constructed abstractly, starting withthe original work [LP07].Sometimes a surface X as above has an embedded extremal P -resolution, which in addition admits another extremal P -resolution overthe same cyclic quotient singularity. One performs the corresponding“extremal P -resolution surgery” on X to obtain another normal pro-jective surface X ′ with only Wahl singularities. Let us assume that X ′ admits a Q -Gorenstein smoothing ( X ′ ⊂ X ′ ) → (0 ∈ D ). (Thisautomatically holds under a cohomological condition on X , which isused in all Lee-Park type of surfaces.) If K X ′ is ample, then one caneasily show that X and X ′ live in the same M K ,χ . If K X ′ is only nef,then the canonical model of X ′ and X belong to the same M K ,χ aswell. But if K X ′ is not nef, then one needs to run the minimal modelprogram (MMP) on the family ( X ′ ⊂ X ′ ) → (0 ∈ D ) to find the KSBAreplacement. This MMP requires flips and/or divisorial contractionsas studied in [HTU13] (see also [U16b]). If it has a minimal model (i.e.canonical class becomes nef) and MMP only requires flips, then theKSBA replacement is again on the same M K ,χ . Conjecture 1.1 (Wormhole conjecture) . The MMP requires only flipsand gives a minimal model. The KSBA replacement of ( X ′ ⊂ X ′ ) → (0 ∈ D ) lives on the same moduli space as the original ( X ⊂ X ) → (0 ∈ D ) . It is not clear if the smooth surfaces in the wormhole are deformationequivalent, i.e. they belong to the same connected component of M K ,χ .For example, M. Reid conjectures that there is one component fortorsion free Godeaux surfaces, and we do have wormholes there bymeans of Lee-Park type of examples, which we do not know how toconnect. On the other hand, wormholes applied to elliptic surfacesmay change the topology. We show examples of that in Section 6. n this paper, we prove the wormhole conjecture for a wide range ofcases. We point out that a fixed extremal P -resolution in X can pro-duce at most one wormhole, because in [HTU13, Section 4] it is provedthat a cyclic quotient singularity can admit at most two extremal P -resolutions. Additionally, when that happens, both share the same δ invariant. In this paper, we give simplified and new proofs of both ofthese facts.We now state the main theorems, which will imply positive evidencefor the wormhole conjecture as a corollary. For all the definitions werefer to Section 2 and Section 3. Theorem 1.2.
Let Y be a nonrational normal projective surface withone cyclic quotient singularity ( Q ∈ Y ) , which is smooth everywhereelse. Assume that Q admits two extremal P -resolutions f + i : ( C i ⊆ X i ) → ( Q ∈ Y ) , i = 1 , , so that the following is satisfied: • The strict transform in the minimal resolution of X of theexceptional curve C for the extremal P -resolution in X is a P with self-intersection − . • The canonical class K X is nef. • Both surfaces X i admit Q -Gorenstein smoothings ( X i ⊆ X i ) → (0 ∈ D ) .Then, we have that K X is nef. Theorem 1.3.
Let Y be a nonrational normal projective surface withone cyclic quotient singularity ( Q ∈ Y ) , which is smooth everywhereelse. Assume that Q admits two extremal P -resolutions f + i : ( C i ⊆ X i ) → ( Q ∈ Y ) , i = 1 , , so that the following is satisfied: • The strict transform in the minimal resolution of X of theexceptional curve C for the extremal P -resolution in X is a P with self-intersection − . • The extremal P -resolution in X has only one singularity. • The canonical class K X is nef. • Both surfaces X i admit Q -Gorenstein smoothings ( X i ⊆ X i ) → (0 ∈ D ) .Then, we only need flips to run MMP on ( X ⊂ X ) → (0 ∈ D ) . We can show via an explicit example that one might indeed need toperform flips in a situation as in Theorem 1.3 (see Section 3). Finally,in Section 6 we briefly show and discuss certain topological aspects ofwormholes, ending with some open questions and with what is left toprove the wormhole conjecture. We also discuss combinatorial potentialcounterexamples. orollary 1.4. Let X be a normal projective surface with only Wahlsingularities and K X ample. We assume: • The surface X is not rational. • There is an embedded extremal P -resolution in X such that itscontraction ( C ⊂ X ) → ( Q ∈ Y ) admits another extremal P -resolution ( C ′ ⊂ X ′ ) → ( Q ∈ Y ) as in Theorem 1.2 or Theorem1.3. • The cohomology group H (cid:0) e X, T e X ( − log( E + e C )) (cid:1) vanishes, where e X → X is the minimal resolution of X , E is the exceptionaldivisor, and e C is the strict transform of C . Hence, there are Q -Gorenstein smoothings ( X ⊆ X ) → (0 ∈ D ) and ( X ′ ⊆ X ′ ) → (0 ∈ D ) .Then, the KSBA replacement of ( X ′ ⊂ X ′ ) → (0 ∈ D ) lives on thesame moduli space as the original ( X ⊂ X ) → (0 ∈ D ) .Notation. • A ( − m )-curve is a curve Γ isomorphic to P with Γ = − m . • If φ : X → W is a birational morphism, then exc( φ ) is theexceptional divisor. • A KSBA surface in this paper is a normal projective surfacewith log-canonical singularities and ample canonical class [KSB88]. • Under a birational map, we may keep the notation for a curveand its strict transform. • For a normal projective surface Z , the tangent sheaf is denotedby T Z := H om O Z (Ω Z , O Z ). If Z is not singular and D is asimple normal crossings divisor on Z , then T Z is the usual rank2 tangent bundle and T Z ( − log( D )) is the dual of the rank 2vector bundle of differentials with simple poles along D . Acknowledgments.
The results of this article are mainly based on themaster’s thesis [V20]. We would like to thank Jonny Evans for usefuldiscussions and comments.2.
A review on continued fractions and extremalP-resolutions
Continued fractions.Definition 2.1.
Given a , a , . . . , a r positive integers, we define the Hirzebruch-Jung continued fraction recursively. If r = 1, then [ a ] := a . If r ≥ a , . . . , a r ] = 0, then we define[ a , . . . , a r ] := a − a , . . . , a r ] . ote that not every list of positive integers makes sense as continuedfraction, for an example take [5 , , , a i ≥ i , the continued fraction automatically makes sense, and[ a , . . . , a r ] > r . If 0 < q < n are coprime numbers,then there exists unique a i ≥ a , . . . , a r ] = nq . To analyze these continued fractions, given a , . . . , a r , we define se-quences p = 1 , p = a , q = 0 , q = 1, and for 2 ≤ i ≤ r , p i = a i p i − − p i − , q i = a i q i − − q i − . Inductively, one can show that (cid:18) a −
11 0 (cid:19) · · · · · (cid:18) a i −
11 0 (cid:19) = (cid:18) p i − p i − q i − q i − (cid:19) , and also p i q i = [ a , . . . , a i ] for every 1 ≤ i ≤ r . We say that { a , . . . , a r } is admissible if p i > i < r . A sequence is admissible if and only ifthe matrix − a − a − a . . . 11 − a r (1)is seminegative definite of rank ≥ r − a i ≥ i , then the sequence is admissible. If some a i is 1 and r ≥
2, then { a − , . . . , a r } , i = 1; { a , . . . , a i − − , a i +1 − , . . . , a r } , ≤ i ≤ r − { a , . . . , a r − − } , i = r. is also admissible. We call this procedure a blow-down . If the originalfraction was nq , then the new one is nq ′ with q ′ ≡ q (mod n ).Given an admissible continued fraction [ a , . . . , a r ], after blowing-down every possible entry, we may get two different results, accordingto the rank of the matrix (1). If its rank is r , then we get either [1]or a continued fraction [ b , . . . , b s ] with b j ≥ ≤ j ≤ s .Otherwise, we get [1 ,
1] as a final fraction.We define zero continued fraction as an admissible continued fraction[ a , . . . , a r ] whose value is equal to zero. Equivalent, the rank of itsmatrix (1) is r − iven a fraction [ a , . . . , a r ] = nq with a i ≥ < q < n coprime,the dual fraction is nn − q = [ b , . . . , b s ] , with b j ≥ j . We have a visual way to compute them, c.f.[Rie74]. Draw a − a − = [2 , , Figure 1.
Dot diagram for [2 , , b − b − = [3 , , , a , . . . , a r ] = nq with 0 < q < n are coprime and a i ≥ (cid:18) a −
11 0 (cid:19) · · · · · (cid:18) a i −
11 0 (cid:19) = (cid:18) n − q ′ q − qq ′ n (cid:19) , where q ′ is the inverse of q modulo n , since every matrix on the lefthas determinant 1. Thus, if [ b , . . . , b s ] = nn − q is the unique continuedfraction with b j ≥
2, then[ a , . . . , a r , , b , . . . , b s ] = 0 . Zero continued fractions.
Now we will focus on zero contin-ued fractions, following [Ste91]. Consider a zero continued fraction[ a , . . . , a r ]. Blowing down every possible 1 until the length is 2, weget [1 , ,
1] through the blow-ups { a , . . . , a r } 7→ { , a + 1 , a , . . . , a r } , { a , . . . , a i − , a i − + 1 , , a i + 1 , a i +1 , . . . , a r } , { a , . . . , a r − , a r + 1 , } . We will show an explicit bijection with triangulation of polygons. Atriangulation of a convex polygon P P . . . P r is given by drawing somenon-intersecting diagonals on it which divide the polygon into triangles.For a fixed triangulation, we define v i as the number of triangles whichhave P i as one of its vertices. Note that r X i =0 v i = 3( r − . (2) sing induction, one can show that [ a , . . . , a r ] is a zero continuedfraction if and only if there exists a triangulation of P P . . . P r suchthat v i = a i for every 1 ≤ i ≤ r . In this way, the number of zerocontinued fractions of length r is the Catalan number r (cid:0) r − r − (cid:1) . Alsoby induction, every triangulation has at least two v i equal to 1. Theycannot be adjacent unless r = 2.2.3. Cyclic quotient singularities.Definition 2.2.
Given coprime numbers 0 < q < n , the cyclic quotientsingularity n (1 , q ) is the germ at 0 of the quotient of C by the action ζ · ( x, y ) = ( ζ x, ζ q y ), where ζ is a primitive n -root of unity.The minimal resolution of X = n (1 , q ) can be recover from the con-tinued fraction of nq . If nq = [ e , . . . , e r ] with e i ≥ σ : ˜ X → X isthe minimal resolution, the exceptional divisor consists of a chain of r nonsingular rational curves E , . . . , E r with E i = − e i . This is picturedin Figure 2. Q X ˜ XE E E r Figure 2.
Minimal resolution of n (1 , q ).Note that if we do a blow-up at the intersection of E i and E i +1 ,we get a new chain E , . . . , E i , F, E i +1 , E r of self-intersections E i = − ( e i + 1) , E i +1 = − ( e i +1 + 1) , F = −
1. A similar remark can be madefor blow-downs. This justifies the terminology blow-down for continuedfractions. We note that we can compare the canonical divisor on X and ˜ X as follows K ˜ X ≡ σ ∗ K X + r X i =1 k i E i , (3)where − < k i ≤ E i . Definition 2.3. A Wahl singularity is a cyclic quotient singularity m (1 , ma − < a < m are coprime numbers.An alternative description can be made by looking at the continuedfraction (see [KSB88, Lemma 3.11]). Every Wahl singularity arises rom [4] by applying the operations[ a , . . . , a r ] ( [2 , a , . . . , a r − , a r + 1][ a + 1 , a , . . . , a r , . (4)From this algorithm and by induction on r , it is clear that every Wahlsingularity m ma − = [ a , . . . , a r ] satisfies P ri =1 a i = 3 r + 1.Let [ a , . . . , a r ] be a Wahl continued fraction. We define integers δ , . . . , δ r in the following inductive way. If r = 1 then δ := 1. If wealready defined δ , . . . , δ r for [ a , . . . , a r ], then we assign δ , . . . , δ r , δ + δ r to [ a + 1 , . . . , a r , δ + δ r , δ , . . . , δ r to [2 , a , . . . , a r + 1] . These numbers compute the discrepancies in Equation (3). If m ma − =[ a , . . . , a r ] has numbers δ , . . . , δ r , then K ˜ X ≡ σ ∗ K X + r X i =1 (cid:18) − δ i δ + δ r (cid:19) E i . (5)This gives us an explicit control on discrepancies, which will be usedto bound them later in this paper.2.4. Extremal P-resolutions and wormhole singularities.
To studythe components of the deformation space of quotient singularities, Koll´ar–Shepherd-Barron introduced P-resolutions in [KSB88, Section 3]. Weonly need a particular class of them, which we now recall.
Definition 2.4.
Let 0 < Ω < ∆ be coprime integers, and let ( Q ∈ Y )be a cyclic quotient singularity (1 , Ω). An extremal P-resolution of( Q ∈ Y ) is a partial resolution f +0 : ( C + ⊂ X + ) → ( Q ∈ Y ), such that X + has only Wahl singularities, there is one exceptional curve C + andisomorphic to P , and K X + is relatively ample.Following [HTU13, § X + has at most two Wahl singu-larities m i (1 , m i a i − m i = a i = 1.If their associated continued fractions are given by m m a − e , . . . , e r ] , m m a − f , . . . , f r ] , and ( C + ) = − c on the minimal resolution of X + , then∆Ω = [ f r , . . . , f , c, e , . . . , e r ] . We denote the extremal P-resolution as [ f r , . . . , f ] − c − [ e , . . . , e r ].The intersection K + · C + can be computed as δm m , where δ = cm m − a − m a . The self-intersection − c of C + can be computed in termsof the continued fraction of ∆Ω . Theorem 2.5.
Consider a cyclic quotient singularity Y = (1 , Ω) ,with ∆Ω = [ b , . . . , b r ] . Suppose that we have an extremal P-resolution ( C + ⊂ X + ) over (1 , Ω) with l singularities ( l = 0 , or ). Then, theself-intersection of the exceptional curve C + on the minimal resolutionof X + is − ( P ri =1 b i − r + 3 − l ) . As a direct consequence, note that if P ri =1 b i < r , there are noextremal P-resolutions. If P ri =1 b i = 3 r , then c can be − − − l = 0), and so on. Proof. If l = 0, then r = 1 and the result is trivially true. Suppose that l = 2; the proof for l = 1 is similar. Consider the extremal P-resolution[ f r , . . . , f ] − c − [ e , . . . , e r ] . Note that P r i =1 e i + c + P r j =1 f j = 3( r + r ) + c + 2, since we have twoWahl singularities.From [KSB88, Lemma 3.13, Lemma 3.14] we know that, from theminimal resolution of Y , one has to blow-up only on the intersectionpoints of exceptional curves. This shows that r X i =1 b i − r = r X i =1 e i + c + r X j =1 f j ! − r + r + 1) , since every blow-up subtracts 3 to the sum of the multiplicities, and 1to the length of the chain. It follows that P ri =1 b i − r = c − (cid:3) Given a coprime pair 0 < Ω < ∆, one can find all possible extremalP-resolutions by looking at the dual fraction ∆∆ − Ω . More precisely, wehave the following result (see [HTU13, Prop 4.1.]). Proposition 2.6. If ∆∆ − Ω = [ c , . . . , c s ] , then there is a bijection be-tween extremal P-resolutions and pairs ≤ α < β ≤ s such that [ c , . . . , c α − , c α − , c α +1 , . . . , c β − , c β − , c β +1 , . . . , c s ] = 0 . (6) Moreover, we have m a = [ c , . . . , c α − ] and m a = [ c s , . . . , c β +1 ] (if α = 1 or β = s , the associated points are smooth). Also, δε = [ c α +1 , . . . , c β − ] ,where < ε < δ (or δ = 1 if α + 1 = β ). It will be useful to denote the expression in Equation (6) with twobars as [ c , . . . , c α , . . . , c β , . . . , c s ] . oreover, if it admits a second extremal P-resolution, then we willdenote it with two underlines. For instance, if ∆ = 36 , Ω = 13, thenwe write 3636 −
13 = [2 , , , , , and so we know it admits two extremal P-resolutions, and we knowhow to obtain them. In this example, [2 , , , ,
4] is associated tothe extremal P-resolution [3 , , −
2, and [2 , , , ,
4] corresponds to[4] − − [6 , , Definition 2.7.
As in [HTU13, Section 4], a sequence { a , . . . , a r } , a i > of WW type if there exists 1 ≤ α < β ≤ r such that[ a , . . . , a α , . . . , a β , . . . , a r ] = 0 . A wormhole singularity is a cyclic quotient singularity (1 , Ω) whichadmits at least two extremal P-resolutions. Equivalently, the contin-ued fraction of ∆∆ − Ω is of WW type by means of at least two pairs( α, β ) , ( α ′ , β ′ ).A consequence of Theorem 2.8 is that a wormhole singularity admitsprecisely two extremal P-resolutions.If { a , . . . , a r } is a sequence of WW type, there is a triangulation ofa polygon P P . . . P r such that v i = a i . Thus, we define a := v . Notethat by Equation (2) we have a = 3 r − − P ri =1 a i , so it does notdepend on the pair ( α, β ). Note also that a may be 1.Therefore, we have two cases: (A) a > a = 1 (as in[HTU13, § P from the polygon P P . . . P r , and repeat until all entries are greater than 1.Our next goal is to give a simplified proof of [HTU13, Thm. 4.3.],and a new proof of Theorem [HTU13, Thm 4.4.]. Theorem 2.8 ([HTU13, Thm 4.3.]) . A cyclic quotient singularity hasat most two distinct extremal P-resolutions.
Theorem 2.9 ([HTU13, Thm 4.4.]) . If a cyclic quotient singularityadmits two extremal P-resolutions, then the δ ’s are equal. To prove Theorem 2.8, note that it suffices to show, by Proposition2.6, that a sequence of WW type { a , . . . , a r } admits at most two pairs( α, β ) such that[ a , . . . , a α − , . . . , a β − , . . . , a r ] = 0 . et a = 3 r − − P ri =1 a i as before, and assume that we are in the case(A), i.e. a >
1. Since the triangulation of P P . . . P r needs to have atleast two vertices with v i = 1, we must have a α − a β − a α = a β = 2.Note then that [ a α +1 , . . . , a β − , . . . , a r , a , a , . . . , a α − ] = 0, sincewe have a triangulation. A matrix computation shows that mm − a = [ a β − , . . . , a α +1 ] , m ′ m ′ − a ′ = [ a β +1 , . . . , a r , a , . . . , a α − ]has m = m ′ and a + a ′ = m . In this way, we have that[ a α +1 , . . . , a r , a , . . . , a α − ] = m ma + 1is the dual of a Wahl singularity. All of them are obtained from [2 , , § , , a .To simplify the proof of Theorem 2.8, we will use the following se-quences of 0s and 1s. Definition 2.10.
Given { a , . . . , a r } , a i ≥
2, we define its indicatorsequence as { , , . . . , | {z } a − , , , . . . , | {z } a − , , . . . , , , . . . , | {z } a r − } . We think on { a , . . . , a r } and its indicator sequence indexed by acyclic groups. As an example, the indicator sequence of { , , , , } is { , , , , , , , , } . Note that we can completely recover the se-quence { a , . . . , a r } from the indicator sequence, and so we can studysequences of WW type from its indicator sequence, in the case (A).We also note that, for every i , there are two indices l i , m i such that e l i +1 , . . . , e m i − are all the zeroes induced by a i . The main advantageof this is that it makes more symmetric the procedure from Equation(4). We start with 1 ,
1; then, we add 1 to one side and 0 to the other,as follows { , } → { , , , } → { , , , , , } → { , , , , , , , } → . . . e repeat and then add 1 , Proof of Theorem 2.8.
We assume a > { a , . . . , a r } , and { e , . . . , e m − } its indicatorsequence. Consider p < q such that [ a , . . . , a p , . . . , a q , . . . , a r ] = 0.Thus, if t, t + 1 and t + m, t + m + 1 are the corresponding indices for a p = 2 and a q = 2, the construction yields e j = ( − e t +1 − j , j = r, r + 1 , r + m, r + m + 1;1 − e t +1 − j , j = r, r + 1 , r + m, r + m + 1 . (7)Given the indicator sequence, it suffices then to show that there areat most two pairs { t, t + m } which makes Equation (7) true. Since t and t + m gives the same pair, we will define f j = ( e j + e j + m ) /
2, as asequence indexed by Z /m Z . The Equation (7) translates to f j = ( − f t +1 − j , j = r, r + 1;1 − f t +1 − j , j = r, r + 1 . (8)Now We are going to use the same trick as in [HTU13, § µ a primitive m -root of unity, and define F = m − X j =0 µ j f j . (9)Adding Equation (8) multiplied by µ j from j = 0 to m , we get( µ t ) · µF − ( µ t ) · ( µ + 1) + F = 0 . For m >
2, this is an equation of degree at least 1 on µ t . Thus,there are at most two valid values of µ t . Note that m = 2 happensonly for the indicator sequence { , , , } , associated to { , , , } .By the correspondence between sequences of type WW in case (A) andindicator sequences, this shows that there are at most two pairs in thiscase.Suppose now that a = 1, this is, we are now on case (B). Notethat for every pair α < β such that [ a , . . . , a α , . . . , a β , . . . , a r ] = 0,the corresponding triangulation on P P . . . P r must have a triangle P P P r . We can remove then vertex P , and look to pairs for the newsequence a − , a , . . . , a r − , a r −
1, since it is easy to show that theyare in bijection with pairs for the original sequence. Inductively, thisreduces case (B) to case (A). (cid:3) roof of Theorem 2.9. We will use Proposition 2.6. Consider a se-quence { a , . . . , a r } with a >
1, i.e. we are in case (A). If p < q is apair such that [ a , . . . , a p , . . . , a q , . . . , a r ] = 0, then [ a p +1 , . . . , a q − ] = δε for some ε . Thus,[ a p +1 , . . . , a r , a , . . . , a p − ] = δ δλ + 1 , for some λ < δ . Since all entries are ≥
2, we can compute (cid:18) a p +1 −
11 0 (cid:19) . . . (cid:18) a r −
11 0 (cid:19) (cid:18) a −
11 0 (cid:19) . . . (cid:18) a p − −
11 0 (cid:19) = (cid:18) δ − δ ( δ − λ ) − δλ + 1 − λ ( δ − λ ) − (cid:19) . Since a p = 2, we obtain that the matrix (cid:18) a p +1 −
11 0 (cid:19) . . . (cid:18) a r −
11 0 (cid:19) (cid:18) a −
11 0 (cid:19) . . . (cid:18) a p − −
11 0 (cid:19) (cid:18) a p −
11 0 (cid:19) is (cid:18) δ i + δ j λ i − − δ i δ i λ i + λ i + 1 − δ i λ i − (cid:19) . Its trace is exactly δ −
2. But recall that the trace of a multiplicationis invariant under cyclic permutations of the factors, which shows thattr (cid:18)(cid:18) a −
11 0 (cid:19) . . . (cid:18) a r −
11 0 (cid:19)(cid:19) = δ − . The left hand side does not depend on the pair ( p, q ), thus δ is thesame for every pair. This proves Theorem 2.9 for case (A).The case (B) is handled by induction to reduce it to case (A), justlike in the proof of Theorem 2.8. If a = 1, we can blow-down thesequence there. Note that (cid:18) a r −
11 0 (cid:19) (cid:18) −
11 0 (cid:19) (cid:18) a −
11 0 (cid:19) = (cid:18) a r − −
11 0 (cid:19) (cid:18) a − −
11 0 (cid:19) , which shows that the trace remains constant. Also, since the interval a p +1 , . . . , a q − is not affected by blow-downs, this inductively reducesit to case (B). (cid:3) General set-up and the wormhole conjecture
In this section we will look at singular surfaces together with asmoothing over a smooth analytic curve germ D . This point of viewwas used in [U16a] under the name of W-surfaces, and it works betterfor the set-up of the wormhole conjecture. We start by recalling it. .1. W-surfaces and their MMP.Definition 3.1. A W-surface is a normal projective surface X togetherwith a proper deformation ( X ⊂ X ) → (0 ∈ D ) such that(1) X has at most Wahl singularities.(2) X is a normal complex 3-fold with K X Q -Cartier.(3) The fiber X is reduced and isomorphic to X .(4) The fiber X t is nonsingular for t = 0.The W-surface is said to be smooth if X is nonsingular.Various basic properties of W-surfaces are shown in [U16a, Section2]. A W-surface X is minimal if K X is nef. This is equivalent to K X nef,as it is shown in [U16a, Lemma 2.3]. If a W-surface X is not minimal,then there is an explicit MMP relative to D which we will review brieflybelow. The outcomes of this MMP are discussed in [U16a, Section2]. We note that invariants such as irregularity, geometric genus, K ,and Euler topological characteristic are constant for the fibers in a W-surface. An invariant that may not remain constant is the topologicalfundamental group. We have that K X ample implies K X t ample forall t , and in this case we may think of a W-surface X as a disk in theKSBA compactification of the moduli space of surfaces of general typewith K = K X and χ = χ ( O X ).Let σ : e X → X be the minimal resolution of X . Lemma 3.2.
Let X be a minimal W-surface such that the minimalresolution of X is ruled. Then X is rational.Proof. Assume that e X is ruled but not rational. Then there is a fibra-tion e X → C whose general fiber is P and C is a nonsingular projec-tive curve of positive genus. Then all curves in the exceptional divisorof σ must be contained in fibers. But if F is a general fiber, then F · K e X = σ ( F ) · K X and by adjunction F · K e X = −
2, which is contraryto the assumption K X nef. (cid:3) When a W-surface X has K X not nef, then there is a smooth rationalcurve C with C · K X <
0. The cases C ≥ C <
0. Then the W-surface X defines an extremal neighborhood oftype mk1A or mk2A, and we need to run MMP. For details we refer to[U16b, Section 2]. Below we describe these two distinct situations onthe surface X . Let ( C ⊂ X ) → ( Q ∈ Y ) be the contraction of C . k1A: In this situation X has one Wahl singularity m (1 , ma − m ma − = [ e , . . . , e s ]. Let E , . . . , E s be the corresponding ex-ceptional curves in e X , so that E j = − e j . The proper transform of C is a ( − E i transversally at onepoint. The curve C contracts to ( Q ∈ Y ), which is the cyclic quotientsingularity (1 , Ω) where∆Ω = [ e , . . . , e i − , e i − , e i +1 , . . . , e s ] . We will denote this situation as [ e , . . . , e i , . . . , e s ]. If we write K e X ≡ σ ∗ ( K X ) + P sj =1 ( − δ j m ) E j , then δ := δ i and we have C · K X = − δm and C = − ∆ m . mk2A: In this situation X has two Wahl singularities m j (1 , m j a j − j = 1 , m m a − = [ e , . . . , e s ] and m m a − = [ f , . . . , f s ]. Let E , . . . , E s and F , . . . , F s be the corresponding exceptional curveswith E j = − e j and F j = − f j . The strict transform of C is a ( − E and F , transversally at one point each. Wehave that ∆Ω = [ f s , . . . , f , , e , . . . , e s ]where ( Q ∈ Y ) is (1 , Ω). Let δ := m a − m ( m − a ), then we have C · K X = − δm m and C = − ∆ m m . We will denote this situation as[ f s , . . . , f ] − [ e , . . . , e s ].To know whether a W-surface X with C · K X < C < Divisorial contraction:
In this case the general fiber of the W-surface X contains a ( − C . That is the divisorwhich is contracted to a new W-surface Y . The contraction of C ⊂ X produces a Wahl singularity ( Q ∈ Y ). Flip:
In this case the contraction of C produces a cyclic quotientsingularity (1 , Ω) = ( Q ∈ Y ). This singularity admits an extremalP-resolution ( C + ⊂ X + ) → ( Q ∈ Y ) so that a suitable W-surface X + is the flip of the W-surface X . The general fibers of the W-surfaces X and X + are isomorphic. f a multiple of K X has sections, then after finitely many mk1A andmk2A we will arrive to a minimal W-surface (see [HTU13, Theorem5.3]). This is the case that matters to us, since we will be working inthe KSBA boundary of the moduli space of surfaces of general type.3.2. Wormholes.
The following is the set-up for a wormhole. Wetake a W-surface X with K X ample, and we assume that X has anextremal P-resolution ( C ⊂ X ) → ( Q ∈ Y ) over a WW singularity( Q ∈ Y ). We also assume that if E is the exceptional (reduced) divisorof the minimal resolution f X → X and f C is the strict transform of C , then H ( f X , T f X ( − log( E + f C ))) = 0 . By [LP07, Section 2], this condition can be used to prove that thereare no-local-to-global obstructions to deform X (which in particu-lar shows existence of W-surfaces X ). If f C is a ( − H ( f X , T f X ( − log( E + f C ))) = 0 is the same as H ( f X , T f X ( − log( E ))) =0, and this is the same as requiring H ( X , T X ) = 0 by [LP07, The-orem 2]. Let X be the surface resulting of contracting the extremalP-resolution in X , and then partially resolving with the second ex-tremal P-resolution of ( Q ∈ Y ). Lemma 3.3.
We have that X defines a W-surface, and K X = K X and χ ( O X ) = χ ( O X ) .Proof. We need to prove existence of a Q -Gorenstein smoothing for X . We know that H ( f X , T f X ( − log( E + f C ))) = 0. Let A be thechain formed by the exceptional curves of the extremal P-resolutionand f C . Let A be the chain formed by the exceptional curves of thethe second extremal P-resolution together with the corresponding curve f C . We know that to obtain A we perform blow-downs until reachingthe exceptional chain of ( Q ∈ Y ), and then we perform blow-ups atthat chain to obtain A . (We may not need blow-downs and/or blow-ups of course.) By the addition/deletion principle of ( − H ( f X , T f X ( − log( E + f C ))) = H ( f X , T f X ( − log( E ′ + f C )))where E ′ is the exceptional divisor of the minimal resolution f X → X .Therefore, by our hypothesis, we have H ( f X , T f X ( − log( E ′ + f C ))) = 0.By using the standard short exact sequence0 → T f X ( − log( E ′ + f C )) → T f X ( − log( E ′ )) → N f C / f X → , e have that H ( f X , T f X ( − log( E ′ ))) = 0. Hence, by [LP07, Theorem2], we have that there are no-local-to-global obstructions to deform X ,and so we have a W-surface X .In relation to invariants, since Wahl singularities are rational, weclearly have χ ( O X ) = χ ( O X ). As for K , we note that if X is anormal projective surface with only Wahl singularities and e X → X is the minimal resolution, then K X = K e X + l where l is the amountof exceptional curves. As A is obtained by blow-downs and blow-upson A and we contract all curves except one, we obtain that K X = K X . (cid:3) Therefore we have a W-surface X with same invariants. However K X may not be nef. Conjecture 3.4 (Wormhole conjecture) . The MMP on the new W-surface finishes in a minimal model and it requires only flips, this is,both punctured W-surfaces live in the same moduli space.
A main purpose of this paper is to show that Conjecture 3.4 is truewhen X is not rational and for a wide range of WW singularities. Onemay hope that perhaps in the case when X not rational we do havethat K X is nef. We will prove that true in many situations, but thefollowing example shows that it is not always the case. Example . We consider an Enriques surface with the configurationof ( − A A A A Figure 3.
Special curves in an Enriques surfaceWe do five blow-ups to get the configuration in Figure 4. The ex-ceptional curves E , . . . , E are indexed according to the order of theblow-ups.We have E = E = E = − , E = E = − , A = − , A = A = − , A = − E − E − A − A − A − E − A , after contracting E corresponds to the minimal resolution of the singu-larity (1 , , , , , ,
3] = . This is a WW singularity,which define surfaces X and X . In both cases we have W-surfaces E A A A E E A E Figure 4.
After five blow-ups. X and X because we can prove they do not have obstructions (see[DRU20, Lemma 2.4]).If we contract A and A − A − A − E − E to singularties P and P , then we obtain the surface X with the extremal P-resolution[2 , , , , − − [4] . It can be proved that a general X has K X ample.If we contract E − A − A − A − A − E to a point P , then weget the surface X with the extremal P-resolution2 − [2 , , , , . But in this case we have K · E = − . The curve E induces a mk1Aneighborhood. The numerical data for this mk1A is[2 , , − , ,
3] = 12979 , which is not a Wahl singularity, and so this is a flipping mk1A. Theextremal P-resolution which does the flip is[2 , , , − − [2 , , . This is in Figure 5, where F , F , F are the new curves from the newblow-ups. We note that E = E = − K of the new singularsurface. E A A F F F A A E E Figure 5.
After the flip.However we compute K · E = , K · E = . In this way K is nownef. We only used one flip to obtain the nef model, and we have awormhole in the moduli space of Z / e now prove a relevant reduction step towards Conjecture 3.4. Letus consider W-surfaces X and X as in Conjecture 3.4. Let ( X ⊂X ′ ) → (0 ∈ D ) be a partial Q -Gorenstein deformation which keeps thedistinguished extremal P-resolution in all fibers but smooths all otherWahl singularities. This is possible since we have H ( f X , T f X ( − log( E + f C ))) = 0. We denote the general fiber by X ′ . Let ( X ⊂ X ′ ) → (0 ∈ D ) be the Q -Gorenstein deformation obtained by first contracting theextremal P-resolution of all fibers in ( X ′ ⊂ X ′ ) → (0 ∈ D ) (where thisdeformation is trivial), and then gluing the other extremal P-resolution.The general fiber is denoted by X ′ . Since we do not have local-to-global obstructions, there are W-surfaces X ′ and X ′ as in the set-upof Conjecture 3.4. Lemma 3.6.
If Conjecture 3.4 is true for the W-surfaces X ′ , X ′ , thenit is also true for the W-surfaces X , X .Proof. The point is that the Q -Gorenstein deformation space of X i ’sand X ′ i ’s is smooth (see [H12, Section 3]). We have that the W-surface X ′ has minimal model and requires only flips to obtain the KSBAreplacement. Then the W-surface X must satisfy the same since its Q -Gorenstein deformation space is smooth and contains the one of X ′ . (cid:3) All in all, to verify that Conjecture 3.4 is true, we only need to verifyit for W-surfaces X which contain an extremal P-resolution over a WWsingularity, so that it contains no other Wahl singularities out of thisextremal P-resolution. That is the importance of Theorem 1.2 andTheorem 1.3, which will be proved in the next two sections.4. Proof of Theorem 1.2
In this section we essentially prove that the wormhole conjectureis valid for non rational surfaces with nef canonical class, and withan extremal P-resolution whose middle curve becomes a ( − − m )-curve with m ≥
2. At first they seem to be too special over a wormhole singularity,but they turn out to be chaotic. In the next section we manage to proveit only for m = 2 in a special situation.Throughout this section we assume the hypothesis of Theorem 1.2,which we now recall. Let Y be a normal projective surface with onecyclic quotient singularity ( Q ∈ Y ), which is smooth everywhere else.We assume that the minimal resolution of Y is not ruled, and that is a wormhole singularity, i.e. it admits two extremal P -resolutions f + i : ( C i ⊆ X i ) → ( Q ∈ Y ) , i = 1 ,
2. In addition we assume: • The strict transform in the minimal resolution of X of theexceptional curve C for the extremal P -resolution in X is a P with self-intersection − • The canonical class K X is nef. • Both surfaces X i admit Q -Gorenstein smoothings ( X i ⊆ X i ) → (0 ∈ D ), i.e. they are W-surfaces.We want to prove that K X is nef. This implies that the family( X i ⊆ X i ) → (0 ∈ D ) has nef canonical class (see e.g. [U16a, Sect.2]).Let (1 , Ω) = ( Q ∈ Y ), and ∆Ω = [ f s , . . . , f ] − − [ e , . . . , e r ] be thenumerical data of the extremal P-resolution X → Y . Let σ : ˜ X → X be the minimal resolution of X over the singularities P and P . Let E i , F j be the P ’s which resolve them respectively. In this way we have E i = − e i and F j = − f j .Let us assume that K X is not nef. By hypothesis we have existenceof ( X ⊆ X ) → (0 ∈ D ), and so we know that there is a curve Γ ≃ P in X such that K X · Γ < X is notruled, we can assume that Γ <
0, and (Γ ⊂ X ⊆ X ) → ( Q ∈ Y ⊂ Y )is an extremal nbhd of type mk1A or mk2A. In this way, the curve Γhas a very special position in relation to the singularities of X . Alsothe assumption that K X is nef puts more constraints, which can besummarized as: • Necessarily Γ intersects ( f +1 ) − ( Q ), since otherwise Γ would benegative for K X . • The curve Γ cannot intersect C out of the singularities P , P ,since otherwise we can contract Γ in ˜ X producing a surface X ′ and a curve C with K X ′ · C < −
1. But this is contrary toour assumption that ˜ X is not ruled (and so it has a minimalmodel). • As we have an mk1A or mk2A situation, the curve Γ in ˜ X can touch one Wahl chain transversally at one point, or bothchains transversally at the ends of each. The first option is notpossible since either it becomes a negative curve for K X or wehave contradiction with the not ruled assumption.Therefore the curve Γ can only intersect the F , F s and the E , E r ina mk2A situation (four possibilities). In the next arguments, we willstrongly use the discrepancies of the two Wahl singularities. We recall hat K X · Γ = (cid:16) K ˜ X − X a k a E a − X b l b F b (cid:17) · Γ = − − k i − l j where k a , l b are the discrepancies of the corresponding divisors, and i = 1 , r and j = 1 , s are the only possibilities. We can easily discardtwo of the four possibilities: • If Γ intersects E and F , then K X · Γ = K X · C > − • If Γ intersects E r and F s , then Γ also intersects the extremecurves of the two chains (one chain would produce an immediatecontradiction) from the minimal resolution of X , as shown onthe proof of Theorem 2.5. By the same result, we know that inthis case the strict transform of C in the minimal resolution of X is a ( − −
1. Therefore we obtain K X · Γ + K X · C = 0but K X · C >
0, and so a contradiction.The third and fourth possibilities are symmetric, so without loss ofgenerality we assume that Γ is intersecting E and F s as in Figure 6. F s F C E E r Γ Figure 6.
The potential bad curve Γ in ˜ X .We note that we must have r >
1, since r = 1 would give a Γintersecting E r = E and F s , but this case was ruled out above. Proposition 4.1.
Let Z be a normal projective surface, P , P ∈ Z the only singular points, which are Wahl singularities, and let σ : ˜ Z → Z be the minimal resolution of Z , which is not ruled. Assume thatthere exists ( − -curves C and Γ , such that on the minimal resolutionwe have the configuration given by Figure 6 (taking C = C ), where E , . . . , E r and F , . . . , F s are the resolutions of P and P . Assumealso that r > . Then, we cannot have simultaneously K Z · C > and K Z · Γ < . We note that the proof of this proposition will finish the proof ofTheorem 1.2. Proposition 4.1 will be also used in the next section. he proof of this proposition will be achieved by means of the next fewlemmas. Lemma 4.2.
We must have s > .Proof. If s = 1, then K Z · Γ = K Z · C > (cid:3) Lemma 4.3.
We must have e r = f s = 2 .Proof. As r, s >
1, have have that exactly one of the values e , e r is 2,the same for f , f s . We will verify that the other 3 cases for e r , f s areimpossible: • If e = f = 2, then contracting in the configuration E , C, F we obtain a P with self-intersection equal to 0. But this is acontradiction with the not ruled assumption on ˜ Z . • If e = f s = 2, then the argument against is analogue to theprevious with E , Γ , F s . • Let e r = f = 2. We have K Z · Γ = − − k − l s . When wecompute the values of δ i for [ e , . . . , e r ], we obtain that δ < δ r ,and so k = − δ δ + δ r < − δ δ + δ = − . An analogue argument shows that l s < − , and so K Z · Γ > − . This shows that the only option is e r = f s = 2. (cid:3) The previous argument used a very simple observation on discrep-ancies of Wahl singularities. To continue the proof of Proposition 4.1we need a more general statement on these discrepancies.
Lemma 4.4.
Let [ b , . . . , b t ] be a Wahl singularity, assume t ≥ and b t = 2 , and let us denote its discrepancies by m , . . . , m t . Then we havethe following bounds: (Type M ) If b = b = · · · = b t , then m = − b − and m t = − b − . (Type B ) Otherwise m = − µ and m t = − µ , where b < µ < b − .Proof. We will use again the δ i as in Equation 5.(Type M ): Every such singularity comes from [4] adding 2’s to theright. In this way δ = 1 , δ = 2 , . . . , δ t = t . Then the discrepanciesare m = − t +1 and m t = − tt +1 . As b = t + 3, we get what wewanted. Type B ): Let us say that b = p + 2. Eliminating the 2’s on theright, we fall into [2 , b , . . . , b t − p ], with b t − p >
2. In this way, δ > δ t − p ,because the first entry is a 2. Adding back the 2’s on the right, we get δ t − p + i = δ t − p + iδ , i = 0 , . . . , p. In particular, δ t = δ t − p + pδ . Hence, if µ = δ δ + δ t , then m = − δ δ + δ t = − µ and m t = − δ t δ + δ t = − δ δ + δ t = − µ. It is enough then to bound µ . As δ > δ t − p , we have µ > δ δ + ( δ + pδ ) = 1 p + 2 = 1 b . On the other hand, as all δ i are positive, µ = 1 p + 1 · ( p + 1) δ ( p + 1) δ + δ t − p < p + 1 = 1 b − . These two bounds give b < µ < b − . (cid:3) We now continue the proof of Proposition 4.1.
Lemma 4.5.
Necessarily [ e , . . . , e r ] must be of type M . Moreover, if [ f , . . . , f s ] is of type B , then e = f + 1 ; If [ f , . . . , f s ] is of type M ,then e = f − .Proof. The basic idea is to see what happens to E in ˜ Z after we con-tract all possible ( − C , and then all( − F j chain F s , . . . , F s ′ . This will imposeconditions to e and f , which will allow to bound the discrepanciesinvolved in K Z · Γ.We are going to analyze the four possible case, which depend on thetype B or M of the singularities. (BB): If [ e , . . . , e r ] and [ f , . . . , f s ] are of type B , then we have f − f s , and so the curve E will have self-intersection − e + 1 + 1 + ( f −
2) = − e + f after we contract Γ , C and { F s , . . . , F s ′ } . Because of our not ruled assumption on ˜ Z , we musthave e ≥ f + 1. By Lemma 4.4, we have k < − e − , l s < − f . herefore, K Z · Γ > − − e − f = e − − f ( e − f ≥ , since e ≥ f + 1. (MB): If [ f , . . . , f s ] is of type M and [ e , . . . , e r ] is of type B , thenwe have s − f − f s , and so the curve E will have self-intersection − e + 1 + 1 + ( f −
4) = − e + f − , C and { F s , . . . , F s ′ } . Again, because of our not ruledassumption on ˜ Z , we must have e ≥ f −
1. The bound for k is asabove, while l s = − f − . In this way, K Z · Γ > − − e − f − e − f + 1( e − f − ≥ , since e ≥ f − (BM): If [ f , . . . , f s ] is of type B and [ e , . . . , e r ] is of type M , thenwe have f − f s , and so the curve E willhave self-intersection − e + f after the contractions as above, and so e ≥ f + 1. By Lemma 4.4 we can write k = − e − , l s < − f , and so K Z · Γ > − − e − f = e − − f ( e − f . If e ≥ f + 2, then K Z · Γ >
0, and so we necessarily get e = f + 1. (MM): If [ e , . . . , e r ] and [ f , . . . , f s ] are of type M , we have f − f s , and so the curve E will have self-intersection − e + f − e ≥ f −
1. ByLemma 4.4 we have k = − e − , l = − f − , and so K Z · Γ = − − e − f − e − f ( e − f − . If e ≥ f , then K Z · Γ >
0, and so we necessarily get e = f − (cid:3) Lemma 4.6.
Necessarily [ e , . . . , e r ] is of type M , and [ f , . . . , f s ] isof type B . roof. By Lemma 4.5, the only other possibility is that [ e , . . . , e r ] and[ f , . . . , f s ] are of type M , together with e = f −
1. Let q := f ≥ , . . . , | {z } q − , q ] − − [ q − , , . . . , | {z } q − ] . We have r = q − , s = q −
3. In ˜ Z we have a situation as in Figure7, where C = Γ = − , F = − q, E = − ( q − , and all the rest are( − E E E E r F F F s − F s C Γ Figure 7.
Situation in ˜ Z , case M M .After we contract Γ, C , F q − , . . . , F , we obtain that F and E forma cycle followed by the chain E , . . . , E r , where E = − , F = − ( q − − E , so F becomesa nodal rational curve and F = − ( q − E , . . . , E r obtaining that K · F = ( q − − · ( r −
1) = ( q − − q −
5) = 4 − q < , since q ≥
5. This gives a contradiction since ˜ Z is not ruled. (cid:3) With the previous lemma, the only option is [ e , . . . , e r ] of type M ,and [ f , . . . , f s ] of type B . Hence we can write[ f , . . . , f s ] = [ p, f , . . . , f t , , . . . , | {z } p − ] , where t ≥
2. Cancelling the 2s on the right, we get [2 , f , . . . , f t ] whichis a Wahl chain again. As its length is at least two, we have that[ f t , . . . , f ,
2] is of type M or B . In the next final lemmas, we will saythat [ f , . . . , f s ] is of type BM or BB respectively. Lemma 4.7. If [ e , . . . , e r ] is of type M , then [ f , . . . , f s ] must be oftype BB .Proof. We assume that [ e , . . . , e r ] is of type M , and [ f , . . . , f s ] is oftype BM . Let p = f = e − ≥
4. Then we have[ e , . . . , e r ] = [ p + 1 , , . . . , | {z } p − ] nd [ f , . . . , f s ] = [ p, , . . . , | {z } q − , q, , . . . , | {z } p − ] , where q ≥
5. Let t = q − q . Inthis way E = − ( p + 1) , F = − p, F t = − q, C = Γ = −
1, and allother curves in this situation are ( − K · F t is eventuallynegative with F t singular, which goes against the assumption ˜ Z is notruled. We first contract C , Γ, F s , . . . , F t +1 . Then E becomes a ( − E , . . . , E r and so F becomes a ( − q >
5, then F t intersects F only at one point with F t · F = p − F , . . . , F t − we get a singular curve F t with K · F t = − ( p − q − − <
0. If q = 5, then F t intersects F at two points,with F t · F = 1 + ( p − F , we get a singular F t with K · F t = 5 − p < (cid:3) Lemma 4.8.
The case [ e , . . . , e r ] of type M and [ f , . . . , f s ] of type BB is impossible.Proof. Let p = f = e − ≥
4. We can write[ e , . . . , e r ] = [ p + 1 , , . . . , | {z } p − ]and [ f , . . . , f s ] = [ p, , . . . , | {z } q − , . . . , q, , . . . , | {z } p − ] , where q ≥
3. Let t = s − ( p −
2) be the position of the entry q . Thecontractions that will come are exactly the contractions we perform inthe previous lemma, but at the end we are contracting F , . . . , F q − .The relevant intersection now is K · F t = ( q − p − − ( q − p −
2) = − ( p − q − − < . (cid:3) With Lemma 4.8 we finish the proof of Proposition 4.1, and so The-orem 1.2. 5.
Proof of Theorem 1.3
Throughout this section we assume the hypothesis of Theorem 1.3,which we now recall. Let Y be a normal projective surface with onecyclic quotient singularity ( Q ∈ Y ), which is smooth everywhere else.We assume that the minimal resolution of Y is not ruled, and that is a wormhole singularity, i.e. it admits two extremal P -resolutions f + i : ( C i ⊆ X i ) → ( Q ∈ Y ) , i = 1 ,
2. In addition we assume: • The strict transform in the minimal resolution of X of theexceptional curve C for the extremal P -resolution in X is a P with self-intersection −
2, and X has only one singularity. • The canonical class K X is nef. • Both surfaces X i admit Q -Gorenstein smoothings ( X i ⊆ X i ) → (0 ∈ D ), i.e. they are W-surfaces.We want to prove that we only need flips to run MMP on ( X ⊂X ) → (0 ∈ D ). Here we cannot guarantee that K X is nef, we indeedmay need some flips, as shown by Example 3.5. The proof will besubstantially different to the proof of Theorem 1.2, and lemmas willtake a more general situation than the one we started with. Lemma 5.1.
Let ˜ Z be a smooth projective surface which is not ruled.Suppose that ˜ Z has some chain of smooth rational curves C, E , . . . , E r ,with C = − , E i = − b i , b i ≥ , and [ b , . . . , b r ] = m ma − is a Wahlchain. Suppose also that we have a ( − -curve Γ which transverselyintersects only one E j at one point, and also intersects C . Then, itfollows that Γ intersects C transversely at one point, b j = 2 , and j = r . This lemma will be useful when we have a mk1A neighborhood viaΓ over an extremal P-resolution with only one Wahl singularity and a( − Z as the minimal resolution of the singularity,and E , . . . , E r the exceptional divisor. Proof.
Note first that K · C = 0 by adjunction. If we blow-down Γ,then the intersection K · C decreases in Γ · C . But canonical class mustbe eventually nef, and so the only possibility is Γ · C = 1.After blowing-down Γ, we can blow-down C , and so E j ≤ − Z is not ruled, i.e. b j = 2.Suppose now that j = r . Since b j >
2, we have b = 2 (or r = 1,where b = b j = 4, which leads to a straightforward contradiction). Wehave two options. • If [ b r , . . . , b ] is a Wahl singularity of type M , so that [ b , . . . , b r ] =[2 , . . . , , r + 3], we can blow-down Γ , C , and E , E , . . . , E r − .We get a nodal curve E r with K · E r = −
1, which is a contra-diction, since ˜ Z is not ruled. • If [ b r , . . . , b ] is a Wahl singularity of type B , so that[ b , . . . , b r ] = [2 , . . . , , b s +1 , . . . , b r − , s + 2] , we can blow-down Γ , C , and E , . . . , E s . Thus, we get K · E r = −
2, which is again a contradiction. t follows that j = r . (cid:3) The next lemma will be useful to control mk1A neigborhoods whichwill appear on the proof of Theorem 1.3. Roughly speaking, whenwe run MMP to ( X ⊂ X ) → (0 ∈ D ), we can get an extremal P-resolution with either two singularities and a ( − − Lemma 5.2.
Let us assume same hypothesis as in Lemma 5.1. Let Z be contraction of E , . . . , E r , and assume that Z admits a Q -Gorensteinsmoothing ( Z ⊂ Z ) → (0 ∈ D ) . Then Γ ⊆ Z induces a mk1A neighbor-hood which must be of flipping type. The resulting extremal P-resolutionafter the flip must have either two Wahl singularities with a ( − -curvein the middle, or one Wahl singularity with a ( − -curve.Proof. Let σ : ˜ Z → Z be the minimal resolution of Z . The curve Γinduces a mk1A neighborhood on Z . Note that Lemma 5.1 says that b j must be >
2. Hence we have the mk1A neigborhood (Γ ⊂ Z ) → ( Q ∈ Y ) = (1 , Ω) where∆Ω = [ b , . . . , b j − , . . . , b r ]has every entry ≥
2. Since [ b , . . . , b r ] is a Wahl singularity, P b i =3 r + 1, and then the sum of the entries of ∆Ω is 3 r . This proves that (1 , Ω) is not a Wahl singularity, and then we have a flipping mk1A(see Subsection 3.1).After we flip, we can apply Theorem 2.5 to show that if the newextremal P-resolution has one singularity, then self-intersection of theflipping curve (on the new minimal resolution) must be −
2; if there aretwo singularities, then this self-intersection must be − (cid:3) The proof of Theorem 1.3 will be based on a repeated use of Lemma5.2. We will need to control the new outcomes from Lemma 5.2. Forthat, we give a definition for these two cases.
Definition 5.3.
Let Y be a normal projective surface with one cyclicquotient singularity Q ∈ Y . We name the following extremal P-resolutions ( C ⊂ Z ) → ( Q ∈ Y ) as follows: Type(-1) : The surface Z has two singularities, and the strict transform of C in the minimal resolution of Z is a ( − Type(-2) : The surface Z has one singularity, and the strict transform of C in the minimal resolution of Z is a ( − K Z is ample, it is only required C · K Z > emma 5.4. Let us consider the hypothesis of Theorem 1.3. Let Z be the W-surface X . Assume we have run the MMP on W-surfaces Z , . . . , Z m so that the flip from Z i to Z i +1 comes always from a Type(-2) extremal P-resolution as in Lemma 5.2. In addition, assume that K Z m is not nef. Then the only possible mk1A for Z m is the one de-scribed in Lemma 5.1.Proof. If not, we have a curve Γ m ⊂ Z m so that it is a ( − Z m , it is disjoint from C m , and it intersectsonly E i transversally at one point. Let Γ i ⊂ Z i be the flipping curvefor each of the flips, so it satisfies Lemma 5.1. Note that Γ i in ˜ Z i +1 isa ( − Z i are ruled, in particular Z m , the curvesΓ m − , Γ m are disjoint in ˜ Z m . Hence Γ m is again a ( − Z m − .Inductively we obtain a ( − m in ˜ Z which is disjoint from C and only intersects some E j transversally at one point. We now go to X . Since X is not ruled, the curve Γ m must be a ( − X intersecting only one exceptional curve of ˜ X → X transversally atone point. But that is not possible since then K X · Γ m <
0. Thereforethe curve Γ m must intersect C m , and so we are as in Lemma 5.1. (cid:3) Lemma 5.5.
Let us consider the hypothesis of Theorem 1.3. Let Z be the W-surface X . Assume we have run the MMP on W-surfaces Z , . . . , Z m so that the flip from Z i to Z i +1 comes always from a Type(-2) extremal P-resolution as in Lemma 5.2 for i = 1 , . . . , m − , and thelast one is Type(-1). In addition, assume that K Z m is not nef. Then itis not possible to have a mk1A nbhd for Z m .Proof. The proof is similar to the proof of Lemma 5.4. A potentialΓ m ⊂ Z m defining a mk1A will survive untouched until reaching Z ,giving a mk1A nbhd to Z , but we know that this is not possible. It iskey that the surfaces involved are not ruled, so that ( − K X is nef. (cid:3) We now show a key step to rule out certain mk2A nbhd. After thatwe will have everything to give a proof for Theorem 1.3.
Lemma 5.6.
Let Z be a normal projective surface, Q , Q the onlysingular points on Z . Assume that there is a ( − -curve D passingthrough Q and Q , such that ( D ⊂ Z ) → ( Q ∈ Y ) is an extremal P -resolution.Let σ : ˜ Z → Z be the minimal resolution of Z , which is not ruled,with F , . . . , F s and G , . . . , G t the exceptional divisors for Q and Q .Suppose that we have two ( − -curves B and Γ on Z , such that on ˜ Z the configuration is as on Figure 8, and B intersect transversally other urve in the set { F i , G j } (different of G t and F s ) in a point. Then Γ · K Z ≥ . G t G D F F s B Γ Figure 8.
Configuration on ˜ Z . Proof.
Let us assume we have such a configuration of curves. Let f , . . . , f s , g , . . . , g t be so that F i = − f i , G j = − g j . We have g t ≥ B y Γ the curve G t is a P with G t = 0, but ˜ Z is not ruled. If f s >
2, then K Z · Γ > f s = 2. In particular s ≥ t = 1, and so g t = 4. Then if we contract B, D,
Γ, we have G t = −
1. But also F s = −
1, and they intersect, a contradiction with˜ Z not ruled. Therefore we have t ≥
2, and with that g = 2.Note that in all contractions below, we can never have a chain[1 , , . . . , ,
1] by the not ruled hypothesis.Let us denote by p := f , q := g t , and { k , . . . , k s } , { l , . . . , l t } thecorresponding discrepancies. We have the following four cases: • If [ f , . . . , f s ] and [ g t , . . . , g ] are of type B , then there are p − − F s . Because of the change of self-intersection of G t after contracting B , Γ and the p − − q ≥ p + 1. By Lemma 4.4, we have K · Γ = − − k s − l t > − p + 1 − q − q − p − p ( q − ≥ , • If [ f , . . . , f s ] is of type B and [ g t , . . . , g ] is of type M , wehave p − − F s , and so q ≥ p + 1 justas before. But in addition we can blow-down D, G , . . . , G t − which gives the better restriction q ≥ p + 2. Lemma 4.4 givesin this case K · Γ = − − k s − l t > − p + 1 − q − q − p − p ( q − ≥ . • If [ f , . . . , f s ] is of type M and [ g t , . . . , g ] is of type B , then wehave p − − F s , and so q ≥ p −
1. Thisimplies K · Γ = − − k s − l t > − p − − q − q + 1 − p ( p − q − ≥ . If both [ f , . . . , f s ] and [ g t , . . . , g ] are of type M , we obtain q ≥ p − D and G , . . . , G t − ,we get q > p −
1. We also have K · D = − k + l = − p − q − p − q ( p − q − > , and so p > q . But then p > q > p − (cid:3) We now finish the proof of Theorem 1.3.Let Z := X . If K Z is nef, then we are done. If not, then byLemma 5.4 we must have a mk1A as in Lemma 5.1. Using Lemma5.2, we can now apply the flip and get Z that sits in two possiblesituations: Type(-1) or Type(-2).We now assume that we have a chain of flips giving only Type(-2),or a chain of Type(-2) followed by one Type(-1).If only Type(-2), then by Lemma 5.4 we have that the new mk1Anbhd can only be as in Lemma 5.1, and we continue, or K is nef.If only Type(-2) and one last Type(-1), then we cannot have a mk1Anbhd by Lemma 5.5. And so either K is nef, or we have a mk2A nbhd.Then by Lemma 5.6 we can only have a Γ intersecting G , F (notpossible since K · D > F , G t or G , F s . Note that in both caseswe have t > s > K · D >
0. Therefore we can apply Proposition 4.1 to deduce thata mk2A nbhd is impossible, and so K must be nef. This process mustend in finitely many steps, so we are done.6. Open questions
Topological type of surfaces in a wormhole.
Let us startwith a couple of examples. Consider a general rational elliptic surface Z → P with sections. Let F E and F G two nodal I fibers, and let S be a section of Z → P . We blow-up s times over the node in F E ,and r times over the node in F G to obtain a surface ˜ X with the Wahlchains [ F E , E , . . . , E s − ] = [3 + s, , . . . ,
2] and [ F G , G , . . . , G r − ] =[3 + r, , . . . , X (see [U16b, Theorem 4.2]), and the general fibers are either Enriquessurfaces (if r = s = 1) or elliptic surfaces of Kodaira dimension 1. Infact, one can prove that the general fiber is an elliptic fibration over P with p g = q = 0 and two multiple fibers of multiplicities s and r ,and so its fundamental group is Z / gcd( r, s ). Hence, although these arenot degenerations of surfaces of general type, they will be useful to seethat wormholes may change the topology of the general fibers. he curve S defines an extremal P-resolution on X . Let us con-sider the chain of curves E s − , . . . , E , F E , S, F G , G , . . . , G r − . Theircontraction defines the cyclic quotient singularity ( Q ∈ Y ) given by[2 , . . . , , s, r, , . . . , , whose dual continued fraction is [ s + 1 , ¯2 , . . . , , , , . . . , ¯2 , r + 1], wherethe numbers of 2’s are s − r − s = 1 or r = 1 are a bit differentas the reader may check.) We want to check whether ( Q ∈ Y ) is awormhole singularity, and so we are looking for another pair. A quickverification shows that r > s > , ¯2 , , , ¯2 , , ¯2 , , ¯3 , , , − − [5 ,
2] and (II) [2 , , − − [4] respectively. We have: (I). In this case, the new extremal P-resolution is [2 , , , −
2. Let X be the corresponding W-surface. The curve G is now a flippingcurve, and after the flip we obtain a W-surface X ′ with extremal P-resolution [2 , , − − [4]. Therefore the canonical class now is nef. Thegeneral fiber of X gives an elliptic surface with fundamental group oforder gcd(4 ,
3) = 1, but the general fiber of X ′ has fundamental groupof order gcd(4 ,
2) = 2. Thus they are not homeomorphic. (II).
In this case, the new extremal P-resolution is 2 − [2 , , X be the corresponding W-surface. The curve E is now a flippingcurve, and after the flip we obtain a W-surface X ′ with extremal P-resolution [2 , − − [4], and so the canonical class now is nef. Thegeneral fiber of X gives an elliptic surface with fundamental group oforder gcd(4 ,
2) = 2, but the general fiber of X ′ is simply connected,since gcd(3 ,
2) = 1. Thus they are not homeomorphic as well.However, in many cases wormholes produce surfaces with isomor-phic fundamental groups. Let us consider a wormhole situation from X to X , where both have two Wahl singularities corresponding tothe extremal P-resolutions. Let d i be the greatest common divisor ofthe indices of the Wahl singularities in X i . (If there is one or zerosingularities, then d i = 1.) Proposition 6.1. If d = d , then the fundamental groups of the gen-eral fiber of X and X are isomorphic.Proof. Let f + i : ( C i ⊂ X i ) → ( Q ∈ Y ) be the contractions to a worm-hole singularity. We are going to use the Seifert–Van-Kampen theorem o compare the fundamental groups of the general fibers X ,t and X ,t .Let L be the link of ( Q ∈ Y ). Let M i be the Milnor fiber of thesmoothing of ( Q ∈ Y ) corresponding to X i (i.e. the blowing-downdeformation of the Q -Goresntein smoothing corresponding to the ex-tremal P-resolution in X i ). Then π ( M i ) ≃ Z /d i , and π ( L ) ≃ Z / ∆where (1 , Ω) = ( Q ∈ Y ). Let X i be the complement of C i . Then X = X =: X , and we have π ( X i,t ) ≃ (cid:0) π ( X ) ⋆ π ( M i ) (cid:1) /N ( αβ − )where α generates π ( L ) in π ( X ), β generates π ( L ) in π ( M i ), and N ( αβ − ) is the smallest normal subgroup containing αβ − . By [LW86,Lemma 5.1], we have that the morphism induced by the inclusion π ( L ) → π ( M i ) is onto. Therefore, if π ( X ) = G/R , where G aregenerators and R are relations, then π ( X i,t ) ≃ G/ ( R, α d i = 1). Theclaim follows when d = d . (cid:3) Corollary 6.2. If d = d = 1 , then the general fibers of X and X have isomorphic fundamental groups and equal to π ( ˜ X ) = π ( ˜ X ) = π ( Y ) . In particular, if in addition ˜ X is rational, then wormholesproduce simply connected surfaces.Proof. Here, by applying the Seifert–Van-Kampen theorem, we havethat π ( X i ) = π ( X i ) / ( α = 1), but this is what we just computed for π ( X i,t ) when d i = 1 (alternatively one can use [LP07, Theorem 3]).The other claim is because we are dealing with rational singularities. (cid:3) Let us consider rational W-surfaces X i with d = d = 1. Let us as-sume K = 1, and so their general fibers are simply connected Godeauxsurfaces. There are plenty of such wormholes in the KSBA compacti-fication of the moduli space of Godeaux surfaces (see e.g. [LP07, Fig.6] for the [2 , , − − [4]). By Freedman’s classification theorem, thegeneral fibers are homeomorphic as oriented 4-manifolds. On the otherhand, Miles Reid conjectures that the moduli space of torsion zeroGodeaux surfaces is irreducible, and so all of these wormholed surfacesshould be diffeomorphic. Very recently, Dias and Rito proved Reid’sconjecture for Z / d = d (as in Example 3.5) givesdiffeomorphic surfaces. Question 6.3.
For a wormhole with d = d , are the general fibersalways diffeomorphic? homeomorphic? n fact one can show that d = d keeps the homology together withthe intersection form, and so if they are simply connected, then Freed-man’s theorem produces an homeomorphism. More on the topologyaspects will be part of a sequel work. On the other hand and as we sawabove, for the case d = d we may have non homeomorphic surfaces(although the example was not of general type). In [DRU20, Figure 5]we have a wormhole defined by [2 , − − [2 , , ,
3] in X ( d = 1),whose Q -Gorenstein smoothing is a Z / X has extremal P-resolution [2 , , , − − [4], and so d = 2. If its Q -Gorenstein smoothing has π = Z /
2, then it would be Z / What is left to prove the conjecture.
In this paper we intro-duced the wormhole conjecture, and we proved it for many situationsunder the assumption that the singular surfaces involved were not ra-tional. Hence we divide the final discussion in two parts:
Nonrational:
Let X and X be the W-surfaces in a wormhole, bothwith an extremal P-resolution over a fixed wormhole singularity, andnonsingular out of them. In the next list, we write Wahl- m -Wahl for anextremal P-resolution with two Wahl singularities (distinct or equal)and a middle curve whose self-intersection in the minimal resolution is − m . If Wahl is dropped, then the point is nonsingular. Using Theorem2.5, and because we already have Theorems 1.2 and 1.3, the list ofpairs of extremal P-resolutions where we do not know the validity ofthe wormhole conjecture is:(a) Wahl- m -Wahl and Wahl- m -Wahl for m ≥ m -Wahl and Wahl-( m − m ≥ m and m -Wahl for m ≥ Example . Let us consider a chain of P ’s E , . . . , E in a nonsingularsurface Z with nef minimal model, where E i = − e i , and[ e , . . . , e ] = [5 , , , , , , , , . Assume that there is a ( − E twice and transver-sally, and disjoint from the rest. This ( − ontradiction with the minimal model of Z . The wormhole singular-ity [5 , , , , , , , ,
5] admits two obvious extremal P-resolutions:[5 , , , , , , , − X and 5 − [2 , , , , , , ,
5] in X , so weare in case (c). The curve Γ is positive for K X , and it is not only nega-tive for K X but it induces a divisorial contraction on the deformationof X .In fact, we have infinitely many examples via[ n + 2 , , . . . , | {z } n , n + 5 , , . . . , | {z } n , n + 2]with a ( − n −
1. Are any of these counterexamples realizable?
Rational:
Here do not have a feasible strategy to prove the conjecture.But we have many examples verifying it for the invariants p g = q = 0and K = 1 , , ,
4. These examples are constructed as in [LP07] andthey have two singularities, they will be part of some future work.We finish the paper with another open question. Note that a WWsingularity (i.e. it admits at least one pair of indices to be a zerocontinued fraction) has complete freedom on the values of δ . But itseems that this freedom is lost for wormhole singularities. Question 6.5.
What are the possible values for δ in a wormhole sin-gularity? For instance, δ = 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , δ for wormholes singularities (1 , Ω) whoseminimal continued fraction have at most 25 entries, and with one oftheir extremal P-resolutions being of type m -Wahl with m = 2. Thefollowing infinite family has the value δ = 2:∆Ω = 2 − [2 , . . . , | {z } k − , , k ] = [2 , . . . , | {z } k − , k + 3] − − [2 , . . . , | {z } k − , k + 1] . ne can compute[2 , . . . , | {z } k − , , k ] = (2 k − (2 k − k − − , which gives ∆ = 4 k , Ω = (2 k − , δ = 2. Actually, the case δ = 2 canbe classified through the use of triangulations of polygons, where onechanges one diagonal in a “corner quadrilateral” by the other diagonal.A better understanding of the wormhole phenomenon on singularitiesis wanted, to potentially solve the wormhole conjecture and to showtopological implications. References [CU18] S. Coughlan, G. Urz´ua, On Z /3-Godeaux surfaces , Int. Math. Res. Not.IMRN 2018, no. 18, 5609–5637.[DR20] E. Dias, C. Rito, Z / -Godeaux surfaces , pre-print arXiv:2009.12645[math.AG].[DRU20] E. Dias, C. Rito, G. Urz´ua, On degenerations of Z / -Godeaux surfaces ,pre-print arXiv:2002.08836 [math.AG].[EU18] J. D. Evans, G. Urz´ua, Antiflips, mutations, and unbounded symplecticembeddings of rational homology balls , arXiv:1807.06073 [math.SG], to appearin Annales de l’Institut Fourier.[H12] P. Hacking,
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Email address : [email protected] Facultad de Matem´aticas, Pontificia Universidad Cat´olica de Chile,Campus San Joaqu´ın, Avenida Vicu˜na Mackenna 4860, Santiago, Chile.