Birational geometry of sextic double solids with a compound A_n singularity
BBirational geometry of sextic double solids with acompound A n singularity Erik Paemurru ∗† January 5, 2021
Abstract
Sextic double solids, double covers of P branched along a sextic surface, are the lowestdegree Gorenstein Fano 3-folds, hence are expected to behave very rigidly in terms ofbirational geometry. Smooth sextic double solids, and those which are Q -factorial withordinary double points, are known to be birationally rigid. In this article, we study sexticdouble solids with an isolated compound A n singularity. We prove a sharp bound n ≤ n explicitly and prove that sextic double solids with n > Contents
1. Introduction 22. Preliminaries 5
3. Constructing sextic double solids with a cA n singularity 11 cA n singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3. Other cA n singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4. Weighted blowups 21 ∗ The author was supported by the Engineering and Physical Sciences Research Council (grant 1820497)while in Loughborough University, and the London Mathematical Society Early Career Fellowship(grant ECF-1920-24) while in Imperial College London. † Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel,Switzerland. a r X i v : . [ m a t h . AG ] J a n . Birational models of sextic double solids 26 Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.1. Singularities after divisorial contraction . . . . . . . . . . . . . . . . . . . . 265.2. cA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3. cA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4. cA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.5. cA family 1 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.6. cA family 2 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.7. cA family 3 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.8. cA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 A. Computer code 43References 46
1. Introduction
We work with projective varieties over C . Classification of algebraic varieties is one of thefundamental goals in algebraic geometry. The Minimal Model Program says that everyvariety is birational to either a minimal model or a Mori fibre space. Two Mori fibre spacesare birational if they are connected by a sequence of Sarkisov links (see [Sar89], [Rei91],[Cor95], [HM13]). In the extreme case, the Mori fibre space is birationally rigid , meaningthat it is essentially the unique Mori fibre space in its birational class.Examples of Mori fibre spaces include Fano varieties. The first birational rigidity resultwas in the seminal paper by Iskovskikh and Manin [IM71] for smooth quartic 3-folds in P .A wealth of examples of birationally rigid varieties was given in [CPR00] and [CP17],by showing that every quasismooth member of the 95 families of Fano 3-folds that arehypersurfaces in weighted projective spaces is birationally rigid. One major consequence ofbirational rigidity is nonrationality. Birational rigidity remains an active area of research(see [Pro18], [Kry18], [AO18], [dF17], [CG17], [CS19], [EP18]).Among smooth Fano 3-folds, the projective space has the highest degree (64), andsextic double solids, double covers of P branched along a sextic surface, have the leastdegree (2). In [Isk80], it is proved that smooth sextic double solids are birationally rigid.It is interesting to see how this changes as we impose singularities on the variety. Thepaper [Puk97] proved that sextic double solids stays birationally rigid if we impose anordinary double point, meaning the 3-fold A singularity x + x + x + x . A sextic doublesolid can have up to 65 singular points (see [Bar96], [Bas06], [JR97], [Wah98]), and foreach n ≤
65, there exists a sextic double solid with exactly n ordinary double points andsmooth otherwise (see [CC82]). A sextic double solid with only ordinary double points isbirationally rigid if and only if it is factorial, which is true for example if it has at most 14ordinary double points (see [CP10]).The next natural question is to consider more complicated singularities in the Moricategory. We study sextic double solids with an isolated compound A n singularity, alsocalled a cA n singularity, meaning that the general section through the point is the Du Val A n singularity x x + x n +13 . A cA n singularity is locally analytically given by x x + h ( x , x )where the least degree among monomials in h is n + 1. The first main result of the paperis describing sextic double solids with an isolated cA n singularity. Theorem (see Theorem A) . If a sextic double solid has an isolated cA n point, then n ≤ . cA n singularity for every n ≤
8. These form 11 families, as there are 4 familiesfor cA . A very general member of every family, except for the family 7.4, is a Mori fibrespace.We say a few words on bounding the number of cA n singularities. It is clear that anisolated cA n singularity has Milnor number at least n . Since the third Betti numberof a smooth sextic double solid is 104 (see [IP99, Table 12.2]), an argument similar to[AK16, Section 3.2] shows that the total Milnor number of a sextic double solid which isa Mori fibre space is at most 104. This gives the bounds that a Mori fibre space sexticdouble solid can have up to 1 cA singularity, or up to 2 cA singularities, or up to 2 cA singularities, . . . , or up to 26 cA singularities. We do not expect these bounds tobe sharp, as already for ordinary double points it gives an upper bound of 104, far fromthe actual 65. Using Theorem A, it is possible to construct sextic double solids with a cA point, a cA point and two ordinary double points with both total Milnor and totalTjurina number at least 66.The second main result is the following theorem: Theorem (see Theorem B and Section 5.3) . A general sextic double solid which is a Morifibre space with an isolated cA n singularity where n ≥ is not birationally rigid. Birational non-rigidity for a sextic double solid X is proved by describing a birationalmodel, meaning a Mori fibre space T → S such that X and T are birational. We findthe birational models by explicitly constructing a Sarkisov link for each family of sexticdouble solids, under the generality conditions described in Condition 5.1. Table 1 gives anoverview of the Sarkisov links X ← Y (cid:57)(cid:57)(cid:75) Y → Z and the birational models, which areeither fibrations Y → Z or Fano varieties Z . In the latter case, Y → Z is a divisorial tothe given singular point. The morphism Y → X is a divisorial contraction with centrethe cA n point. The birational maps Y (cid:57)(cid:57)(cid:75) Y are isomorphisms in codimension 1.Note that when we say that a birational map Y (cid:57)(cid:57)(cid:75) Y is k Atiyah flops, then we meanthat algebraically it is one flop, contracting k curves to k points and extracting k curves,and locally analytically around each of those points, it is an Atiyah flop. Similarly forflips. Also note that the Sarkisov link to a sextic double solid with a cA singularitywas already described in [Oka14, Section 9, No. 9], starting from a general quasismoothcomplete intersection X , ⊆ P (1 , , , , , X is a divisorial contraction Y → X . Kawakita described divisorial contractionsto cA n points locally analytically, showing that they are certain weighted blowups. Toconstruct Sarkisov links, we need a global description. In Proposition 4.5 and Lemma 4.8,we show how to construct divisorial contractions to cA n points algebraically on affinehypersurfaces, and use this in Section 5 to construct divisorial contractions Y → X for(projective) sextic double solids X . Using unprojection techniques (see [PR04] for a generaltheory of unprojection), we find an embedding of Y inside a toric variety T , such that the2-ray link of T restricts to a Sarkisov link for X (following [BZ10] and [AZ16]).If we try the same methods as in the proof of Theorem B on sextic double solids witha cA n singularity where n ≤
3, then we do not find any new birational models. Moreprecisely: a (3 , , , cA singularity on a general Mori fibre spacesextic double solid initiates a Sarkisov link to itself X (cid:57)(cid:57)(cid:75) X . A (2 , , , cA singularity, a (2 , , , x x + x + x singularityand the (usual) blowup for an ordinary double point on a general Mori fibre space sextic3able 1. Birational models for general sextic double solids that are Mori fibre spaces withan isolated cA n singularity Y Y cA n ∈ X ⊆ P (1 , Z ϕ ψ cA n weightedblowup ϕ weightedblowup orfibration ψ ZcA (3 , , ,
1) 10 Atiyah flops ( , , ) (1 , , ∈ Z , ⊆ P (1 , , , cA (3 , , ,
1) 4 Atiyah flops (3 , , , cA ∈ Z ⊆ P (1 , X (cid:29) Z if general cA (4 , , ,
1) 2 Atiyah flops, then(4 , , , − , −
1; 2)-flip (3 , , , cA ∈ Z ⊆ P (1 , cA , 1 (4 , , ,
1) two (4 , , , − , −
1; 2)-flips (1 , , ,
1) ODP ∈ Z , ⊆ P (1 , ) cA , 2 (4 , , ,
1) Atiyah flop, thentwo (4 , , − , − , , , cA ∈ Z , ⊆ P (1 , cA , 3 (4 , , ,
1) 2 Atiyah flops dP -fibration P cA (5 , , ,
1) (4 , , , − , −
1; 2)-flip (3 , , , , cD ∈ Z , ⊆ P (1 , K -trivial curves are contracted, leaving the Mori category. We expect that generalMori fibre space sextic double solids with a cA singularity are birationally rigid, and withcertain cA or cA singularities are birationally superrigid. Organization of the paper
In Sections 2.1, 2.2 and 2.3, we give known results that we use respectively in Sections 3, 4and 5. In Section 3, we construct a parameter space of sextic double solids in Theorem Awith an isolated cA n singularity. In Section 4, we explain the relationship betweenalgebraic and local analytic weighted blowups, and in Proposition 4.5 and the technicalLemma 4.8, show how to construct divisorial contractions to cA n points algebraicallyon affine hypersurfaces. In Section 5, we construct birational models for general sexticdouble solids which are Mori fibre spaces with an isolated cA n singularity where n ≥ . Preliminaries All varieties we consider are irreducible and over C .We study sextic double solids, which are double covers of the projective 3-space branchedalong a sextic surface. We use the following equivalent characterization: Definition 2.1. A sextic double solid is an irreducible hypersurface given by the zerolocus of w + g in the weighted projective space P (1 , , , ,
3) with variables x, y, z, t, w ,where g ∈ C [ x, y, z, t ] is a homogeneous polynomial of degree six. We recall some results from the singularity theory of complex analytic spaces and onterminal singularities.We denote the variables on C n by x = ( x , . . . , x n ), where n is a positive integer. Let C { x } denote the convergent power series ring. The zero set of an ideal I is denoted by V ( I ), where I is either an ideal of regular functions or holomorphic functions, dependingon context. Given a regular or holomorphic function f on a variety or space X , denotethe non-zero locus of f by X f . Given positive integer weights w = ( w , . . . , w n ) for x , wecan write a non-zero polynomial or power series f as a sum of its weighted homogeneousparts f i . Then, the weight of f , denoted wt( f ), is the least non-negative integer d suchthat f d = 0. We define wt(0) = ∞ . If w = (1 , . . . , d is called the multiplicity ,denoted mult( f ). A hypersurface singularity is a complex analytic space germ (notnecessarily irreducible or reduced) that is isomorphic to ( X, ) where X ⊆ C n is givenby the zero set of some f ∈ C { x } . A singularity is isolated if it has a smooth analyticpunctured neighbourhood. Definition 2.2 ([GLS07, Definition 2.9]) . Let f, g ∈ C { x } .(a) We say f and g are right equivalent if there exists a biholomorphic map germ ϕ : ( C n , ) → ( C n , ) such that g = f ◦ ϕ .(b) We say f and g are contact equivalent if there exists a biholomorphic map germ ϕ : ( C n , ) → ( C n , ) and a unit u ∈ C { x , . . . , x n } such that g = u ( f ◦ ϕ ). Remark . Two convergent power series f, g ∈ C { x } arecontact equivalent if and only if the complex analytic space germs ( X, ) and ( Y, ) areisomorphic, where X ⊆ C n is given by the zeros of f and Y ⊆ C n is given by the zerosof g .We use the following proposition in Section 3 to parametrize sextic double solids with a cA singularity: Proposition 2.4 ([GLS07, Remark 2.50.1]) . Let f, g ∈ C { x , . . . , x n } be two contactequivalent power series with zero constant term. Then their multiplicity m is the sameand furthermore, f m and g m are the same up to an invertible linear change of coordinates. We use the following proposition in Section 3 to construct sextic double solids with a cA n singularity where n ≥
2, as well as in Section 4 to describe weighted blowups of cA n points: Proposition 2.5.
Let F = x + . . . + x k + f and G = x + . . . + x k + g , where f and g are convergent power series in C { x k +1 , . . . , x n } with zero constant term. Then F and G are contact (respectively, right) equivalent if and only if f and g are contact (respectively,right) equivalent. roof. By a result of Mather and Yau [MY82] (see also [GLS07, Theorem 2.26]), f and g are contact equivalent if and only the Tjurina algebras T f and T g are isomorphic. Asimple computation shows that T f ∼ = T F and T g ∼ = T G , which proves the proposition forcontact equivalence.The proof for right equivalence is similar. Namely, we use a statement analogous to[MY82]: two elements h, k ∈ C { x } with zero constant term are right equivalent if andonly if the Milnor algebras M h and M k are isomorphic as algebras over the ring C { t } ,where t acts on M h , respectively M k , by multiplying by h , respectively k (see [GLS07,Theorem 2.28]).Reid defined in [Rei80, Definition 2.1] that a compound Du Val singularity is a3-dimensional singularity where a hypersurface section is a Du Val singularity, also calleda surface ADE singularity. The singularity is denoted cA n , cD n or cE n , respectively, if thegeneral hyperplane section is an A n , D n or E n singularity, respectively. Reid showed in[Rei83, Theorem 0.6] that a 3-dimensional hypersurface singularity is terminal if and onlyif it is an isolated compound Du Val singularity.In this paper, we focus on the most general class of compound Du Val singularities,namely cA n singularities. Since a surface A n singularity is given by x + y + z n +1 , wehave the following almost immediate corollary: Corollary 2.6.
Let n be a positive integer. A singularity is of type cA n if and only if it isisomorphic to the complex analytic space germ ( X, ) where X ⊆ C is given by the zeroset of x + y + g ≥ n +1 ( z, t ) with variables x, y, z, t where g ∈ C { z, t } is a convergent powerseries of multiplicity n + 1 . The simplest example of a cA singularity is the ordinary double point , given by x + y + z + t . Note that terminal sextic double solids have only hypersurface singularities,therefore only cA n , cD n and cE n singularities. The first step in a Sarkisov link for a Fano variety is a divisorial contraction.
Definition 2.7.
Let ϕ : Y → X be a proper morphism with connected fibres between vari-eties with terminal singularities. We say ϕ is a divisorial contraction if the exceptionallocus of ϕ is a prime divisor and − K Y is ϕ -ample.Theorem 2.10 says that divisorial contractions to cA n points are weighted blowups.First, we recall the definition of a weighted blowup in both the algebraic and the analyticcategories. Definition 2.8.
We say that two birational morphisms of varieties (or bimeromorphicholomorphisms of complex analytic spaces) ϕ : Y → X and ϕ : Y → X are equivalent if there exist isomorphisms X ∼ = X and Y ∼ = Y such that the diagram Y Y X X ϕ ϕ commutes. We say ϕ and ϕ are locally equivalent if there exist isomorphic open subsets U ⊆ X and U ⊆ X containing the centres of the morphisms ϕ and ϕ such that therestrictions ϕ | ϕ − U : ϕ − U → U and ϕ | ϕ U : ϕ U → U are equivalent.6f we consider the complex analytic space corresponding to a variety or when we wish toemphasize that we are working in the category of complex analytic spaces, we sometimessay analytically equivalent or locally analytically equivalent . Definition 2.9.
Let w = ( w , . . . , w n ) be positive integers, called the weights of theblowup. Define a C ∗ -action on C n +1 by λ · ( u, x , . . . , x n ) = ( λ − u, λ w x , . . . , λ w n x n ) anddefine T by the geometric quotient ( C n +1 \ V ( x , . . . , x n )) / C ∗ (or its analytification). Then,the morphism ϕ : T → C n , [ u, x , . . . , x n ] ( x u w , . . . , x n u w n ) is called the w -blowupof C n at the origin . If Z ⊆ C n is a subvariety (or a complex analytic subspace) containing and ˜ Z is its strict transform, then the restriction ϕ | ˜ Z : ˜ Z → Z is called the w -blowupof Z at . Let ψ : Y → X be a birational morphism of varieties (or bimeromorphicholomorphism of complex analytic spaces). Given an open subset U ⊆ X containing thecentre of ψ and an isomorphism U ∼ = X ⊆ C n taking a point P ∈ X to , ψ is called the w -blowup of X at P if the restriction ψ | ψ − U : ψ − U → U is equivalent, through thegiven isomorphism U ∼ = X , to the w -blowup of X at .Note that a weighted blowup crucially depends on the choice of coordinates, that is, onthe isomorphism U ∼ = X , even though it is not explicit in the notation.Kawakita [Kaw03] described divisorial contractions to cA n points. Notational differencesfrom [Kaw03, Theorem 1.13] are that below we have left out the description for cA singularities and an exceptional case for cA . Also, we have written out the conversestatement more explicitly (that being a Kawakita blowup implies that it is a divisorialcontraction). Theorem 2.10 ([Kaw03, Theorem 1.13]) . Let P be a cA n point where n ≥ of a variety X with terminal singularities. Let ϕ : Y → X be a morphism of varieties such thatthe restriction ϕ | Y \ E : Y \ E → X \ { P } is an isomorphism, where the reduced closedsubvariety E is given by ϕ − { P } . If ϕ is a divisorial contraction, then ϕ is locallyanalytically equivalent to the ( r , r , a, -blowup of V ( x x + g ( x , x )) ⊆ C at withvariables x , x , x , x where1. a divides r + r and is coprime to both r and r ,2. g has weight r + r , and3. the monomial x ( r + r ) /a appears in g with non-zero coefficient.Moreover, any ϕ which is locally analytically equivalent to a weighted blowup as above is adivisorial contraction, even for n = 2 . Any weighted blowup that is locally analytically equivalent to ϕ in Theorem 2.10 for n ≥ r , r , a, -Kawakita blowup , or simply a Kawakita blowup. One of the possible outcomes of the minimal model program is a Mori fibre space:
Definition 2.11. A Mori fibre space is a morphism of normal projective varieties ϕ : X → S with connected fibres such that1. X is Q -factorial and has terminal singularities,2. the anticanonical class − K X is ϕ -ample,3. X/S has relative Picard number 1, and4. dim
S < dim X . 7f dim S >
0, then we say ϕ is a strict Mori fibre space.The main examples of Mori fibre spaces we see in this paper are Fano 3-folds that areprojective, Q -factorial, with terminal singularities and Picard number 1, considered as amorphism over a point.Any birational map between two Mori fibre spaces is a composition of Sarkisov links (see[Cor95] or [HM13]). Below, we describe the two possible types of Sarkisov links startingfrom a Fano variety. Definition 2.12. A Sarkisov link of type I (respectively II) between a Fano variety X and a strict Mori fibre space Y k → Z (respectively Fano variety Z ) is a diagram of theform Y . . . Y k X Z ϕ ψ where X , Y , . . . , Y k , Z are normal, projective and Q -factorial, the varieties X , Y , . . . , Y k have terminal singularities, Z has terminal singularities if it 3-dimensional, X hasPicard number 1, ϕ : Y → X is a divisorial contraction, Y (cid:57)(cid:57)(cid:75) . . . (cid:57)(cid:57)(cid:75) Y k is a sequence ofanti-flips, flops and flips, and ψ : Y k → Z is a strict Mori fibre space (respectively divisorialcontraction). If we do not require the varieties X, Y , . . . Y k (respectively X, Y , . . . Y k , Z )to be terminal and we do not require − K Y to be ϕ -ample and we do not require − K Y k tobe ψ -ample but all the other properties hold, then the diagram above is called a ([BZ10, Definition 2.1]). Definition 2.13.
A Fano 3-fold X that is a Mori fibre space is birationally rigid if forany Mori fibre space Y → S such that X and Y are birational, we have that S is a pointand X and Y are isomorphic.In Section 5, we show that a general sextic double solid X with a cA n singularity with n ≥ X and another Mori fibre space. We find the Sarkisovlink by restricting from a toric 2-ray link, as described in Construction 2.14.See [Cox95] for the definition of Cox rings for toric varieties (where it is called the homogeneous coordinate ring ), and [HK00, Definition 2.6] for the definition of Cox ringsfor Mori dream spaces. Note that isomorphic varieties can have different Cox rings. By[Cox95, Theorem 3.7], closed subschemes of a toric variety T with only cyclic quotientsingularities are given by homogeneous ideals in the Cox ring Cox T , which is a polynomialring. Construction 2.14.
Let X be a Fano variety embedded in a weighted projective space P ,where X is a Mori fibre space, and let Y → X be a divisorial contraction from a projective Q -factorial variety Y . By [AK16, Lemma 2.9], the divisorial contraction Y → X is partof a Sarkisov link only if Y is a Mori dream space.By [HK00, Proposition 2.11], we can embed a Mori dream space Y into a projectivetoric variety T with cyclic quotient singularities such that the Mori chambers of Y areunions of finitely many Mori chambers of T . Moreover, we can embed Y in such a waythat Y is given by a homogeneous ideal I Y in Cox T , and the toric 2-ray link T T · · · T r P T X · · · S T Y Y · · · Y r X X · · · S, where each Y i ⊆ T i is given by the same ideal I Y ⊆ Cox T = . . . = Cox T r , and X i ⊆ T X i is given by the ideal I Y ∩ C [ ν , . . . , ν s ], where T X i is given by Proj C [ ν , . . . , ν s ] for somepolynomials ν j ∈ Cox T that depend on i (see [AZ16, Remark 4]). In this case, Cox( T ) /I Y is a Cox ring for Y and it is said that I Y T ([AZ16, Definition 3.5]).Note that some of the small birational maps T i (cid:57)(cid:57)(cid:75) T i +1 may restrict to isomorphisms Y i → Y i +1 . If all the varieties Y i are terminal and the anticanonical divisor − K Y of Y isinside the interior int(Mov Y ) of the movable cone, then the 2-ray link for Y is a Sarkisovlink (see [AK16, Lemma 2.9]), otherwise it is called a bad link .In Section 5, where X is a sextic double solid and the centre of Y → X is a cA n point,we use a projective version of Corollary 4.9 to construct the divisorial contraction Y → X ,which is the restriction of a toric weighted blowup ¯ T → P . This gives us an embedding Y → V ( I ¯ Y ) ⊆ ¯ T where I ¯ Y might not 2-ray follow ¯ T . We use unprojection to modify¯ T to find an embedding Y → V ( I Y ) ⊆ T such that I Y T . See [Rei00,Section 2.1] for a simple example of unprojection, and Sections 5.2, 5.5, 5.6 and 5.8 forapplications of unprojection.To explain the notation we use for 2-ray links, we do an example in detail, namelythe 2-ray link for the ambient space of the sextic double solid with a cA singularity inSection 5.2. Example 2.15 (2-ray link for P (1 , , , , , . Denote the variables on P (1 , , , , , x, y, z, t, α, ξ . We perform the weighted blowup T → P (1 , , , , ,
5) with weights(1 , , , ,
6) for variables y, z, t, α, ξ , where the centre is the point P x = [1 , , , , , T as a geometric quotient. By a slight abuse of notation we denote thevariables on C by u, x, y, z, α, ξ, t , repeating the symbols for P (1 , , , , , C ∗ ) -action on C for all ( λ, µ ) ∈ ( C ∗ ) by( λ, µ ) · ( u, x, y, z, α, ξ, t ) = ( µ − u, λx, λµy, λµz, λ µ α, λ µ ξ, λµ t ) . Define the irrelevant ideal I = ( u, x ) ∩ ( y, z, α, ξ, t ), and define T by the geometricquotient C \ V ( I ) / ( C ∗ ) . We use the notation u x y z α ξ tT : (cid:18) (cid:19) . − T . Note that we order the variables u, x, . . . , t such thatthe corresponding rays ( − ), ( ), . . . , ( ) are ordered anti-clockwise around the origin.The vertical bar indicates that the irrelevant ideal is ( u, x ) ∩ ( y, z, α, ξ, t ). The Cox ring of T is given by Cox T = C [ u, x, y, z, α, ξ, t ]. The weighted blowup T → P (1 , , , , ,
5) isgiven by [ u, x, y, z, α, ξ, t ] [ x, uy, uz, u t, u α, u ξ ] . (2.1)We describe the cones of the toric variety T . By [HK00], T is a Mori dream space.The Picard group of T is generated by V ( u ), the reduced exceptional divisor, and V ( x ),9 xy, z, αξt (a) Rays in N ( T ) P (1 , , T T X T P (1 , , , (b) Ample models of T Figure 1. Cones of T the strict transform of a plane not passing through P x , which have bidegree ( − ) and ( ),respectively. The variety T is Q -factorial, and any two divisors with the same bidegreeare linearly equivalent. As in [BZ10, Section 4.1.3], the effective cone Eff( T ) is given by h V ( u ) , V ( x ) i , a cone in the group N ( T ) of divisors of T up to numerical equivalence withcoefficients in R . As in [AZ16, Section 3.2], the movable cone Mov(T ) is h V ( x ) , V ( ξ ) i ,and it is divided into the nef cone Nef( T ) = h V ( x ) , V ( y ) i of T and h V ( y ) , V ( ξ ) i , which isthe pull-back of the nef cone of the small Q -factorial modification T of T . The cones h V ( x ) , V ( y ) i and h V ( y ) , V ( ξ ) i are called Mori chambers . The variety T is defined by u x y z α ξ tT : (cid:18) (cid:19) . − T is the geometric quotient ( C \ I ) / ( C ∗ ) , where the irrelevant ideal I is givenby ( u, x, y, z, α ) ∩ ( ξ, t ), which is indicated by the position of the vertical bar in theaction-matrix. The Cox ring of T is equal to the Cox ring of T , namely Cox T = C [ u, x, y, z, α, ξ, t ].The weighted blowup morphism T → P (1 , , , , ,
5) can be read off from the action-matrix of T . Consider the ray given by V ( x ) in N ( T ). The union of the linear systems | ( n ) | where n ≥ C -algebra basis x, uy, uz, u t, u α, u ξ . So, the ample model (see[BCHM10, Definition 3.6.5]) of the divisor class V ( x ) is the morphism T → Proj M n ≥ H ( T , O T ( n ( ))) = Proj C [ x, uy, uz, u t, u α, u ξ ] = P (1 , , , , , u, x, y, z, α, ξ, t ] [ x, uy, uz, u t, u α, u ξ ] , which is precisely the weighted blowup T → P (1 , , , , ,
5) given in Equation (2.1).As in [BZ10, Section 2.1], there are two projective morphisms of relative Picard number1 from T up to isomorphisms, corresponding to the ample models of divisors in the twoedges of the nef cone of T . The ample model of any divisor in the interior of the nef coneof T gives an embedding of T into a weighted projective space. The ample model of V ( y ) ∈ N ( T ) is given by T → Proj C [ y, z, α, uξ, ut, xξ, xt ] ⊆ P (1 , , , , , , u, x, y, z, α, ξ, t ] [ y, z, α, uξ, ut, xξ, xt ] . T X = Proj C [ y, z, α, uξ, ut, xξ, xt ], we see that the morphism T → T X contracts V ( ξ, t ) to the surface P (1 , , ⊆ T X and is an isomorphism elsewhere. The ample modelof V ( y ) ∈ N ( T ) is given similarly by T → Proj C [ y, z, α, uξ, ut, xξ, xt ] = T X , contracting V ( u, x ) to P (1 , , T (cid:57)(cid:57)(cid:75) T , a small Q -factorial modification, given by[ u, x, y, z, α, ξ, t ] [ u, x, y, z, α, ξ, t ] . The diagram T → T X ← T is a flop.Note that multiplying the action-matrix of T or T with a matrix in GL(2 , Q ) isequivalent to choosing a different basis for the group ( C ∗ ) , so the geometric quotients T and T stay the same (see [Ahm17, Lemma 2.4]). If we multiply with a matrix withnegative determinant, then we change the order of the rays in N ( T ) from anti-clockwiseto clockwise.Similarly, there are only two projective morphisms of relative Picard number 1 from T :the contraction T → T X and the ample model of V ( ξ ). We multiply the action-matrix of T by the matrix (cid:16) − − (cid:17) with determinant 4 to find u x y z α ξ tT : (cid:18) − (cid:19) . V ( ξ ) is given by T → P (1 , , , , , u, x, y, z, α, ξ, t ] h t u, t y, t z, t x, t α, ξ i . Note that this is a morphism of varieties despite having fractional powers (see [BB13]).The 2-ray link that we have found for P (1 , , , , ,
5) is summarized by the diagrambelow. T T P (1 , , T X P (1 , , ,
3. Constructing sextic double solids with a cA n singularity In this section, we give a bound n ≤ cA n singularity on a sextic doublesolid, and we explicitly describe all sextic double solids that contain an isolated cA n singularity where n ≤
8. The main tool we use for this is the splitting lemma fromsingularity theory, first introduced in [Tho72], which is used for separating the quadraticterms and the higher order terms of a power series.11 .1. Splitting lemma from singularity theory
Before we go into details, let us recall the statement of the splitting lemma. Here thestatement is taken from [GLS07, Theorem 2.47], with a slight modification in notation.Specifically, we write v ( x + p ) instead of x + g , where v is a unit in the power series ringand p does not depend on x , as we use this form in Section 5 for constructing birationalmodels. Theorem 3.1 (Splitting lemma) . Let m be a positive integer and let y denote variables ( y , . . . , y m ) . Let f ∈ C { x, y } be a convergent power series of multiplicity two, withdegree two part of the form x + (terms in y ) . Then, there exist unique v ∈ C [[ x, y ]] and p, h ∈ C [[ y ]] , where v is a unit and the multiplicity of p is at least two, such that f = ( v ( x + p )) + h. Moreover, the power series h, p and v are absolutely convergent around the origin, and themultiplicity of h is at least two. It follows immediately that f is right equivalent to x + h .Proof. It is proved in [GLS07, Theorem 2.47] that there exist unique g ∈ C [[ x, y ]] and h ∈ C [[ y ]], where the multiplicity of g is at least two, such that f = ( x + g ) + h . Moreover,it is proved that the power series g and h are absolutely convergent around the origin, andthe multiplicity of h is at least two.By the Weierstrass preparation theorem (see [GLS07, Theorem 1.6]), there exists aunique unit v ∈ C { x, y } and a unique p ∈ C { y } such that x + g = v ( x + p ).Below we give explicit recurrent formulas for g, h, p, v of the splitting lemma in terms ofthe coefficients of f , which is implemented in Listing 1 in Appendix A. Proposition 3.2 (Explicit splitting lemma) . Below, we use the same notation as in thesplitting lemma Theorem 3.1 and its proof. Denote f = X i,d ≥ x i f i,d , g = X i,d ≥ x i g i,d , h = X d ≥ h d , p = X d ≥ p d , v = X i,d ≥ x i v i,d where f i,d , g i,d , h d , p d , v i,d ∈ C [ y ] are homogeneous of degree d . Then, g , = 0 ,g i,d = 12 f i +1 ,d − d X k =0 min( i +1 ,i + d − k − X j =max(0 , − k ) g j,k g i +1 − j,d − k , if ( i, d ) = (1 , , (3.1) h d = f ,d − d − X j =2 g ,j g ,d − j , (3.2) p d = g ,d − d − X j =2 v ,d − j p j , (3.3) v , = 1 ,v i,d = g i +1 ,d − d X j =2 ( v i +1 ,d − j p j ) , if ( i, d ) = (0 , . (3.4) Proof.
Taking the degree d part of the coefficient of x i +1 in f = ( x + g ) + h where i ≥
0, we find Equation (3.1). Taking all degree d terms of f = ( x + g ) + h that are notdivisible by x , we find Equation (3.2). Taking the degree d part of the coefficient of x i +1 in x + g = v ( x + p ) where i ≥
0, we find Equation (3.4), and taking all degree d terms notdivisible by x , we find Equation (3.3). 12 xample 3.3. Using the notation of Proposition 3.2, the first few homogeneous parts of h are given in terms of coefficients of f by h = f , h = f , h = f , − f , h = f , − f , f , − f , f , h = f , − f , f , f , f , − f , f , f , f , f , − f , f , − f , . cA n singularity Now, we apply the explicit splitting lemma (Proposition 3.2) to the case we are mostinterested in, that is, sextic double solids. First, we describe the equation of a sextic doublesolid X ⊆ P (1 , , , ,
3) that has a singular point P . The following argument shows thatwithout loss of generality, we can move the singular point to P x = [1 , , , ,
0] using anautomorphism of P (1 , , , , P = [ P , P , P , P , P ]. Since X does not containthe point [0 , , , , ≤ i ≤ P i = 0. After interchanging thevariables if necessary, we find P = 0. Now, the automorphism of P (1 , , , ,
3) given by( x, y, z, t, w ) x, y − P P x, z − P P x, t − P P x, w − P P x ! takes P to P x .Below, the subindices denote degree and V ( f ) denotes the zero locus of a polynomial f . Notation 3.4.
Define the variety X by X : V ( f ) ⊆ P (1 , , , , , with variables x, y, z, t, w where f = − w + x t + x ξ + x (4 t a + 4 t a + 2 ta + a )+ x (2 t b + 2 t b + 2 t b + 2 tb + b )+ x (2 t c + 2 t c + 2 t c + 2 t c + 2 tc + c )+ t d + 2 t d + t d + 2 t d + t d + 2 td + d , where the polynomials ξ j , a j , b j , c j , d j ∈ C [ y, z ] are homogeneous of degree j .Now, define the following technical conditions:2. ξ = 0.3. a = 0.4. b = a .5. c = 2 a b − a a .6. d = 2 a c + b − a a b − a b + 4 a a + 16 a a .13. There exist unique q, r, s, e ∈ C [ y, z ] where r and s do not have a common primedivisor, and q and e do not have a common prime divisor, such that a = qrb = qs + 4 a qrc = 2 a qs − a q r + 8 a qr + erd = 2 b qs − a qs − es − b q r + c qr, and q, r, s, e are respectively homogeneous of degrees(7.1) 0 , , , q = 1,(7.2) 1 , , , q under the ordering y < z is one,(7.3) 2 , , , r = 1, or(7.4) 3 , ∗ , , r = 0 and s = 1 (since r = 0, ‘ ∗ ’ denotes that r is homogeneous ofany non-negative degree).8. Condition (7.1) holds and there exists a unique A ∈ C and a unique polynomial B ∈ C [ y, z ] homogeneous of degree 1 such that e = 4 A r + b − a c = 6 a s − A s + 4 a a r − A a r + B r + 2 a b − a d = − s B + 16 r A − b r A + 16 a r A + 4 b s − a a s − b r + 2 c r + b − a b + 4 a . Notation 3.4 describes 11 families of sextic double solids, namely when conditions 1 to n are satisfied for some n ≤
8. For n = 7, there are 4 families. Below, general means ‘ina Zariski open dense subset of the family’, and very general means ‘outside a countableunion of proper Zariski closed subsets of the family’. Using the above notation, we statethe main theorem of this section, describing sextic double solids with an isolated cA n singularity. Theorem A. (a) If a sextic double solid has an isolated cA n singularity, then n ≤ .Furthermore, for every positive integer n ≤ :(b) Every sextic double solid with an isolated cA n singularity P is isomorphic to some X satisfying conditions to n in Notation 3.4, with the isomorphism sending P P x = [1 , , , , .(c) Every X that satisfies conditions to n in Notation 3.4 and has otherwise generalcoefficients is smooth outside a cA n singularity at P x .(d) A very general sextic double solid with an isolated cA n singularity is factorial andhas Picard number , except for the family (7.4) in Notation 3.4. No member of thefamily (7.4) is Q -factorial.Remark . Note that if n = 1, then the set of conditions 2 to n is empty, so everyvariety with a cA singularity is isomorphic to some X in Notation 3.4, and every X in Notation 3.4 that has general coefficients has a cA singularity at P x and is smoothelsewhere. Note that zero is homogeneous of every non-negative degree, so for example incondition (7.1) of Notation 3.4, the term e can be zero. Also, note that in conditions (7.1)and (7.2), the terms r and s must both be non-zero, otherwise either r or s is a commondivisor of both r and s . 14efore we prove Theorem A, we state a few lemmas needed for the proof. Lemma 3.6. If X in Notation 3.4 satisfies conditions to and P x is an isolatedsingularity of X , then either a = 0 or b = 0 .Proof. If conditions 2 to 6 hold and a = b = 0, then a = b = c = d = 0. Let C bethe curve defined by the ideal ( t, w, xc + 2 d ). Taking partial derivatives, we see thatevery point of C is a singular point of X . Since C passes through P x , X does not have anisolated singularity at P x , a contradiction. Lemma 3.7.
Let r, s ∈ C [ y, z ] have no common prime divisors, and let q ∈ C [ y, z ] benon-zero. Let h n ∈ C [ y, z ] be of the form h n = q α ( r β C r − s γ C s ) where C r , C s ∈ C [ y, z ] and α, β, γ are non-negative integers. Then h n = 0 ⇐⇒ there exists C ∈ C [ y, z ] such that C r = s γ C and C s = r β C. Lemma 3.8. If X in Notation 3.4 satisfies conditions to and P x is an isolatedsingularity of X , then q and e do not have a common prime divisor in C [ y, z ] .Proof. If q and e have a common prime divisor D , then D divides a , b , c , d , and D divides a , b , c , d . Let C be the curve defined by the ideal ( D, t, w ). Taking partialderivatives, we see that X is singular at every point of C , so P x is not isolated, acontradiction. Lemma 3.9.
Denote the parameter space of all possible f in Notation 3.4 satisfyingconditions to n by P n . Denote the parameter space of all f ∈ P n where V ( f ) has asingular point with x , t and w -coordinate zero by A n . Then dim A n ≤ dim P n − .Proof. Let P = [0 , β, γ, ,
0] be a singular point of V ( f ) where f ∈ A n and β, γ ∈ C . Wefind f ( P ) = d ( P ) , ∂f∂x ( P ) = c ( P ) , ∂f∂y ( P ) = ∂d ∂y ( P ) , ∂f∂z ( P ) = ∂d ∂z ( P ) , ∂f∂t ( P ) = 2 d ( P ) . Define the polynomial l = γy − βz . Since P is a singular point, we have l divides c , d and d , and l divides d . (3.5)We use the divisibility constraint (3.5) repeatedly below.1. If n ≤
5, then there are no restrictions on d or d in P n . For A n , we have therestrictions that l | d and l | d . In particular, d has a square factor which is alsoa factor of d . We find that dim A n ≤ dim P n − n = 6, then there are no restrictions on d , a , b or c in P n . We have c = a (2 b − a a ) and d = a · ( . . . ) + b . Below we consider f ∈ A n .If l divides a , then using the divisibility constraint (3.5), we find that l divides b .So, there are at least two less degrees of freedom in choosing a , b and d .If l does not divide a , then l divides 2 b − a a , and from l | d we find that l alsodivides 8 c − a b + 8 a a + a a . So, after fixing a , a , a and b , there are at leasttwo less degrees of freedom in choosing b , c and d .In either case, we see that dim A n ≤ dim P n − n = 7, then c = 4 q r (2 s + a r ) d = − es + q (2 b s − a s − b qr + c r ) d = 4 q ( er + q ( s + a rs − a qr − b r + a r )) . Let us consider f ∈ A n . If l | q , then since q and e are coprime, we have l | r and l | s , a contradiction. If l | r , then since l | d , we find l | s , a contradiction.Therefore, l divides neither q nor r .So, l divides 2 s + a r . Using l | d , we see that l divides − a q r − b q + 3 a q + 4 e .After fixing a , a , b , q and r , we see that there are at least two less degrees of freedomin choosing s and e . So, we have dim A n ≤ dim P n − n = 8, then c = 2 r ( s + 2 a r ) d = r ( r B − s A − a r A + 6 a s − b r + 4 a a r + 2 a b − a )+ s ( b − a ) d = r (8 r A + 4 a s − a r + 4 a r ) + s We consider f ∈ A n . If l | r , then l | s , a contradiction. So, l divides s + 2 a r .Since l divides d , we have l | r ( A − a ). So, A = a . Since l divides d , wesee that l | r ( B − b ). We find that the coefficients of f have at least three lessdegrees of freedom, namely A = a , and the polynomials B , b and s + 2 a r havea common prime divisor. So, we have dim A n ≤ dim P n − Lemma 3.10.
Denote the parameter space of all possible f in Notation 3.4 satisfyingconditions to n by P n . Denote the parameter space of all f ∈ P n such that V ( f ) has asingular point at P t = [0 , , , , by B n . Then, dim B n = dim P n − .Proof. We find f ( P t ) = d , ∂f∂x ( P t ) = 2 c , ∂f∂y ( P t ) = 2 ∂d ∂y , ∂f∂z ( P t ) = 2 ∂d ∂z , ∂f∂t ( P t ) = 6 d . For B n , we have d = c = d = 0. So, there are 4 less degrees of freedom in choosingcoefficients for f ∈ B n , therefore dim B n = dim P n − Q -factoriality and that the Picard number is 1, we use the following propositionby Namikawa: Proposition 3.11 ([Nam97, Proposition 2]) . Let X be a Fano 3-fold with Gorensteinterminal singularities and D its general anti-canonical divisor. Then, the natural homo-morphism Pic( X ) → Pic( D ) is an injection. Corollary 3.12.
In the notation of Proposition 3.11, let X be smooth along D . Then Cl( X ) → Pic( D ) is an injection.Proof. Let U be the smooth locus of X . Since X \ U is of codimension at least 2 in X ,we have Cl( X ) ∼ = Cl( U ). It follows from the proof of Proposition 3.11 that Pic( U ) injectsinto Pic( D ). 16 roof of Theorem A . Note that every X above with a cA n singularity at P x is irre-ducible.(b) We prove that every sextic double solid Y ⊆ P (1 , , , ,
3) with a cA n singularityis isomorphic to some X above, with the isomorphism sending the cA n point to P x =[1 , , , , P (1 , , , , cA n point isat P x . From Corollary 2.6, we know that a cA n singularity is isomorphic to a complexanalytic space germ ( V ( α + β + H ) , ) with variables α, β, γ, δ where H ∈ C { γ, δ } hasmultiplicity n + 1. Consider the affine patch Y x , given by inverting x . Using Proposition 2.4,we find that after a suitable invertible linear coordinate change, Y x is given by V ( − w + t + g ≥ ( y, z, t )) in C with variables y, z, t, w , where g ∈ C [ y, z, t ] has multiplicity at leasttwo and the degree 2 part g is contained in C [ y, z ]. So, Y has the required form V ( f ),proving the case n = 1.Applying the splitting lemma on the affine patch where x is non-zero, we find that X islocally analytically of the form V ( − w + t + h ( y, z )) in C with variables y, z, t, w for some h ∈ C { y, z } of multiplicity at least two. Any cA n singularity is isomorphic to a complexanalytic space germ ( V ( α + β + H ) , ) with variables α, β, γ, δ where H ∈ C { γ, δ } hasmultiplicity n + 1. By Proposition 2.5, X has a cA n singularity at P x if and only if h = h = . . . = h n = 0 and h n +1 = 0 as polynomials in C [ y, z ]. We have h = ξ , so h = 0 is equivalent to condition 2, namely ξ = 0. We can show that h = a , so h = 0is equivalent to condition 3, namely a = 0. Similarly, using the explicit splitting lemma(Proposition 3.2), it is straightforward to compute that h = . . . = h n = 0 is equivalent tosatisfying conditions 2 to n when n ≤
6, even if P x is not isolated. This proves part (b)for n ≤ P x of X isisolated, we show that h = . . . = h n = 0 is equivalent to satisfying conditions 2 to n forany n ≤ a = 0 or b = 0. We write a = qr and b = qs + 4 a qr , where q ∈ C [ y, z ] is a (homogeneous) greatest common divisor of a and b , and r and s ∈ C [ y, z ]have no common prime divisor. In the rest of the proof of parts (a) and (b), we repeatedlyuse Lemma 3.7.If conditions 2 to 6 hold, then using the explicit splitting lemma (Proposition 3.2), wecompute in Listing 3 of Appendix A that h = q ( r ( − a q rs + 4 b qs − b q r + 2 c qr − d ) − s (2 c − a qs )) . Using Lemma 3.7, we find that h = 0 is equivalent to the existence of a homogeneous e ∈ C [ y, z ] such that c = 2 a qs − a q r + 8 a qr + erd = 2 b qs − a qs − es − b q r + c qr. We defined q as a homogeneous greatest common divisor of a and b . Every non-zerocomplex multiple of q is another greatest common divisor. Therefore, there is redundancyin choosing q, r, s, e . We eliminate this redundancy by choosing q = 1 in condition (7.1),leading coefficient of q equal to one in condition (7.2), r = 1 in condition (7.3), and s = 1in condition (7.4). By Lemma 3.8, q and e have no common prime divisor in C [ y, z ].This proves part (b) for n = 7, namely that h = . . . = h = 0 is equivalent to conditions2 to 7 if P x is an isolated singularity of X .Now, we show that if h = . . . = h = 0 and one of the conditions (7.2) to (7.4) holds,then the singularity P x is not isolated. In condition (7.2), we calculate that h + e r is17ivisible by q , giving r = Cq for some C ∈ C . Substituting into h , we compute that h − qes is divisible by q . Therefore q and s have a common prime divisor, giving that r and s have a common prime divisor, a contradiction. So, P x is not an isolated singularityof X . Conditions (7.3) and (7.4) are similar.Hence, if h = . . . = h = 0 and P x is an isolated singularity, then condition (7.1) holds.Using the explicit splitting lemma, we calculate h in Listing 3 in Appendix A, and usingLemma 3.7, we can show that h = . . . = h = 0 is equivalent to conditions 2 to 8.(a) If conditions 2 to 8 are satisfied, then similarly to proof of part (b), P x being a cA n singularity where n ≥ h = 0. Using the explicit splitting lemma, wecompute h in Listing 3 in Appendix A, and using Lemma 3.7, we find that this impliesthat there exists B ∈ C such that A = a B = b d = − s B + 2 b s − a s + c r − a b r + 16 a a r + b b − a a b − a b + 8 a a c = r B − a r + 2 a b + 2 a b − a a . Substituting into f gives x a + x b + xc + d = ( s + 2 a r + xr ) x a + x b + xc + d = ( s + 2 a r + xr )( − a r + b − a + 2 xa + x ) . Define the curve C by the ideal ( w, t, s + 2 a r + xr ). Taking partial derivatives, we findthat X is singular at every point of C . Therefore, P x is not an isolated singularity of X .(c) We consider varieties X satisfying conditions 1 to n . We show that a general X hasno other singular points apart from P x = [1 , , , , f in Notation 3.4 satisfying conditions 2 to n by P n .If P = P x is a singular point of V ( f ) with t -coordinate zero, then one of y or z -coordinateis non-zero. A suitable change of coordinates of the form x x + αy or x x + αz ,where α ∈ C , takes the point P to P with x , t and w -coordinate zero. Note that thiscoordinate change fixes the point P x , keeps the form of f given in Notation 3.4, and f willcontinue to satisfy conditions 2 to n after this coordinate change. Using Lemma 3.9, wefind that the parameter space of all f such that V ( f ) has a singular point P = P x with t -coordinate zero is at most (dim P n − P is a singular point of V ( f ) with t -coordinate non-zero and n ≥
2, then a suitablechange of coordinates given by x x + α x t , y y + α y t and z z + α z t , where α x , α y , α z ∈ C , takes the point P to P t = [0 , , , , P x , keeps the form of f given in Notation 3.4, and f will continue tosatisfy conditions 2 to n after this coordinate change. Using Lemma 3.10, we find that theparameter space of all f such that V ( f ) has a singular point P with t -coordinate non-zerois at most (dim P n − n ≥
2. If n = 1, then instead weperform a suitable coordinate change given by x x + α x t , y y + α y t , z z + α z t and t t or t t , composed with a coordinate change of the form t β y y + β z z + β t t where β y , β z and β t ∈ C depend only on α x , α y , α z and the coefficients of x ξ , such that thiscomposition takes the point P to P t , fixes the point P x and keeps the form of f given inNotation 3.4. This extra coordinate change t β y y + β z z + β t t is needed to keep the formof f in Notation 3.4, namely to diagonalize the quadratic part with respect to t , that is, toremove the quadratic monomials yt and zt , and set the coefficient of t to one. Similarly,18sing Lemma 3.10, we find that the parameter space of all f such that V ( f ) has a singularpoint P with t -coordinate non-zero is at most (dim P n − n = 1.This shows that a general X satisfying 2 to n is smooth outside a cA n singularity at P x .(d) Since X has local complete intersection singularities, it is Gorenstein ([Eis95,Corollary 21.19]). A terminal Gorenstein Fano 3-fold is Q -factorial if and only if it isfactorial [Kaw88, Lemma 6.3]. To see that in family (7.4) we do not have any Q -factorialmembers, suffices to note that V ( t, q − w ) and V ( t, q + w ) are not Cartier on the sexticdouble solid.In all other families apart from (7.4), a very general sextic double solid X satisfies thatthe hyperplane section V ( x ) is a very general sextic double plane. More precisely, fix apositive integer n ≤ cA n singularity described in Remark 3.13, other than cA family 4.Consider the 28-dimensional parameter space of sextic double planes V ( − w + g ) ⊆ P (1 , , , y, z, t, w where g ∈ C [ y, z, t ] is homogeneous of degree 6, where the sexticdouble plane corresponds to a point in C given by the coefficients of y , y z , . . . , t . For cA , cA , cA and cA families 1–3, the image of the parameter space of sextic doublesolids, under taking the hyperplane section V ( x ), contains a Zariski open dense set ofthe parameter space of sextic double planes. We show this by computing the rank of theJacobian matrix corresponding to this projection morphism, and showing it is 28 for someparticular point. For cA family 4, respectively cA , it gives set which is open dense in asubvariety of codimension 3, respectively 1. For cA , using additionally the coordinatetransformation t αy + βz + t where α, β ∈ C on the image of the parameter spaceof sextic double solids, we get a Zariski open dense set of the parameter space of sexticdouble planes.Since a very general sextic double plane has Picard number 1, by Corollary 3.12 a verygeneral sextic double solid X has Class number 1, except for cA family 4. Therefore, X is factorial and has Picard number 1. Remark . Consider the tuples η of coefficients of ξ , a i , b i , c i , d i in Notation 3.4. Theseform the parameter space C . As in the proof of Theorem A, we can locally analyticallywrite X with a cA n singularity by V ( wt + h ) ⊆ C where h ∈ C { y, z } has multiplicity n + 1. Requiring that all the coefficients of h n +1 are zero gives us n + 2 conditions. If theseconditions are algebraically independent and if h n +2 is not zero after imposing these, thenwe would expect the parameter space for cA n +1 to have dimension n + 2 less. Countingparameters in Notation 3.4, we see that this is precisely the case. Table 2 shows thedimensions of the parameter spaces.The parameter space for n +1 lies in the Zariski closure of the parameter space for n . Notethat the parameter space for cA has four connected components, each of dimension 44.The automorphisms of P (1 , , , ,
3) that keep the form of a general f when n ≥ xyztw α α α α α α α α α α α − ± xyztw . These automorphisms form a 10-dimensional algebraic group. When n = 1, instead ofpolynomials f , we can consider polynomials F of the form − w + x E + x A + x B +19 C + D where E i , A i , B i , C i , D i ∈ C [ y, z, t ] are homogeneous of degree i . The parameterspace of polynomials F is 80-dimensional, and the automorphisms of P (1 , , , ,
3) thatkeep the form of a general F form a 13-dimensional algebraic group. If the course modulispace of sextic double solids with an isolated cA n singularity exists, then we expect it tohave dimension 10 less than the parameter space in Notation 3.4.Table 2. Dimension of the space of sextic double solids with an isolated cA n n Remark . In some cases it suffices if X is general in Theorem A part (d), as opposedto very general. For example, if n = 1, then a general X has only one singularity whichis an ordinary double point, and every such X is factorial and has Picard number 1,that is, X is a Mori fibre space. If n = 2, then a general X has the singularity given by x x + x + x , and is a Mori fibre space by [CM04, Remark 1.2]. If n = 4, a general X is a Mori fibre space, since in Section 5.2 we construct a Sarkisov link to a completeintersection Z , ⊆ P (1 , , , , ,
4) which is a Mori fibre space if it is general. cA n singularities Although the primary interest is in isolated cA n singularities since these are terminal, it isalso possible to study non-isolated singularities with the same methods. Remark . It follows from the proof of Theorem A that every f that satisfies conditions2 to n defines a sextic double solid with a singularity at P x which is either cA m (possiblynon-isolated) where m ≥ n , or it is the germ ( Z, ) where Z = V ( x + x ) ⊆ C withvariables x , x , x , x .We describe a family of examples of sextic double solids with a non-isolated cA n singularity for all 9 ≤ n ≤ Proposition 3.16.
Let ≤ n ≤ . If X in Notation 3.4 satisfies conditions to ,and in addition satisfies conditions to n and does not satisfy condition n + 1 from thefollowing:9. there exists B ∈ C such that A = a B = b d = − s B + 2 b s − a s + c r − a b r + 16 a a r + b b − a a b − a b + 8 a a c = r B − a r + 2 a b + 2 a b − a a . B = b d = 2 c r − a b r + 16 a r + 2 b b − a b + b − a a b − a b + 24 a a c = 2 a b + 2 a b − a a , c = 2 a b − a d = b b − a b − a a b + 8 a a , d = b − a b + 4 a , then P x is a non-isolated cA n singularity of a non-terminal sextic double solid X .Proof. Repeatedly applying the divisibility condition (3.5) similarly to the proof of part (b)of Theorem A.
Remark . If 9 ≤ n ≤
11, then the variety X in Proposition 3.16 is singular along thecurve C : V ( t, w, s +2 a r + xr ) passing through P x (see the proof of part (a) of Theorem A).We can compute that at a general point of C , the singularity is locally analytically C × ODP,that is, it is isomorphic to the germ ( Z, ) where Z is V ( x + x + x ) ⊆ C with variables x , x , x , x . Remark . Translating the point P t = [0 , , , ,
0] to [1 , , , , cA n singularity at P t ∈ X , which can be usedto construct general sextic double solids with two cA n singularities. It is also easy toconstruct simple examples with only two cA singularities, such as the variety below with cA singularities at P x and at P t , V ( − w + x t + x t + y + z ) ⊆ P (1 , , , , .
4. Weighted blowups
In this section, we discuss weighted blowups from both algebraic and local analytic pointsof view. In Proposition 4.5 we show that to check whether a weighted blowup is a Kawakitablowup (see Theorem 2.10), it suffices to compute the weight of the defining power series.Using this, in the technical Lemma 4.8 we show how to algebraically construct Kawakitablowups of cA n points on affine hypersurfaces. Let n and m be positive integers. Let x = ( x , . . . , x n ) and y = ( y , . . . , y m ) denote thecoordinates on C n and C m , respectively. Choose positive integer weights for x and y . Definition 4.1.
Let X ⊆ C n , X ⊆ C m be complex analytic spaces containing the origins.We say a biholomorphic map ψ : X → X is weight-respecting if denoting its inverse by θ ,we can locally analytically around the origins write ψ = ( ψ , . . . , ψ m ) and θ = ( θ , . . . , θ n )where for all i and j , the power series ψ j ∈ C { x } and θ i ∈ C { y } satisfy wt( ψ j ) ≥ wt( y j )and wt( θ i ) ≥ wt( x i ).It is known that a biholomorphic map taking the origin to the origin lifts to a uniquebiholomorphic map of the blown-up spaces under the usual weights (1 , . . . ,
1) (see forexample [GLS07, Remark 3.17.1(4)]). It is easy to come up with examples where abiholomorphic map does not lift under weighted blowups. We give one example below.
Example 4.2.
Let X ⊆ C be the complex analytic space given by V ( f ) where f = x x + x + ax x + bx for some a, b ∈ C ∗ . Define X ⊆ C by V ( f ) where f = f ( x , x , − x + x ). Chooseweights (1 , ,
2) for ( x , x , x ). Then, X and X are biholomorphic and wt f = wt f , butthe weighted blowups of X and X are not locally analytically equivalent.21 roof. Let ψ : X → X be any local biholomorphism taking the origin to the origin.Composing with a suitable weight-respecting biholomorphic map and using Lemma 4.3,it suffices to consider the case where ψ is a linear biholomorphism. Since the ellipticcurve defined by f in P with variables x , x , x has only two automorphisms, thereare only four possibilities for a linear biholomorphism X → X , namely ( x , x , x ) ( x , ± x , ± x + x ).Let Y → X and Y → X be the (1 , , X and X respectively. Then Y isgiven by V ( g ) where g ( u, x , x , x ) = ux x + x + au x x + bu x . Denoting the points of Y and Y by [ u, x , x , x ], the lifted map ψ Y : Y → Y is givenby [ u, x , x , x ] [ u, x , ± x , ± x /u + x ], which is not holomorphic on the exceptionallocus V ( u ).On the other hand, a weight-respecting coordinate change does lift to weighted blowups: Lemma 4.3.
The weighted blowups of X ⊆ C n and X ⊆ C m at the origin are analyticallyequivalent if there exists a weight-respecting biholomorphic map X → X taking to .Proof. Let ϕ : Y → X and ϕ : Y → X be the weighted blowups at the origin and let ψ : X → X be a weight-respecting biholomorphic map. We define the holomorphism ψ Y = ( ψ Y, , ψ Y, , . . . , ψ Y,m ) : Y → Y by choosing ψ Y, = u and for all j ≥ ψ Y,j =( ψ j ◦ ϕ ) /u wt( y j ) . Similarly, we define θ : Y → Y by θ = u and θ i = ( ψ − i ◦ ϕ ) /u wt( x i ) .Since ψ and ψ − are weight-respecting, the maps ψ Y and θ are indeed holomorphic.The map θ ◦ ψ Y coincides with the identity map on a dense open subset of Y : namely,for all [1 , x ] ∈ Y , we have ( θ ◦ ψ Y )[1 , x ] = θ [1 , ψ ( x )] = [1 , x ]. Since coincidence sets areclosed, the map θ ◦ ψ Y is the identity. Similarly, ψ Y ◦ θ is the identity, giving θ = ψ − Y .Also, we have ( ϕ ◦ ψ Y )[1 , x ] = ψ ( x ) = ( ψ ◦ ϕ )[1 , x ], showing that the diagram Y Y X X ψ Y ϕ ϕ ψ commutes. Therefore, ϕ and ϕ are analytically equivalent. In the following, we focus on Kawakita blowups (see Theorem 2.10). Unlike Example 4.2,for cA n singularities, having the correct weight for the defining power series is enough forthe local analytic equivalence of weighted blowups. Notation 4.4.
We choose positive integer weights w = ( r , r , a,
1) for variables x =( x , x , x , x ) on C and define n = ( r + r ) /a − a divides r + r and is coprime to both r and r ,• r ≥ r , and• n ≥ Proposition 4.5.
Using Notation 4.4, let f ∈ C { x } be such that V ( f ) has an isolated cA n singularity at the origin and f has weight r + r . Then, the w -blowup of V ( f ) ⊆ C is a w -Kawakita blowup. roof. First, we remind that the terms homogeneous , degree and multiplicity are withrespect to the standard weights (1 , . . . , quadratic part of f denote the homoge-neous part of f of degree 2. After a suitable invertible linear weight-respecting coordinatechange, the quadratic part of f is x x .We find that f = x x + x G + H , where G ∈ C { x , . . . , x } has weight at least r and multiplicity m ≥
2, and H ∈ C { x , x , x } . The coordinate change x x − G m ,where G m is the homogeneous degree m part of G , takes f to x x + x G + H , where G has multiplicity at least m + 1. By induction, this defines the unique formal powerseries K ∈ C [[ x , . . . , x ]] of multiplicity at least 2 and weight at least r such that thetransformation x x + K takes f to the form x x + H where H ∈ C [[ x , x , x ]].Similarly, we transform f into x x + h where h ∈ C [[ x , x ]], using x x + L where L ∈ C [[ x , x , x ]]. At the end of the proof, we show how to find a convergent weight-respecting coordinate change.Since the singularity is cA n where n = ( r + r ) /a − h must contain a monomialof degree ( r + r ) /a . Since x x + h has weight r + r , if a >
1, then the coefficientof x ( r + r ) /a in h is non-zero. If a = 1, then after a suitable invertible linear coordinatechange on C { x , x } , the coefficient of x ( r + r ) /a in h is non-zero.We found that we can transform f into the form x x + h where the coefficient of x ( r + r ) /a in h is non-zero, by using only weight-respecting coordinate changes. By Lemma 4.3, theweighted blowup of f is locally analytically equivalent to the weighted blowup of x x + h ,which is precisely a Kawakita blowup.Lastly, we discuss convergence. Instead of the coordinate changes x x + K , x x + L , which might not be convergent, we do a coordinate change with truncatedpower series K ≤ N and L ≤ N of homogeneous parts of K and L of degree at most N . Thecoordinate change Ψ : x x + ix , x x − ix takes x x into x + x . Now we usethe splitting lemma, which gives a convergent coordinate change Φ which respects theweighting when N is large enough, to give f the form x + x + h ( x , x ) where h converges.Applying Ψ − , we get x x + h . Note that the coordinate changes Ψ and Ψ − might notrespect the weighting w , but the total coordinate change Ψ − ◦ Φ ◦ Ψ is weight-respectingif N is large enough.Given a variety X with an isolated cA n point P , we show that any two w -Kawakitablowups Y → X and Y → X of the point P are locally analytically equivalent. Notethat they need not be globally algebraically equivalent. For example, [CM04, Remark 2.4]describes two different (2 , , , cA singularity on a quartic 3-fold. Proposition 4.6.
Any two w -Kawakita blowups of locally biholomorphic singularities arelocally analytically equivalent.Proof. Let f = x x + g ( x , x ) and f = x x + g ( x , x ) be contact equivalent, where g, g ∈ C { x , x } have weight r + r and x ( r + r ) /a appears in both g and in g withnon-zero coefficient. Suffices to show that the w -blowups of V ( f ) ⊆ C and V ( f ) ⊆ C are locally analytically equivalent.Since f and f are contact equivalent, there exists a unit u ∈ C { x } and a localbiholomorphism ψ : ( C , ) → ( C , ) such that f = u ( f ◦ ψ ). Note that f and f ◦ ψ havethe same weight r + r , and x ( r + r ) /a appears in f ◦ ψ with non-zero coefficient. Sincethe germs ( V ( f ) , ) and ( V ( f ◦ ψ ) , ) are equal, it suffices to show that the w -blowups of V ( f ) and V ( f ◦ ψ ) are locally analytically equivalent.23sing arguments similar to the proof of Proposition 4.5, we can find a weight-respectingbiholomorphic map germ θ : ( C , ) → ( C , ) such that f ◦ ψ ◦ θ is of the form x x + g where g ∈ C { x , x } contains x ( r + r ) /a and has weight r + r . It suffices to show thatthe w -blowups of V ( f ◦ ψ ◦ θ ) and V ( f ) are locally analytically equivalent.By Proposition 2.5, g and g are right equivalent, meaning there exists an automorphismΦ of C { x , x } such that Φ( g ) = g . Since x ( r + r ) /a has non-zero coefficient in both g and g , and both g and g have weight r + r , the image of x has weight a underboth Φ and Φ − . Define the biholomorphic map germ ϕ : ( V ( f ◦ ψ ◦ θ ) , ) → ( V ( f ) , )by x ( x , x , Φ( x ) , Φ( x )). By Lemma 4.3, the w -blowups of V ( f ◦ ψ ◦ θ ) ⊆ C and V ( f ) ⊆ C are locally analytically equivalent. In this section, we see how to construct weighted blowups for affine hypersurfaces with a cA n singularity where n ≥ cA n singularities do not admit ( r , r , a, a ≥
2. Belowwe define the type of an isolated cA n singularity, which for n ≥ a such that it admits some ( r , r , a, cA n singularity have a type 1 cA n singularity. Definition 4.7.
Let (
X, P ) be the complex analytic space germ of an isolated cA n singularity. Let a be the largest integer such that ( X, P ) is isomorphic to some germ( V ( x x + g ) , ) where g ∈ C { x , x } has weight a ( n + 1) under the weighting ( a,
1) for( x , x ). Then, we say that the cA n singularity is of type a .It is not obvious how to globally algebraically construct a Kawakita blowup for varietywith a cA n singularity. We show this for affine hypersurfaces in the technical Lemma 4.8.We use a projectivization of Corollary 4.9 in Section 5 for constructing Kawakita blowupsof sextic double solids.We describe the notation for Lemma 4.8. Choose n ≥ w = wt( α, β, x , x ) = ( r , r , a, F ∈ C [ x , x , x , x ] have multiplicity at least 3, and let f = − x + x + F be such that V ( f ) ⊆ C has terminal singularities and has a cA n singularity of typeat least a at the origin. Let q, w be the power series when splitting with respect to x (Theorem 3.1), and p, v be the power series when splitting with respect to x , that is, f = − (( x + q ) w ) + (( x + p ) v ) + h (4.1)where q ∈ C { x , x , x } and p ∈ C { x , x } both have multiplicity at least 2, and w ∈ C { x , x , x , x } and v ∈ C { x , x , x } are units, and h ∈ C { x , x } has multiplicity atleast 3. Choose weights w = wt( α, β, x , x , x , x ) = ( r , r , m, min( r , mult p ) , a, C , where m = min( r , mult q ). If a >
1, then perform a coordinatechange on x , x for f such that h has weight r + r . Writing a power series s ∈ { x , x , x , x } as a sum of its w -weighted homogeneous parts s = P ∞ i =0 s i , let s Using the notation above, the w -blowup of V ( I ) is a w -Kawakita blowup.Proof. The morphism ϕ : C → C ( x , x , x , x ) (( x + q Using the notation above, if F ∈ C [ x , x , x ] , or equivalently, if q = 0 and w = 1 , then define the ideal J ⊆ C [ α, β, x , x , x ] by J = ( − ( α − ( x + p 5. Birational models of sextic double solids In this section, we prove Theorem B on birational non-rigidity of certain sextic doublesolids. First, we give generality conditions we use. Condition 5.1. Let the sextic double solid X be given as in Notation 3.4, and let P (1 , , y, z, w and P have variables y, z . Then we have the following conditions,depending on the family that X lies in:( cA ) V (2 wa + c , w − d ) ⊆ P (1 , , 3) is 10 distinct points,( cA ) V ( a , − w + d ) ⊆ P (1 , , 3) is 4 distinct points,( cA ) c − a b − a b + 2 a a + 6 a a ∈ C [ y, z ] is non-zero, and V ( a ) ⊆ P is twodistinct points, and for both of these points P , either b ( P ), c ( P ) or d ( P ) isnon-zero,( cA , 1) V ( − e + 4 a r + b − a ) ⊆ P is two distinct points,( cA , 2) r and q are coprime in C [ y, z ],( cA , 3) q ∈ C [ y, z ] is not a square,( cA ) a = A . Theorem B. A sextic double solid, which is a Mori fibre space containing an isolated cA n singularity with n ≥ and satisfying Condition 5.1, has a Sarkisov link starting with aweighted blowup of the cA n point. We treat each of the 7 families separately. We use the notation in Construction 2.14and Example 2.15 for the 2-ray links. We write the cA case in more detail. Below, whenwe say that a birational map is k Atiyah flops, then we mean that the base of the flopis k points, above each we are contracting a curve and extracting a curve, and locallyanalytically above each of the points it is an Atiyah flop (see [Rei92, Section 1.3] for Atiyahflop). Similarly for flips. Below, for a morphism Φ : T → P , Φ ∗ : Cox P → Cox T denotesa corresponding C -algebra homomorphism of Cox rings (described explicitly in the proofof Proposition 5.4). Before proving Theorem B, we show that for any Kawakita blowup Y → X (Theorem 2.10)of a sextic double solid X with an isolated cA n singularity, the variety Y has only up totwo singular points if X is general, which are quotient singularities. We do not give thegenerality conditions of Proposition 5.3 explicitly. We do not use this proposition in theproof of Theorem B. First, we give an elementary lemma: Lemma 5.2. Let a, b ∈ C [ y, z ] be non-zero homogeneous polynomials with deg a ≥ deg b such that for every homogeneous polynomial c ∈ C [ y, z ] of degree deg a − deg b , thepolynomial a + bc is divisible by the square of a linear form. Then a and b are both divisibleby the square of the same linear form. roof. Suffices to prove that for non-zero polynomials f, g ∈ C [ x ], if f + λg has a repeatedroot for all λ ∈ C , then f and g have a common repeated root. This holds if there exists x ∈ C which is as a repeated root of f + λg for infinitely many λ . Since g and f /g + λ have only finitely many repeated roots, the claim follows. Proposition 5.3. Let X be a general sextic double solid with an isolated cA n singularity P and Y → X a divisorial contraction with centre P , which is a ( r , r , , -Kawakitablowup. Then, Y has a quotient singularity /r (1 , , r − if r > and a quotientsingularity /r (1 , , r − if r > , and is smooth elsewhere.Proof. By Theorem A, a general X is smooth outside P . So, it suffices to show that Y has only up to two quotient singularities on the exceptional divisor and is smoothelsewhere. Since Y → X is a ( r , r , , P where X is given by wt + h ( y, z ) where h ∈ C { y, z } has multiplicity n + 1. The variety Y is locally analytically around the exceptionaldivisor given by wt + u n +1 h ( uy, uz ) inside the geometric quotient ( C \ V ( w, t, y, z )) / C ∗ where the C ∗ -action is given by λ · ( u, w, t, y, z ) = ( λ − u, λ r w, λ r t, λy, λz ). Taking partialderivatives, the singular locus of Y is given bySing Y = V u, w, t, h n +1 , ∂h n +1 ∂y , ∂h n +1 ∂z , h n +2 ! ∪ { P w } if r > ∪ { P t } if r > , where h i denotes the homogeneous degree i part of h , and P w and P t are the points[0 , , , , 0] and [0 , , , , X is general, then no square of a linear form divides h n +1 , that is, h n +1 is squarefree.Considering the 11 families of Theorem A separately, it is easy to compute using theexplicit splitting lemma (Proposition 3.2) and Lemma 5.2 that h n +1 is squarefree when X is general. For example, for a cA singularity, we compute that h = Q − d r = 8( a − A ) s + r R, where Q, R ∈ C [ y, z ] are homogeneous of degrees 9 and 7 respectively, and Q does notcontain the polynomial d . If the affine cone of Y is not smooth for a general X , then theaffine cone of Y is singular for all X . In that case, Lemma 5.2 shows that a prime factorof r divides h , which implies that it divides s , a contradiction. Similarly for the other10 families. cA model Note that Okada described a Sarkisov link starting from a general complete intersection Z , ⊆ P (1 , , , , , 4) to a sextic double solid (see entry No. 9 of the table in [Oka14,Section 9]). We show the converse: Proposition 5.4. A sextic double solid with a cA singularity satisfying Condition 5.1has a Sarkisov link to a complete intersection Z , ⊆ P (1 , , , , , , starting with a (3 , , , -blowup of the cA point, then Atiyah flops, and finally a Kawamata divisorialcontraction (see [Kaw96]) to a terminal quotient / , , point. Under further generalityconditions (Proposition 5.3), Z is quasismooth. roof. We exhibit the diagram below. Y Y cA ∈ X ⊆ P (1 , X / , , ∈ Z , ⊆ P (1 , , , (3 , , , 1) 10 × (1 , , − , − 1) ( , , ) The corresponding diagram for the ambient toric spaces is given in detail in Example 2.15.First, we describe the sextic double solid X . By Theorem A, any sextic double solid ˆ X with an isolated cA singularity can be given byˆ X : V ( ˆ f ) ⊆ P (1 , , , , x, y, z, t, w whereˆ f = − w + x t + 2 x ta + x t A + x a + x tB + xC + D , where a ∈ C [ y, z ] is homogeneous of degree 2, and A i , B i , C i , D i ∈ C [ y, z, t ] are homoge-neous of degree i . Define the bidegree (5 , 6) complete intersection X , isomorphic to ˆ X ,by X : V ( f, − xξ + α − D ) ⊆ P (1 , , , , , x, y, z, t, α, ξ , where f = − ξ + 2 αa + 2 αxt + x t A + xtB + C . The isomorphism is given byˆ X → X [ x, y, z, t, w ] h x, y, z, t, α , α a + 2 α xt + x t A + xtB + C i where α = w + x t + xa , with inverse[ x, y, z, t, α, ξ ] [ x, y, z, t, α − x t − xa ] . We describe the divisorial contraction ϕ : Y → X . Define the toric variety u x y z α ξ tT : (cid:18) (cid:19) , − V ( x ), that is,Φ : T → P (1 , , , , , u, x, y, z, α, ξ, t ] [ x, uy, uz, u t, u α, u ξ ] . Let Y be the strict transform of X . Let Φ ∗ denote the corresponding C -algebra homomor-phism, namely Φ ∗ : C [ x, y, z, t, α, ξ ] → C [ u, x, y, z, α, ξ, t ]Φ ∗ : x x, y uy, z uz, t u t, α u α, ξ u ξ. A Y = A ( y, z, ut ) , B Y = B ( y, z, ut ) , C Y = C ( y, z, ut ) , D Y = D ( y, z, ut )and define the polynomial g = Φ ∗ f /u , that is, g = − uξ + 2 αa + 2 αxt + x t A Y + xtB Y + C Y . Then, Y is given by Y : V ( I Y ) ⊆ T where I Y = ( g, − xξ + α − D Y ) . We will see later that I Y T . Note that there exist other ideals that definethe same variety Y ⊆ T (see [Cox95, Corollary 3.9]), but where the ideal might not 2-rayfollow T . Also note that we have not (and do not need to) prove that the ideal I Y issaturated with respect to u , although in general, saturating might help in finding the idealthat 2-ray follows T . The morphism Y → X is the restriction of T → P (1 , , , , , Y ) x → X x is the (3 , , , V ( f ) ⊆ C with variables α, t, y, z , where f = − α + 2 αa + 2 αt + t A + tB + C + D . Since wt f = 5, by Proposition 4.5, ( Y ) x → X x is a (3 , , , Y is 10 Atiyah flops, under Condition 5.1. Wedescribe the diagram Y → X ← Y globally. Multiplying the action matrix of T by thematrix ( − ), define u x y z α ξ tT : (cid:18) (cid:19) . − − Y by V ( I Y ) ⊆ T . Define the morphisms Y → X and Y → X as the amplemodels of V ( y ). The exceptional locus of Y → X is E − = V ( ξ, t ) ⊆ Y , the exceptionallocus of Y → X is E +1 = V ( u, x ) ⊆ Y , and the base of the flop is { P i } = V (2 αa + C ( y, z, , α − D ( y, z, ⊆ P (1 , , ⊆ X , where P (1 , , 3) has variables y, z, α . If a , C ( y, z, 0) and D ( y, z, 0) are general enough,that is, if Condition 5.1 is satisfied, then the base of the flop is 10 points { P i } ≤ i ≤ , andboth E − and E +1 are 10 disjoint curves mapping to { P i } ≤ i ≤ .We show that locally analytically, the diagram Y → X ← Y is 10 Atiyah flops. Let P ∈ X be any point in the base of the flop. Then, P has either y or z coordinate non-zero.We consider the case where the y -coordinate is non-zero, the other case is similar. Sincethe base of the flop is 10 points, the point P is smooth in P (1 , , α and z in terms ofthe variables u, x, ξ, t on the patches ( Y ) y , ( X ) y and ( Y ) y . So, the flop Y → X ← Y islocally analytically a (1 , , − , − P .The last morphism Y → Z in the link for X is a divisorial contraction. Multiplying theaction matrix of T by the matrix (cid:16) − − (cid:17) with determinant 4, we see that u x y z α ξ tT ∼ = (cid:18) − (cid:19) . Y → Z be the ample model of V ( ξ ), that is, Y → Z [ u, x, y, z, α, ξ, t ] h t u, t y, t z, t x, t α, ξ i . Then Z is the bidegree (5 , 6) complete intersection Z : V ( h, − xξ + α − D ( y, z, u )) ⊆ P (1 , , , , , u, y, z, x, α, ξ , where the h is given by applying the C -algebra homomorphism t g . The morphism Y → Z contracts the exceptional divisor V ( t ) ⊆ Y to thepoint P ξ = [0 , , , , , Y ) ξ , we can express u and x locally analytically equivariantly in terms of y, z, α, t . So, the morphism Y → Z is locallyanalytically the Kawamata weighted blowdown (see [Kaw96]) to the terminal quotientsingular point P ξ of type 1 / , , Remark . We explain below how we found the variety X . We start with the varietyˆ X , given by Theorem A. Next, we perform the coordinate change ˆ X → ¯ X given inEquation (4.4) of Corollary 4.9, with ( r , r , a, 1) = (3 , , , p = a and v = 1. We seethat ˆ X is isomorphic to ¯ X : V ( ¯ f ) ⊆ P (1 , , , , x, y, z, t, α , where¯ f = α ( − α + 2 x t + 2 xa ) + x t A + x tB + xC + D . We construct a (3 , , , Y → ¯ X . Define the toric variety ¯ T by u x y z α t ¯ T : (cid:18) (cid:19) . − T is given by the geometric quotient¯ T = C \ V (( u, x ) ∩ ( y, z, α, t ))( C ∗ ) . Let ¯Φ be the ample model of V ( x ), and let ¯ Y ⊆ ¯ T be the strict transform of ¯ X . By Corol-lary 4.9, ¯ Y → ¯ X is a (3 , , , Y ) x → ¯ X x to show it is a (3 , , , I ¯ Y does not 2-ray follow ¯ T . We describe the next (and the final) map inthe 2-ray game for ¯ T . Acting by the matrix (cid:16) − − (cid:17) , we can write ¯ T by u x y z α t ¯ T ∼ = (cid:18) − (cid:19) . V ( y ) is the weighted blowup¯ T → P (1 , , , , u, x, y, z, α, t ] [ y, z, ut, xt, α ] , P (1 , , 3) given by V ( u, x ) ⊆ P (1 , , , , 3) with variables y, z, u, x, α . Above every point in P (1 , , P . Define¯ Z : V (¯ h ) ⊆ P (1 , , , , h = α ( − uα + 2 x + 2 xa ) + x A Z + x B Z + xC Z + uD Z , where A Z = A ( y, z, u ) , B Z = B ( y, z, u ) , C Z = C ( y, z, u ) , D Z = D ( y, z, u ) . We show that when restricting the weighted blowup to ¯ Y → ¯ Z , the exceptional lo-cus is 1-dimensional. After restricting to ¯ Y , the exceptional divisor V ( t ) becomes V ( t, x (2 αa + C ( y, z, u ( − α + D ( y, z, P , . . . , P ∈ P (1 , , ⊆ ¯ Z such that 2 αa + C ( y, z, 0) and − α + D ( y, z, 0) havea common solution. Above each of those points, the fibre is P . Above any other point,the fibre is just one point. Therefore, the morphism ¯ Y → ¯ Z contracts 10 curves onto 10points, and is an isomorphism elsewhere. This shows that ¯ Y does not 2-ray follow ¯ T ,since a 2-ray link ends with either a fibration or a divisorial contraction.The problem with the previous embedding was that ¯ g belonged to the irrelevant ideal( u, x ). We “unproject” the divisor V ( u, x ), to embed ¯ Y into a toric variety T such that Y T . The varieties Y ⊆ T are defined as in the proof of Proposition 5.4.We see that ¯ Y is isomorphic to Y through the map[ u, x, y, z, α, t ] " u, x, y, z, α, α − D Y x , t . The map is a morphism, since we have the equality α − D Y x = 2 αa + 2 αxt + x t A Y + xtB Y + C Y u in the field of fractions of C [ u, x, y, z, α, t ], and either x or u is non-zero at every pointof T . For more details on this kind of “unprojection”, see [Rei00, Section 2] or [PR04,Section 2.3].Now, the coordinate change ¯ Y → Y induces a coordinate change ¯ X → X , where X isdefined as in the proof of Proposition 5.4. cA model Proposition 5.6. A sextic double solid X which is a Mori fibre space with a cA singularitysatisfying Condition 5.1 has a Sarkisov link to a sextic double solid Z with a cA singularity,starting with a (3 , , , -blowup of the cA point in X , then four Atiyah flops, and finallya (3 , , , -blowdown to a cA point. If in addition c is general after fixing a i , b i and d in Notation 3.4, then X and Z are not isomorphic. Under further generality conditions,both X and Z are smooth outside the cA point.Proof. We exhibit the diagram below. Y Y cA ∈ X ⊆ P (1 , X cA ∈ Z ⊆ P (1 , (3 , , , 1) 4 × (1 , , − , − 1) (3 , , , 31e construct X and a (3 , , , Y → X . Using Theorem A, andperforming the coordinate change in Equation (4.5) of Corollary 4.9 (with p = a ), wecan write a sextic double solid X with a cA singularity by X : V ( f, − β + xt + a ) ⊆ P (1 , , , , , , with variables x, y, z, t, β, w where f = − w + xβ (2 b − βa + 8 xta + xβ ) + 4 x t a + x t B + xtC + D , where B i , C i , D i ∈ C [ y, z, t ] are homogeneous of degrees i . Define T by u x y z w β tT : (cid:18) (cid:19) . − T → P (1 , , , , , 3) be the ample model of V ( x ), Y ⊆ T the strict transformof X , and Y → X the restriction of Φ. Then, Y is given by Y : V ( I Y ) ⊆ T where I Y = (Φ ∗ f /u , − uβ + xt + a ) , and Y → X is a (3 , , , Y is a flop, locally analytically 4Atiyah flops, under Condition 5.1. Acting by the matrix ( − ), we find u x y z w β tT ∼ = (cid:18) (cid:19) . − − P (1 , , ⊆ X is given by V ( a , − w + D ( y, z, ⊆ P (1 , , a and D ( y, z, 0) are general, that is, Condition 5.1 is satisfied, then this is exactly 4points. In this case, any such point P is a smooth point in P (1 , , y -coordinate of P is non-zero, the case where z is non-zero is similar. Locallyanalytically equivariantly, we can express z and w in terms of u, x, β, t in Y , X and Y .So, the diagram Y → X ← Y is locally analytically four Atiyah flops.The last map in the 2-ray game of Y is a weighted blowdown Y → Z . After acting by (cid:16) − − (cid:17) on the initial matrix of T , we find that T is given by u x y z w β tT : (cid:18) − (cid:19) . Z ⊆ P (1 , , , , , 3) with variables β, u, y, z, x, w is given by the ideal I Z = ( h, − uβ + x + a ) , where h is given by sending t to 1 in Φ ∗ f /u , namely h = − w + xβ (2 b − uβa + 8 xa + xβ ) + 4 x a + x B Z + xC Z + D Z and B Z = B ( y, z, u ) , C Z = C ( y, z, u ) , D Z = D ( y, z, u ) . x = uβ − a into h , we find that Z is a sextic double solid. Applying theexplicit splitting lemma (Proposition 3.2), we find that the complex analytic space germ( Z, P β ) is isomorphic to ( V ( h ana ) , ) ⊆ ( C , ) with variables w, u, y, z , where h ana = − w + u + d − ( b − a a ) + (h.o.t in y, z ) , where (h.o.t in y, z ) stands for higher order terms in the variables y, z . So, P β ∈ Z is a cA singularity. On the patch where β is non-zero, we can substitute u = xt + a , sothe morphism ( Y ) β → Z β is a weighted blowup of a hypersurface given by a weight 6polynomial. By Proposition 4.5, Y → Z is a (3 , , , X and Z are not isomorphic when a = 0 and c is general, usinga dimension counting argument similar to [GLS07, Theorem 2.55]. Using the explicitsplitting lemma, we find that the complex analytic space germ ( X, P x ) is isomorphic to( V ( f ana ) , ) ⊆ ( C , ) with variables w, t, y, z where f ana = − w + t + d − a c + 2 a b − a a − ( b − a a ) + (h.o.t in y, z ) . If X and Z are isomorphic, then this implies that the complex analytic space germs ( X, P x )and ( Z, P β ) are isomorphic, implying by Propositions 2.5 and 2.4 that the degree 6 partsof f ana (0 , , y, z ) and h ana (0 , , y, z ) are the same up to an invertible linear coordinatechange on y, z . Fixing a , a , a , b , b and d , we see that h ana (0 , , y, z ) is fixed, but f ana (0 , , y, z ) has 5 degrees of freedom. Since there are only 4 degrees of freedom in pickingan element of GL( C , f ana (0 , , y, z ) and h ana (0 , , y, z ) are not relatedby an invertible linear coordinate change when c is general. This shows that if X isgeneral, then the varieties X and Z are not isomorphic. cA model Proposition 5.7. A sextic double solid that is a Mori fibre space with a cA singularitysatisfying Condition 5.1 has a Sarkisov link to a hypersurface Z ⊆ P (1 , , , , with a cA singularity, starting with a (4 , , , -blowup of the cA point, then two (1 , , − , − -flops,then a (4 , , , − , − 1; 2) -flip, and finally a (2 , , , -blowdown to a cA point. Underfurther generality conditions, the singular locus of Z consists of 3 points, namely the cA point, the / , , quotient singularity and an ordinary double point.Proof. We exhibit the diagram below. Y Y Y cA ∈ X ⊆ P (1 , X X cA ∈ Z ⊆ P (1 , (4 , , , 1) 2 × (1 , , − , − 1) (4 , , , − , − , , , We construct X and a (4 , , , Y → X . Using Theorem A andCorollary 4.9 with p = a and p = b − a a , we can write a sextic double solid X witha cA singularity by X : V ( f, − β + xt + a ) ⊆ P (1 , , , , , , with variables x, y, z, t, β, w where f = α ( − α + 2( b − βa + 4 xta + xβ ))+ 2 β ( c − βb + 2 xtb + 2 xβa + 2 β a − xtβa + 6 x t a )+ x t B + xt C + tD B i , C i , D i ∈ C [ y, z, t ] are homogeneous of degree i . Define T by u x y z α β tT : (cid:18) (cid:19) . − T → P (1 , , , , , 3) be the ample model of V ( x ), Y ⊆ T the strict transformof X , and Y → X the restriction of Φ. Then, Y is given by Y : V ( I Y ) ⊆ T where I Y = (Φ ∗ f /u , − uβ + xt + a ) , and Y → X is a (4 , , , Y → X ← Y in the 2-ray game for Y is locallyanalytically two Atiyah flops under Condition 5.1, namely that V ( a ) ⊆ P with variables y, z consists of exactly two points, and for both of the points P , either b ( P ), c ( P ) or d ( P ) is non-zero, where D = t d + 2 t d + t d + 2 t d + td + 2 d . Acting by the matrix (cid:16) − − (cid:17) , we find u x y z α β tT : (cid:18) − − (cid:19) . − − y, z , we find that a = yz . Let P = V ( z ) ∈ P ⊆ X , the case where P = V ( y ) is similar. On the patch where y is non-zero, we can substitute z = uβ − xt . The contracted locus is P ∼ = V ( α, β, t ) ⊆ ( Y ) y ,and the extracted locus is V ( u, x ) = V ( u, x, αb (1 , 0) + βc (1 , 0) + td (1 , ⊆ ( Y ) y . ByCondition 5.1, we can express one of α, β, t equivariantly locally analytically in the othervariables. So, the flop diagram Y → X ← Y is locally analytically a (1 , , − , − Y is a (4 , , , − , − 1; 2)-flip (thisis case (1) in [Bro99, Theorem 8]). The toric variety T is given by u x y z α β tT : (cid:18) − − (cid:19) . − − P α = [0 , , , , , , α is non-zero, we canexpress u locally analytically and equivariantly in terms of x, y, z, β, t . After substitution,the ideal is principal, with generator f = − β · (2 x + . . . ) + xt + a . Under Condition 5.1, a has a non-zero coefficient in f , so the flip diagram corresponds to case (1) in [Bro99,Theorem 8]. The flips contracts a curve containing a 1 / , , 3) singularity and extractsa curve containing a 1 / , , 1) singularity and an ordinary double point. The ordinarydouble point on Y is at [ u , , , , , , 1] for some u ∈ C .We show that the last map in the 2-ray game of Y is a weighted blowup Y → Z , where Z is isomorphic to a hypersurface Z ⊆ P (1 , , , , 2) with variables u, y, z, β, α . Actingby the matrix (cid:16) − − (cid:17) on the initial action-matrix of T , we find that T is given by u x y z α β tT : (cid:18) − (cid:19) . , 2) complete intersection Z : V ( h, a − uβ + x ) ⊆ P (1 , , , , , u, y, z, β, x, α , where h = α ( − uα + 2( b − uβa + 4 xa + xβ ))+ 2 β ( c − uβb + 2 xb + 2 xβa + 2 u β a − uxβa + 6 x a )+ x B Z + xC Z + D Z , where B Z = B ( y, z, u ) , C Z = C ( y, z, u ) , D Z = D ( y, z, u ) . The morphism Y → Z given by the ample model of V ( β ) is a weighted blowdown withcentre P β and exceptional locus V ( t ). Substituting x = uβ − a (5.1)into h , we find that Z is isomorphic to a hypersurface Z ⊆ P (1 , , , , 2) with variables u, y, z, β, α . The substitution (5.1) does not lift onto Y . Instead, on the patch Z β , wecan substitute u = ( a + x ) /β . This substitution lifts to ( Y ) β . By Condition 5.1, P β ∈ Z is a cA singularity and the hypersurface Z β is given by a weight 4 polynomial. ByProposition 4.5, ( Y ) β → Z β is a (3 , , , Z has an ordinary double point at [ u , , , , 2] for some u ∈ C . cA family 1 model Proposition 5.8. A Mori fibre space sextic double solid with a cA singularity in family 1satisfying Condition 5.1 has a Sarkisov link to Z , ⊆ P (1 , , , , , with an ordinarydouble point, starting with a (4 , , , -blowup of the cA point, then two (4 , , , − , − 1; 2) -flips, and finally a blowdown (with standard weights (1 , , , ) to an ordinary doublepoint. Under further generality conditions, Z has exactly five singular points, namely two / , , singularities and three ordinary double points.Proof. We exhibit the diagram below. Y Y Y cA ∈ X ⊆ P (1 , X ODP ∈ Z , ⊆ P (1 , ) (4 , , , ∼ × (4 , , , − , − , , , We construct X and a (4 , , , Y → X . We can write a sexticdouble solid X with an isolated cA singularity in family 1 by X : V ( f, β − xt − r , γ − xβ − s ) ⊆ P (1 , , , , , , x, y, z, t, β, γ, w , where f = − w + γ − tγe + 2 β e + 2 tβc + 4 tγb − β b − tβ b + 4 xt βb + 2 x t b − tγa + 16 β a + 4 βγa − β a + 12 xtβ a + xt C + t D , where C i , D i ∈ C [ y, z, t ] are homogeneous of degree i . Define T by u x y z w γ β tT : (cid:18) (cid:19) . − Y by Y : V ( I Y ) ⊆ T where I Y = (Φ ∗ f /u , uβ − r − xt, uγ − s − xβ ) . The ample model of V ( x ) ⊆ Y is a (4 , , , Y → X .We show that the diagram Y → X ← Y induces an isomorphism Y → Y . Acting bythe matrix (cid:16) − − (cid:17) , we find u x y z w γ β tT ∼ = (cid:18) − − (cid:19) . − − T , respectively T , with the same action as T but with irrelevant ideal ( u, x, y, z ) ∩ ( w, γ, β, t ), respectively ( u, x, y, z, w, γ ) ∩ ( β, t ). Define Y ⊆ T and Y ⊆ T by the sameideal I Y . The base of the flop T → T X ← T restricts to V ( r , s ) ⊆ P ⊆ X , which isempty. Therefore, Y → X and X ← Y are isomorphisms.We show that the next diagram Y → X ← Y in the 2-ray game of Y is locallyanalytically two (4 , , , − , − 1; 2)-flips. The only monomials in Φ ∗ f /u that are not in( u, x, y, z, β, t ) are − w and γ . Therefore, the base of the flip is two points, [1 , 1] and[ − , ∈ P with variables w and γ inside X . We make a change of coordinates w = w − γ ,respectively w = w + γ , for the flip above [1 , − , γ is non-zero, we can substitute u = s + xβ in Φ ∗ f /u , and express w locally analyticallyand equivariantly above [1 , − , x, y, z, β, t . After projectingaway the variables u and w , we are left with the principal ideal ( βs − r + xβ − xt ).Since it contains both r and xt , by case (1) in [Bro99, Theorem 8], it is a terminal(4 , , , − , − 1; 2)-flip above both [1 , 1] and [ − , / , , 3) singularity, and extracts two curves, both containing a 1 / , , cA singularity. The cA points are both ordinary double points if r isnot a square of a linear form, and are both 3-fold A singularities (given by x + x + x + x )otherwise. On Y , the cA singularities are at [0 , , , , , , , 1] and [0 , , , , − , , , X is a divisorial contraction Y → Z . Actingby the matrix (cid:16) − − (cid:17) on the initial action-matrix of T , we see that u x y z w γ β tT ∼ = (cid:18) − (cid:19) . Z ⊆ P (1 , , , , , , 2) with variables u, y, z, β, w, γ, x by the ideal I Z , where I Z is the image of the ideal I Y under the homomorphism t 1. Let Y → Z be the amplemodel of V ( β ). On the affine patch Z β , we can express u and x locally analyticallyand equivariantly in terms of y, z, w, γ, β, t . This coordinate change lifts to Y . ByCondition 5.1, we can compute that P β ∈ Z is an ordinary double point, and Y → Z islocally analytically the (usual) blowup with centre P β .The variety Z is isomorphic to a complete intersection Z , ⊆ P (1 , ), by projectingaway from x . The variety Z is given by Z , : V ( − s + βr + uγ − uβ , h ) ⊆ P (1 , , , , , u, y, z, β, w, γ , where h = − w + γ + 2 b r − βb r − uβb r − β a r − γe + 2 β e + 2 βc + 4 γb − β b + 2 uβ b + 2 u β b − γa + 16 β a + 4 βγa + 4 uβ a + ( uβ − r ) C Z + D Z , C Z = C ( y, z, u ) and D Z = D ( y, z, u ). The variety Z has two cA singularities at[0 , , , , , 1] and [0 , , , , − , Remark . We explain how we found the variety X . Using p = r and p = s , we canwrite a sextic double solid with an isolated cA in family 1 by ¯ X : V ( ¯ f , x t + xr + s − ¯ γ )inside P (1 , , , , , 3) with variables x, y, z, t, w, ¯ γ , where ¯ f is given as in Theorem A. The(1 , , , , Y → ¯ X for variables y, z, w, ¯ γ, t is a (4 , , , Y does not follow the ambient toric variety ¯ T . Namely, the toricanti-flip ¯ T → ¯ T ¯ X ← ¯ T restricts to ¯ Y → ¯ X ← ¯ Y , where ¯ Y → ¯ X is an isomorphismand ¯ X ← ¯ Y extracts P , a divisor on ¯ Y . The reason why ¯ Y was not the correct varietyis that one of the generators of the ideal of ¯ Y is ¯ g = x t + xr + us − u ¯ γ , which isinside the irrelevant ideal ( u, x ). We find the correct variety Y by “unprojecting” ¯ g = 0with respect to u, x . By “unprojection”, we mean the coordinate change ¯ Y → Y , anisomorphism. See [Rei00, Section 2] or [PR04, Section 2.3] for more details on this type ofunprojection. This coordinate change induces the coordinate change ¯ X → X , where X isgiven as in the proof of Proposition 5.8. cA family 2 model Proposition 5.10. A Mori fibre space sextic double solid with a cA singularity in family 2satisfying Condition 5.1 has a Sarkisov link to a complete intersection Z , ⊆ P (1 , , , , , with a cA singularity, starting with a (4 , , , -blowup of the cA point, followed by oneAtiyah flop, then two (4 , , − , − -flips, and finally a (3 , , , -blowdown to a cA point.Under further generality conditions, the variety Z is smooth outside the cA point.Proof. We exhibit the diagram below. Y Y Y Y cA ∈ X ⊆ P (1 , X X cA ∈ Z , ⊆ P (1 , (4 , , , 1) (1 , , − , − 1) 2 × ( − , − , , ∼ (3 , , , We describe the sextic double X . Define X ⊆ P (1 , , , , , , , 3) with variables x , y , z , t , β , w , γ , ξ by the ideal I X = ( f − e ξ, β − q r − xt, γ − q s − xβ, − ξ + ts − βr ) , (5.2)where f = − w + γ + 2 tβc + 4 tγb − β b − tβ b + 4 xt βb + 2 x t b − tγa + 16 β a + 4 βγa − β a + 12 xtβ a + xt C + t D where C i , D i ∈ C [ y, z, t ] are homogeneous of degree i .We describe the weighted blowup Y → X , restriction of Φ : T → P (1 , , , , , , , T by u x y z w γ β ξ tT : (cid:18) (cid:19) . − Y ⊆ T by the ideal I Y with the 6 generators g − e ξ, uβ − q r − xt, uγ − q s − xβ, − uξ + ts − βr , − xξ + βs − γr , − q ξ + tγ − β , g = Φ ∗ f /u . On the affine patch X x , we can express β, t and ξ in terms of w, γ, y, z ,to get a hypersurface in C given by f hyp ∈ C [ w, γ, y, z ]. Note that these coordinate changeslift to ( Y ) x . Since f hyp has weight 8, by Proposition 4.5, Y → X is a (4 , , , Y → X ← Y in the 2-ray game of Y is an Atiyahflop, provided that r and q are coprime in C [ y, z ]. Acting by the matrix (cid:16) − − (cid:17) on theaction-matrix of T , define T by u x y z w γ β ξ tT : (cid:18) − − − (cid:19) . − − Y ⊆ T by the ideal I Y . The base of the flop is V ( q ) ⊆ P with variables y, z ,which is one point. Perform a suitable invertible linear coordinate change on y, z suchthat q = z and r = y . Since uβ − q r − xt is in I Y , we can substitute z = uβ − xt onthe patch where y is non-zero. The coefficients of β in − uξ + ts − βy ∈ I Y and γ in − xξ + βs − γy ∈ I Y are non-zero on the patch where y is non-zero. Therefore, we canlocally analytically equivariantly express β and γ in terms of u, x, w, t . After substituting z, β, γ , we find that the diagram Y → X ← Y is locally analytically the Atiyah flop.The next diagram in the 2-ray game of Y is the flip Y → X ← Y . The base of the flipis V ( γ − w ) ⊆ P with variables w, γ , which is two points [1 , 1] and [ − , P = [1 , w = w − γ . On the patch where γ is non-zero, we find u = q s + xβ and t = q ξ + β .Writing q = z and r = y as before, we find y = − xξ + βs . We are left with theprincipal ideal in C [ x, z, w , β, ξ ] generated by − w (2 + w ) + terms not involving w . So,we can locally analytically equivariantly express w in terms of x, z, β, ξ . So, the diagram Y → X ← Y is locally analytically two ( − , − , , T → T X ← T restricts to isomorphisms Y → X ← Y . The reason is that the base of the toric flip P β restricts to an empty setin X , since I Y contains the polynomial tγ − β − qξ .We show that the last diagram in the 2-ray game of Y is a divisorial contraction Y → Z .Multiplying the action-matrix of T by ( ), we see that T is given by u x y z w γ β ξ tT : (cid:18) − (cid:19) . Z ⊆ P (1 , , , , , , , 2) with variables ξ, u, y, z, β, x, w, γ where Y → Z is the ample model of V ( ξ ). On the patch Z ξ , we can substitute u = s − βr , x = βs − γr and z = γ − β , and compute that Z ξ is a hypersurface given by a weight6 polynomial, with a cA singularity at P ξ ∈ Z ξ , of type at least 2 (see Definition 4.7).These substitutions lift to ( Y ) ξ , showing that Y → Z is a (3 , , , P ξ . If the coefficients are general, namely when − e β + 8 β a r β − β b β + 12 β a β ∈ C [ y, β ]is not a full square, where r β = r ( y, − β ), e β = e ( y, − β ), a β = a ( y, − β ) and b β = b ( y, − β ), then the point P ξ is exactly of type 2.The variety Z is isomorphic to a complete intersection Z , ⊆ P (1 , , , , , 2) withvariables u, y, z, β, ξ, w . We see this by substituting x = uβ − q r and γ = q ξ + β . We38nd that Z is isomorphic to Z , : V ( − uξ + s − βr , h ), where h = − w + ξ q − e ξ + β + 2 b q r − βb q r − uβb q r − β a q r + 4 ξb q − ξa q + 4 βξa q + 2 β ξq + 2 βc + 2 β b + 2 uβ b + 2 u β b + 4 β a + 4 uβ a + ( uβ − q r ) C Z + D Z , where C Z = C ( y, z, u ) and D Z = D ( y, z, u ). Remark . We explain below how we found the embedding of X . Using Theorem Aand the coordinate change in cA family 1, we can write a sextic double solid ¯ X with anisolated cA in family 2 by¯ X : V ( f − e ( ts − βr ) , β − xt − q r , γ − xβ − q s ) ⊆ P (1 , , , , , , x, y, z, t, β, γ, w .We construct a (4 , , , Y → ¯ X . Define ¯ T by u x y z w γ β t ¯ T : (cid:18) (cid:19) . − T → P (1 , , , , , , 3) be the ample model of V ( x ) and Y ⊆ T the strict transformof X . Then ¯ Y is given by the ideal I ¯ Y = (¯ g , . . . , ¯ g ), where¯ g = ug + 2 e ( βr − ts ) , ¯ g = uβ − q r − xt, ¯ g = uγ − q s − xβ, ¯ g = xg + 2 e ( γr − βs ) , ¯ g = q g + 2 e ( β − tγ )) . The morphism ¯ Y → ¯ X is a (4 , , , Y ) x → ¯ X x .Note that we do not prove that I ¯ Y is saturated with respect to u . In fact, the saturationwill not be I Y if we do not use assume some generality conditions, similarly to cA and cA family 1. As a heuristic argument to see why I ¯ Y might be saturated in the general case(“general” meaning a Zariski open dense set of the parameter space), we can use computeralgebra software, pretend that a i , b i , c i , d i , q , r , s , e are algebraically independentvariables of a polynomial ring over Q or Z p for a large prime p , and calculate that thesaturation in that case indeed equals the ideal I ¯ Y .Similarly to the diagram Y → X ← Y in the proof of Proposition 5.10, the diagram¯ Y → ¯ X ← ¯ Y is an Atiyah flop, provided r and q are coprime.We show that I ¯ Y does not 2-ray follow ¯ T , namely that the diagram ¯ Y → ¯ X ← Y contracts a curve and extracts a divisor. Acting by the matrix (cid:16) − − (cid:17) on the action-matrixof ¯ T , define ¯ T by u x y z w γ β t ¯ T : (cid:18) − − (cid:19) , − − Y ⊆ ¯ T by the zeros of I ¯ Y . We consider the toric flip ¯ T → ¯ T ¯ X ← ¯ T andrestrict it to ¯ Y → ¯ X ← ¯ Y . Since I ¯ Y is the zero ideal when restricting to V ( u, x, y, z, β, t ),the base P ⊆ ¯ T ¯ X of the toric flip restricts to P ⊆ ¯ X with variables w, γ . The morphism¯ Y → ¯ X contracts a curve P to both of the points [1 , 1] and [1 , − 1] in the base P ⊆ ¯ X and is an isomorphism elsewhere. The morphism ¯ X ← ¯ Y extracts a curve P for every39oint in the base P ⊆ ¯ X , so extracts a divisor on ¯ Y . The diagram ¯ Y → ¯ X ← ¯ Y is nota step in the 2-ray game of ¯ Y , so I ¯ Y does not 2-ray follow ¯ T . The reason for this wasthat the ideal I ¯ Y is contained in ( u, x, y, z ), so the surface V ( u, x, y, z ) ⊆ ¯ T exists on ¯ Y ,but does not exist on ¯ T .We “unproject” ¯ g = ¯ g = ¯ g = 0 with respect to u, x, y, z in ¯ Y ⊆ ¯ T , to find a variety Y ⊆ T . We explain below what we mean by this. We can write the system of equations¯ g = ¯ g = ¯ g = 0 in the matrix form g βr − ts g γr − βs g β − tγ uxq e = . If the above equations hold, then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) βr − ts g γr − βs g β − tγ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g βr − ts γr − βs g β − tγ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − x = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g βr − ts g γr − βs β − tγ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g g 00 0 g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − e . Calculating the determinants and dividing by − g , we find the equalities ts − βr u = βs − γr x = tγ − β q = g e , (5.3)between elements of the field of fractions of C [ u, x, y, z, w, γ, β, t ] /I ¯ Y . Using the Equa-tions 5.3, we see that the morphism ¯ Y → Y given by[ u, x, y, z, w, γ, β, t ] [ u, x, y, z, w, γ, β, ts − βr u , t ]is an isomorphism, where Y is described in the proof of Proposition 5.10.The coordinate change ¯ Y → Y induces an isomorphism ¯ X → X , giving the variety X . cA family 3 model Proposition 5.12. A Mori fibre space sextic double solid with a cA singularity in family 3satisfying Condition 5.1 has a Sarkisov link to a degree del Pezzo fibration, starting witha (4 , , , -blowup of the cA point and followed by two Atiyah flops.Proof. We exhibit the diagram below. Y Y Y cA ∈ X ⊆ P (1 , X P , , , 1) 2 × (1 , , − , − ∼ dP -fibration First, we define X and a (4 , , , Y → X . Any sextic double solidwith an isolated cA family 3 can be given by a bidegree (6 , 2) complete intersection X : V ( f, − ξ + ts − q − xt ) ⊆ P (1 , , , , , x, y, z, t, ξ, w , where f = − w + x ξ − ξe + ξ ( s + 4 a s + 2 xs − b + 16 a + 4 xa + 8 ξa )+ t ( ts + 4 ta s − t a s − ξs + 2 tb s − t b s − ξa s + 24 tξa s + 12 xt a s − xξs + 2 tc s + 4 tξb s + 4 xt b s − ξa s − xξa s − ξ a s − xtξa s − ξc − xξb − ξ b − xtξb + 2 x t b + 16 xξa + 12 xξ a + xt C + tD ) , where C i , D i ∈ C [ y, z, t ] are homogeneous of degree i . Define u x y z w ξ tT : (cid:18) (cid:19) . − T → P (1 , , , , , 3) by the ample model of V ( x ), and define Y as the stricttransform of X . Then, Y is given by Y : V ( I Y ) ⊆ T where I Y = (Φ ∗ f /u , − u ξ + uts − q − xt ) , Using Proposition 4.5, we see that Y → X is a (4 , , , Y → X ← Y . Multiplying the action-matrix of T by ( − 10 1 ), wefind u x y z w ξ tT ∼ = (cid:18) − − − (cid:19) . − V ( q ) ⊆ P ⊆ X . After a suitable coordinate change on y, z , we find q = yz . Consider the flop over V ( y ), the flop over the other point is similar.Since q and e have no common divisor, on the patch where z is non-zero, we can express y and ξ locally analytically equivariantly in terms of u, x, t, w . So, Y → X ← Y is locallyanalytically two Atiyah flops.The morphisms Y → X ← Y are isomorphisms, since w has a non-zero coefficientin Φ ∗ f /u .We show that Y is a degree 2 del Pezzo fibration. Multiplying the original action-matrixof T by the matrix ( − ) with determinant − 1, we find u x y z w ξ tT : (cid:18) (cid:19) . V ( t ) is Y → P (2 , u, x, y, z, w, ξ, t ] [ ξ, t ] . Since P (2 , 1) is isomorphic to P , we see that Y is a fibration onto P . On the patch ( Y ) t ,we can substitute x = us − q − u ξ , to find that the general fibre is a weighted degree 4hypersurface in P (1 , , , .8. cA model Proposition 5.13. A Mori fibre space sextic double solid with a cA singularity satisfy-ing Condition 5.1 has a Sarkisov link to a complete intersection Z , ⊆ P (1 , , , , , with a cD singularity, starting with a (5 , , , -blowup of the cA point, followed by a (4 , , , − , − 2; 2) -flip, and finally a (3 , , , , -blowdown to the cD singularity. Underfurther generality conditions, the singular locus of Z consists of 3 points, namely the cD point, the / , , singularity and an ordinary double point.Proof. We exhibit the diagram below. Y Y Y Y cA ∈ X ⊆ P (1 , X cD ∈ Z , ⊆ P (1 , (1 , , , ∼ (4 , , , − , − ∼ (3 , , , , First, we describe X and the weighted blowup Y → X . A sextic double solid with a cA singularity can be given by a multidegree (6 , , 3) complete intersection X : V ( f, β − xt − r , γ − xβ − s ) ⊆ P (1 , , , , , , , with variables x, y, z, t, β, γ, ξ where f = 8 β ( A − a ) + ξ ( − ξ + 2 γ − tA r + 2 tb − ta + 4 βa )+ t ( − tβA r + 2 tβc + 4 tγb − β b − tβ b + 4 xt βb − tγa a + 8 β a a + 12 βγa − tγB + 2 β B + 16 tβ A − xt βA − βγA + xt C + t D )where C i , D i ∈ C [ y, z, t ] are homogeneous of degree i . Note that B ∈ C [ y, z ]. Define u x y z γ β ξ tT : (cid:18) (cid:19) . − T → P (1 , , , , , , 3) be the ample model of V ( x ) and let Y ⊆ T be the stricttransform of X . Then Y is given by Y : V ( I Y ) ⊆ T where I Y = (cid:16) Φ ∗ f /u , uβ − xt − r , uγ − xβ − s (cid:17) , and Y → X is a (5 , , , T restricts to a isomorphisms Y → X ← Y ,since r and s are coprime.The second diagram in the 2-ray game of T restricts to a (4 , , , − , − 2; 2)-flip Y → X ← Y . Define the toric variety T by multiplying the action matrix of T by the matrix (cid:16) − − (cid:17) , u x y z γ β ξ tT : (cid:18) − − − (cid:19) . − − γ is non-zero, we have u = xβ + s and we can write ξ locallyanalytically equivariantly in terms of x, y, z, β, t . We are left with the hypersurface givenby xβ + βs − xt − r in C with variables x, y, z, β, t with weights (4 , , , − , − xt and r , so this corresponds to case (1) in [Bro99, Theorem 8], a(4 , , , − , − 2; 2)-flip. Similarly to Proposition 5.8, the flip contracts a curve containinga 1 / , , 3) singularity, and extracts a curve containing a 1 / , , 1) singularity and a cA singularity, which is an ordinary double point if r is not a square and is a 3-fold A singularity otherwise. The cA singularity on Y is at [0 , , , , , , − a , T restricts to isomorphisms Y → X ← Y ,under Condition 5.1, namely that a = A . On the patch where β is non-zero, the base ofthe toric flip restricts to V ( A − a , u, x, y, z, γ, ξ, t ) ⊆ X .We describe the weighted blowdown Y → Z . Multiplying the action matrix of T bythe matrix (cid:16) − − (cid:17) , the toric variety T is given by u x y z γ β ξ tT : (cid:18) − (cid:19) . V ( ξ ) is Y → Z where Z is the tridegree (3 , , 3) complete intersection Z : V ( h, uβ − x − r , uγ − xβ − s ) ⊆ P (1 , , , , , , u, y, z, β, ξ, x, γ , where h = 8 β ( A − a ) + ξ ( − uξ + 2 γ − A r + 2 b − a + 4 βa ) − βA r + 2 βc + 4 γb − β b − uβ b + 4 xβb − γa a + 8 β a a + 12 βγa − γB + 2 β B + 16 uβ A − xβA − βγA + xC Z + D Z where C Z = C ( y, z, u ) and D Z = D ( y, z, u ). Substituting x = uβ − r , we see that Z is isomorphic to a complete intersection of bidegree (3 , 3) in P (1 , 2) with variables u, y, z, β, ξ, γ . The variety Z has a cA singularity at [0 , , , , − a , P ξ ∈ Z is a cD point, by showing the complex analytic space germ ( Z, P ξ )is isomorphic to ( V ( u + 2 βr − s + h . o . t) , ) ⊆ ( C , ) with variables u, β, y, z , whereh . o . t are higher order terms in y, z, β . We can compute that Y → Z is the divisorialcontraction to a cD point described in [Yam18, Theorem 2.3]. A. Computer code The code below is for the computer algebra system Maxima [Max18]. To use the splittinglemma library, copy the code below to a file named “Splitting lemma.mac”, start Maximain the same folder as that file, and load the library using load("Splitting lemma.mac"); .Alternatively, just copy-paste the code below to Maxima.Listing 1. Splitting lemma library /* Language: Maxima 5.42.1 */splitting(str, poly, inDeg, splitVar, dummyVar, outDeg) := block(/* Assume f[0, 0] = f[1, 0] = f[0, 1] = f[1, 1] = 0 and f[2, 0] = 1 */[simpPoly, outFun, outPoly],local(f, h, g, p, v),simpPoly : ratexpand(poly),/* Memoizing functions f[i, d] instead of f(i, d) for performance */f[i, d] := coeff(coeff(simpPoly, splitVar, i), dummyVar, inDeg-i-d), * Use apply + makelist instead of sum to avoid dynamic scoping issues */h[d] := ratexpand(f[0, d] - apply("+", makelist(g[0, j]*g[0, d-j], j, 2, d-2))),g[i, d] := if i = 1 and d = 0 then0elseratexpand(1/2*(f[i+1, d] - apply("+", makelist(apply("+", makelist(g[j, k]*g[i+1-j, d-k], j, max(0, 2-k), min(i+1, i+d-k-1))), k, 0, d)))),p[d] := ratexpand(g[0, d] - apply("+", makelist(v[0, d-j]*p[j], j, 2, d-1))),v[i, d] := if i = 0 and d = 0 then1elseratexpand(g[i+1, d] - apply("+", makelist(v[i+1, d-j]*p[j], j, 2, d))),for i : 1 thru 4 doif str = ["h", "g", "p", "v"][i] then outFun : [h, g, p, v][i],outPoly : if member(str, ["h", "p"]) thenapply("+", makelist(dummyVar^(outDeg-d)*outFun[d], d, 0, outDeg))else if member(str, ["g", "v"]) thenapply("+", makelist(apply("+", makelist(dummyVar^(outDeg-k) * splitVar^i * outFun[i, k-i], i, 0, k)), k, 0, outDeg))else"Splitting error: first argument must be ’h’, ’g’, ’p’ or ’v’.",return(outPoly)); We give an example below how to use the splitting lemma library.Listing 2. Splitting lemma example — quartic surface /** Language: Maxima 5.42.1** Example of a quartic surface in projective space with an* A_{19} singularity. We use the splitting lemma twice to verify that* the singularity type is A_{19}, so it is locally analytically given* by x^2 + y^2 + z^20.** The quartic polynomial is taken from M.~Kato, I.~Naruki, \emph{Depth* of rational double points on quartic surfaces}, Proc.~Japan* Acad.~Ser.~A Math.~Sci.~\textbf{58} (1982), no 2, p 72--75.* doi:10.3792/pjaa.58.72,* \url{https://projecteuclid.org/euclid.pja/1195516147}.** Here t is the dummy homogenizing variable, x and y are the splitting* variables. We check the singularity type of the point [0, 0, 0, 1].*/load("Splitting lemma.mac");f : 1/16*(16*(x^2 + y^2)*t^2 + 32*x*z^2*t - 16*y^3*t + 16*z^4 - 32*y*z^3+ 8*(2*x^2 - 2*x*y + 5*y^2)*z^2 + 8*(2*x^3 - 5*x^2*y - 6*x*y^2 - 7*y^3)*z We use the code below in Section 3 to find the equations of sextic double solids with a cA n singularity.Listing 3. Construct sextic double solids with a cAn singularity /* Language: Maxima 5.42.1 */load("Splitting lemma.mac");splitSDS(poly, n) := subst(1, x, splitting("h", poly + w^2, 6, t, x, n));fGen : -w^2 + x^4*t^2+ x^3*(4*t^3*a_0 + 4*t^2*a_1 + 2*t*a_2 + a_3)+ x^2*(2*t^4*b_0 + 2*t^3*b_1 + 2*t^2*b_2 + 2*t*b_3 + b_4)+ x*(2*t^5*c_0 + 2*t^4*c_1 + 2*t^3*c_2 + 2*t^2*c_3 + 2*t*c_4 + c_5)+ t^6*d_0 + 2*t^5*d_1 + t^4*d_2 + 2*t^3*d_3 + t^2*d_4 + 2*t*d_5 + d_6;h_3 = splitSDS(fGen, 3);sub3(poly) := ratexpand(subst(0, a_3, poly));h_4 = splitSDS(sub3(fGen), 4);sub4(poly) := ratexpand(subst(a_2^2, b_4, sub3(poly)));h_5 = splitSDS(sub4(fGen), 5);sub5(poly) := ratexpand(subst(2*a_2*b_3 - 4*a_1*a_2^2, c_5, sub4(poly)));h_6 = splitSDS(sub5(fGen), 6);sub6(poly) := ratexpand(subst(2*a_2*c_4 + b_3^2 - 8*a_1*a_2*b_3 - 2*a_2^2*b_2+ 4*a_0*a_2^3 + 16*a_1^2*a_2^2, d_6, sub5(poly)));h_7 = splitSDS(sub6(fGen), 7);sub7(poly) := ratexpand(subst(q*r, a_2,subst(q*s + 4*a_1*q*r, b_3,subst(2*a_1*q*s - 6*a_0*q^2*r^2 + 8*a_1^2*q*r + e*r, c_4,subst(2*b_2*q*s - 8*a_1^2*q*s - e*s - b_1*q^2*r^2 + c_3*q*r, d_5,sub6(poly))))));sub71(poly) := ratexpand(subst(1, q, subst(r_2, r, subst(s_3, s, subst(e_2, e,sub7(poly))))));h_8Family1 = splitSDS(sub71(fGen), 8);sub72(poly) := ratexpand(subst(q_1, q, subst(r_1, r, subst(s_2, s,subst(e_3, e, sub7(poly))))));h_8Family2 = splitSDS(sub72(fGen), 8);sub73(poly) := ratexpand(subst(1, r, subst(q_2, q, subst(s_1, s, subst(e_4, e,sub7(poly))))));h_8Family3 = splitSDS(sub73(fGen), 8);sub74(poly) := ratexpand(subst(1, s, subst(0, r, subst(q_3, q, subst(e_5, e,sub7(poly))))));h_8Family4 = splitSDS(sub74(fGen), 8); ub8(poly) := ratexpand(subst(4*A_0*r_2 + b_2 - 6*a_1^2, e_2,subst(r_2*B_1 - 4*s_3*A_0 + 6*a_0*s_3 + 4*a_0*a_1*r_2 - 2*a_1*e_2 + 4*a_1*b_2- 16*a_1^3, c_3,subst(-2*s_3*B_1 + 16*r_2^2*A_0^2 - 8*b_2*r_2*A_0 + 16*a_1^2*r_2*A_0+ 4*b_1*s_3 - 8*a_0*a_1*s_3- 2*b_0*r_2^2 + 2*c_2*r_2 + b_2^2 - 4*a_1^2*b_2+ 4*a_1^4, d_4,sub71(poly)))));h_9 = splitSDS(sub8(fGen), 9); References [Ahm17] Hamid Ahmadinezhad, On pliability of del Pezzo fibrations and Cox rings , J.Reine Angew. 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