Non-rigid quartic 3-folds

Let $X\subset \mathbb{P}^4$ be a terminal factorial quartic $3$-fold. If $X$ is non-singular, $X$ is \emph{birationally rigid}, i.e. the classical MMP on any terminal $\mathbb{Q}$-factorial projective variety $Z$ birational to $X$ always terminates with $X$. This no longer holds when $X$ is singular, but very few examples of non-rigid factorial quartics are known. In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface $X\subset \mathbb{P}^4$. A singular point on such a hypersurface is either of type $cA_n$ ($n\geq 1$), or of type $cD_m$ ($m\geq 4$), or of type $cE_6, cE_7$ or $cE_8$. We first show that if $(P \in X)$ is of type $cA_n$, $n$ is at most $7$, and if $(P \in X)$ is of type $cD_m$, $m$ is at most $8$. We then construct examples of non-rigid factorial quartic hypersurfaces whose singular loci consist (a) of a single point of type $cA_n$ for $2\leq n\leq 7$ (b) of a single point of type $cD_m$ for $m= 4$ or $5$ and (c) of a single point of type $cE_k$ for $k=6,7$ or $8$.


Introduction
A classical problem in algebraic geometry is to determine which quartic hypersurfaces in P 4 are rational. In their seminal paper [IM71], Iskovskikh and Manin prove that a nonsingular quartic hypersurface X 4 ⊂ P 4 is birationally rigid (see the precise definition below) and, in particular, is not rational.
The classical Minimal Model Program (MMP) shows that a uniruled projective 3-fold Z with terminal singularities is birational to a Mori fibre space X/S. More precisely, there is a small morphism f : Z → Z where Z is terminal and Q-factorial (see [KM98,Section 6.3]) and the classical MMP ψ : Z X terminates with a Mori fibre space X/S (see [KM98,Section 3]). Neither the morphism f nor the birational map ψ is unique in general. Mori fibre spaces are end products of the MMP and hence should be seen as distinguished representatives of their classes of birational equivalence. In general, there may be more than one Mori fibre space in a class of birational equivalence. The pliability of a uniruled terminal 3-fold Z is the set of distinguished representatives in its class of birational equivalence, that is: where ∼ denotes the square birational equivalence defined in [Cor95,Definition 5.2]. If X itself is a Mori fibre space, X is called birationally rigid if its pliability is P(X) = {[X]}.
A quartic hypersurface X ⊂ P 4 with terminal singularities is a Mori fibre space precisely when it is factorial, that is when every Weil divisor on X is Cartier. The two conditions on the singularities of a Mori fibre space are quite different: requiring that the singularities of X are terminal is a local analytic condition, while factoriality is a global topological condition. Mella extended Iskovskikh and Manin's result and proved that terminal factorial quartic hypersurfaces with no worse than ordinary double points are birationally rigid (see [Mel04,Theorem 2]). Factoriality is a crucial condition for this to hold. Indeed, general determinantal quartic hypersurfaces are examples of rational nodal quartic hypersurfaces (see the introductions of [Pet98,Mel04]) but are not factorial. Todd discusses several examples of non-factorial rational nodal quartic hypersurfaces: the Burkhardt quartic studied in [Tod36] has 45 nodes (see also [Pet98,Section 5.1]); an example with 36 nodes is mentioned in [Tod33] (see also [Pet98,Example 6.4.2]) and two examples with 40 nodes are studied in [Tod35] (see also [Pet98,Examples 6.2.1, 6.2.2]). In fact, most terminal non-factorial quartic hypersurfaces are rational [Kal12].
However, factoriality alone is not sufficient to guarantee birational rigidity. There are several known examples of non-rigid terminal factorial quartic hypersurfaces: an example with a cA 2 point is studied in [CM04] and entry No. 5 in Table 1 of [Ahm12] is an example with a cA 3 point. In this paper, we show that these examples are not pathological by constructing many examples of non-rigid terminal factorial quartic 3-folds with a singular point of type cA n for n ≥ 2. It is conjectured that a terminal factorial quartic 3-fold with no worse than cA 1 points is rigid; we address this conjecture in forthcoming work.
Terminal 3-fold hypersurfaces have isolated cDV singularities [Rei87, (3.1)]; the local analytic type of a singular point thus belongs to one of two infinite families-cA n for n ≥ 1, or cD m for m ≥ 4-or is cE 6 , cE 7 or cE 8 . The first step in our study is to bound the local analytic type of singularities that can occur on a terminal factorial quartic 3-fold. We use topology and singularity theory to bound the local analytic type in the cA n and cD m case, and we show: Proposition 1.1. If (P ∈ X) is a cA n (resp. cD m ) point on a terminal factorial quartic hypersurface X ⊂ P 4 , then n ≤ 7 (resp. m ≤ 8).
The methods used to prove Proposition 1.1 do not restrict the local analytic type of points of type cE. In fact, all possible local analytic types of cE points are realised: we give examples of terminal factorial quartic hypersurfaces with isolated singular points of type cE 6 , cE 7 or cE 8 (see Example 4.14). As is noted in Remark 3.10, the bound on the local analytic type of cA points is sharp, but we do not believe that the bound is optimal in the cD case .
If X is a terminal factorial quartic 3-fold, the Sarkisov Program shows that any birational map X X ′ to a Mori fibre space X ′ /S ′ is the composition of finitely many Sarkisov links (see Section 2 for definitions and precise statements). Thus X is non-rigid precisely when there exists a link X X ′ where X ′ /S ′ is a Mori fibre space. Such a link is initiated by a morphism f : Z → X, where Z is terminal and Q-factorial, and f contracts a divisor to a singular point or to a curve passing through a singular point. In general, little is known about the explicit form of the morphism f . When f contracts a divisor to a cA n point (P ∈ X), Kawakita shows that the germ of f is a weighted blowup, and classifies possible weights according to the local analytic type of (P ∈ X) [Kaw01, Kaw02,Kaw03].
For each n with 2 ≤ n ≤ 7, we write down the equation of a quartic hypersurface X with a morphism f : Z → X that contracts a divisor to a cA n point and initiates a Sarkisov link. After a suitable embedding of X as a complete intersection in a larger weighted projective space P = P(1 5 , α, β), we recover f as the restriction of a weighted blowup F → P whose weights are determined by Kawakita's classification. The variety F is a toric variety of Picard rank 2, and therefore it is possible to write down explicitly all contracting rational maps F U to a projective variety U . We then check that the birational geometry of F induces a Sarkisov link X X ′ , where X ′ /S ′ is a Mori fibre space. To our knowledge, our construction is the first use of Kawakita's classification to write down explicit global uniruled 3-fold extractions. We prove: Theorem 1.2. If (P ∈ X) is a singular point of type cA n on a terminal factorial quartic 3-fold, then n ≤ 7. There are examples of non-rigid terminal factorial quartic 3-fold with a singular point of type cA n for 2 ≤ n ≤ 7.
We also give examples of non-rigid terminal factorial quartic 3-folds with cD 4 , cD 5 and cE 6 , cE 7 and cE 8 singular points (Examples 4.12, 4.13, and 4.14). We make the following general conjecture, which generalises [CM04, Section 1.3 and Theorem 1.6].
Conjecture 1.3. Let X ⊂ P 4 be a terminal factorial quartic hypersurface. Then P(X) is finite and P(X) = {[X]} precisely when X has no worse than cA 1 singularities. In particular, no terminal factorial quartic hypersurface is rational.
Outline of the paper. Section 2 recalls general results on the Sarkisov program-that is, on the study of birational maps between Mori fibre spaces-in dimension 3 and on the geography of models of Mori dream spaces. When X is a terminal Q-factorial Fano 3-fold with ρ(X) = 1, a Sarkisov link X X ′ is initiated by a morphism f : Z → X that contracts a single divisor. Here, we state precise conditions for f to initiate a Sarkisov link. We urge the reader who is mainly interested in explicit examples and in bounds on singularities to skip this section on a first reading and refer back to it as and when needed.
Section 3 collects results on terminal singularities in dimension 3. We concentrate on the case of terminal Gorenstein singularities, which are those that appear on hypersurfaces. We use the existence of smoothings of terminal Gorenstein Fano 3-folds to bound the local analytic type of singularities on a terminal quartic hypersurface. Last, we recall Kawakita's classification of the germs of divisorial contractions with centre at a cA n point in terms of the local analytic type of that point.
Section 4 presents our examples of non-rigid terminal factorial quartic 3-folds. We consider hypersurfaces X ⊂ P 4 that can be embedded as general complete intersections of type (2, 2, 4) in a weighted projective space P = P(1 5 , 2 2 ). For suitable weighted blowups F : F → P, the restriction f = F |Z : Z → X (where Z is the proper transform of X) is a divisorial contraction with centre at a cA n point, and the birational geometry of F induces a Sarkisov link X X ′ . We give examples of non-rigid quartic hypersurfaces with a cA n point for all 2 ≤ n ≤ 7, and explain our construction in detail in a few cases. We also give examples of non-rigid terminal factorial quartic hypersurfaces with singular points of type cD and cE.
Definition 2.1. Let Z be a normal projective variety and D ∈ Div Q (Z). 1. A birational map f : Z X is contracting if f is proper and f −1 contracts no divisor. The map f is small if both f and f −1 are contracting birational maps. 2. Let D ∈ Div Q (Z) be a Q-Cartier divisor and let f : Z X be a contracting birational map such that f * D is Q-Cartier. The map f is D-nonpositive if for a resolution (p, q) : W → Z × X, where E is effective and q-exceptional. When Supp E contains the strict transform of all f -exceptional divisors, f is said to be D-negative.
normal and projective and D ′ is semiample. If ϕ : X → S is the semiample fibration defined by D ′ , the ample model of D is the composition ϕ • f : Z X → S.
Notation 2.2. If D = K Z is a canonical divisor on Z, we say that a birational contraction is K-nonpositive (resp K-negative) instead of K Z -nonpositive (resp. K Z -negative).
if ϕ is either a morphism whose exceptional locus is a prime divisor on X or a small birational map that fits into a diagram where f and f ′ are morphisms and the Picard ranks of X, X ′ and W satisfy ρ(X) = ρ(X ′ ) = ρ(W ) + 1.
2. Assume that X is Q-factorial and let D be an effective Q-divisor on X. A D-MMP on X is a composition of D-nonpositive elementary contractions X X 1 · · · X n = X D , where X D is a semiample model for D.
Geography of models of Mori dream spaces. We first recall the definition of Mori dream spaces and the properties that will be important in this paper.
Definition 2.4. [HK00] Let Z be a projective Q-factorial variety with Pic(Z) Q = N 1 Q (Z); Z is a Mori dream space if (i) Nef(Z) is the affine hull of finitely many semiample line bundles, (ii) there are finitely many small birational maps f i : . When Z is a Mori dream space, we may run the D-MMP for every Q-divisor D. More precisely, there is a finite decomposition [KKL14, Section 5]: (a) C i is a rational polyhedral cone, (b) there is a birational contraction to a Q-factorial normal projective variety ϕ i : Z Z i that is the ample model of all D ∈ C i and a semiample model of all D ∈ C i . A Mori dream space Z with ρ(Z) = 2 always has a 2-ray configuration, which is defined as follows. Let M 1 , M 2 be Q-divisors such that Denote by ϕ 1 : Z Z 1 (resp. ϕ 2 : Z Z 2 ) the ample model of M 1 + εM 2 (resp. εM 1 + M 2 ) for an arbitrarily small positive rational number ε. Let f i : Z i → X i be the ample model of (ϕ i ) * M i . Then, the birational map ϕ i is small, and f i is a fibration when [M i ] lies on the boundary of Eff Z, and ϕ i is a birational map that contracts a single exceptional divisor otherwise. These maps fit in a diagram which we call a 2-ray configuration: When f : Z → X is a divisorial contraction, we may assume that ϕ 1 is the identity map and that X is the ample model of M 1 (i.e. that f = f 1 ).
The Sarkisov Program. We recall a few notions on birational maps between end products of the classical MMP.
Definition 2.5. Let X be a terminal Q-factorial variety, and p : X → S a morphism with positive dimensional fibres (so that dim S < dim X). Then X/S is a Mori fibre space if p * O X = O S , −K X is p-ample and ρ(X) = ρ(S) + 1.
The classical MMP shows that any uniruled terminal Q-factorial variety Z is birational to a Mori fibre space, so that P(Z) = ∅. The Sarkisov program decomposes any birational map between Mori fibre spaces [X/S], [X ′ /S ′ ] ∈ P(Z) into a finite number of Sarkisov links [Cor95,HM13]. Next, we recall the definition of Sarkisov links.
Definition 2.6. A divisorial contraction f : Z → X is a morphism between terminal Q-factorial varieties such that −K Z is f -ample, f * O Z = O X , and ρ(Z) = ρ(X) + 1.
We sometimes call f an extraction when we study properties of f in terms of its target X.
Definition 2.7. Let X/S and X ′ /S ′ be two Mori fibre spaces. A Sarkisov link is a birational map ϕ : X X ′ of one of the following types: (I) A link of type I is a commutative diagram: where Z → X is a divisorial contraction and Z X ′ a sequence of flips, flops and inverse flips between terminal Q-factorial varieties; (II) A link of type II is a commutative diagram: where Z → X and Z ′ → X ′ are divisorial contractions and Z Z ′ a sequence of flips, flops and inverse flips between terminal Q-factorial varieties; (III) A link of type III is the inverse of a link of type I; (IV) A link of type IV is a commutative diagram: where X X ′ is a sequence of flips, flops and inverse flips between terminal Q-factorial varieties.
Definition 2.8. Let X/S be a Mori fibre space and f : Z → X an extraction; f initiates a link if it fits into an Sarkisov link.
The following lemma is a criterion for a divisorial extraction to initiate a link. It is of little practical use, but we want to highlight some of the subtleties that arise when proving that a 2-ray configuration is indeed a Sarkisov link.
Lemma 2.9. Let X be a terminal Q-factorial Fano variety with ρ(X) = 1 and let f : Z → X be an extraction. Then f initiates a link if and only if the following hold: Proof. Assume that f : Z → X is a divisorial contraction that initiates a link. Then, as X is a Fano 3-fold with rational singularities, h 1 (X, O X ) = 0, and since f * O Z = O X , by the Leray spectral sequence, h 1 (Z, O Z ) = 0 and we have the equality Pic(Z) Q = N 1 Q (Z). Since ρ(Z) = 2, if f initiates a Sarkisov link, then following the notation of Definition 2.7, there are 2 distinct birational contractions Z → X and Z X ′ that are compositions of finitely many elementary maps (flips, flops, inverse flips and divisorial contractions) between terminal Q-factorial varieties. Thus Z has a 2-ray configuration as above, and Z is automatically a Mori dream space. The chambers of the decomposition of Eff Z are indexed by the divisorial contraction Z → X and by the elementary maps that decompose Z X ′ . Furthermore, if X ′ is Fano, then and the class of −K Z is in the interior of Mov Z because X and X ′ have terminal singularities and Z → X and Z ′ → X ′ are not isomorphisms. If X ′ /S ′ is a Mori fibre space with dim S ′ ≥ 1, then K X ′ = ψ * K Z , and where A S ′ is the pullback of a suitable ample divisor on S ′ and the class of −K Z is in the interior of Mov Z because, as before, X is terminal so that We have seen that Z is a Mori dream space so that if D is any movable Q-divisor, the D-MMP terminates with a Q-factorial semiample model for D which we denote Z D . The small birational map Z Z D factors Z X ′ , therefore Z Z D is the composition of finitely many elementary contractions between terminal Q-factorial varieties, and in particular, Z D is terminal. Let Z Z ′ be an arbitrary small birational map, and assume that Z ′ is Q-factorial. Let D be the proper transform of an ample Q-Cartier divisor on Z ′ ; then D is mobile because Z Z ′ is small. By construction, Z ′ ≃ Proj R(Z, D) is the ample model of D, and if Z D is the end product of a D-MMP on Z, then Z D → Z ′ is a morphism and a small map. Since Z ′ and Z D are both Q-factorial, it follows that they are isomorphic, so that Z ′ has terminal singularities.
Conversely, if Z is a Picard rank 2 Mori dream space then be the ample model of M 2 + εM 1 for an arbitrarily small positive rational number ε and let X ′ be the ample of model of M 2 . Then, Z Z ′ is a small birational map, and Z ′ has terminal singularities by assumption (ii). The birational map Z Z ′ is small, hence we may identify divisors on Z and on Z ′ and, under this identification, Mov Z is equal to Mov Z ′ , so that [−K Z ′ ] is in the interior of Mov Z ′ by assumption (iii). It follows that the morphism Z ′ → X ′ is K-negative and that X ′ has terminal singularities. When M 2 = D 2 , let X ′ be the ample model of M 2 + εM 1 for an arbitrarily small positive rational number ε and let S ′ be the ample model of M 2 . Then, X ′ is terminal by assumption (ii) and the fibration X ′ → S ′ is K-negative by assumption (iii). Since M 2 is not a big divisor, dim S ′ < dim X ′ and X ′ /S ′ is a Mori fibre space.
Remark 2.10. Condition (ii) may only fail when the 2-ray configuration on Z involves an antiflip because flips and flops of terminal varieties are automatically terminal. For example, consider then Z is a Mori fibre space (a P 2 -bundle over P 1 ) and a Mori dream space on which (i), (iii) hold but (ii) fails. It follows that the 2-ray configuration on Z does not produce a Sarkisov link. Example 4.11 is a Sarkisov link involving an antiflip, and we check that condition (ii) holds directly.
Remark 2.11. Note that condition (ii) always holds when M 2 is of the form K Z + Θ for Θ a nef divisor. Indeed, in that case, every D ∈ Mov Z ∩ Big Z is of the form K Z + Θ ′ for Θ ′ nef, and every D-negative birational contraction is K-nonpositive (see [Kal13,2.10]). In particular, the ample model ϕ D : Z Z D is a D-negative birational contraction, hence is K Z -nonpositive, so that for any resolution (p, q) : U Z × Z D , we have: where E is an effective q-exceptional divisor. This implies that for any divisor F over Z D , the discrepancy a F (Z D ) ≥ a F (Z), and that Z D has terminal singularities if Z does.

Terminal singularities on quartic 3-folds
In this section, we recall some results on the local analytic description of terminal hypersurface singularities in dimension 3 and we bound the local analytic type of singularities on terminal factorial quartic hypersurfaces in P 4 .
3.1. Local analytic description and divisorial extractions. We first recall a few results on isolated hypersurface singularities.
Singularity theory. Let C[[x 1 , · · · , x n ]] be the ring of complex formal power series in n variables, and C{x 1 , · · · , x n } ⊂ C[[x 1 , · · · , x n ]] the subring of formal power series with nonzero radius of convergence. For F ∈ C{x 1 , · · · x n }, (F = 0) is a germ of a complex analytic set, and the ] be a power series, and d a positive integer. We denote by F d the degree d homogeneous part of F and by F ≥d the series k≥d F k . The multiplicity of F is mult F = min{d ∈ N|F d = 0}. Two power series F, G are equivalent if there exist an automorphism In other words, F and G are equivalent if the singularities (F = 0) and (G = 0) are isomorphic. We denote the equivalence of power series by F ∼ G.
In what follows, as we are only interested in isolated critical points, by [AGZV85, I, Section 6.3] we may (and will) assume that the power series u, F and G all have nonzero radius of convergence and that ϕ ∈ Aut(C{x 1 , · · · x n }).
1. The singularity (h(x, y, z) = 0) is Du Val if h is equivalent to one of the standard forms: ] define a cA singularity, then one of the following holds: Remark 3.3. Up to change of coordinates on P 1 z,t , we may assume that z n+1 appears with coefficient 1 in f ≥n+1 (z, t). Since (F = 0) is an isolated singularity, f ≥n+1 (z, t) has no repeated factor and contains at least one monomial of the form t N or zt N −1 for N ≥ n+1. When N > n+1, as in [AGZV85, Section 12], where f n+1 is a homogeneous form with no repeated factor of degree n + 1.
] define a cD singularity, then one of the following holds: where a ∈ C, r ≥ 3, s ≥ 4 and h s = 0. This has type cD m for m = min{2r, s + 1} if a = 0 and m = s + 1 otherwise.
Definition 3.5. The Milnor number of the singularity (F = 0) is is the Jacobian ideal of F , i.e. the ideal generated by the partial derivatives of F .
If F ∼ G, the Milnor numbers of (F = 0) and of (G = 0) are equal. The Milnor number µ(F = 0) is finite precisely when (F = 0) is an isolated singularity.
Proof. If (F = 0) is a cA n singularity with n ≥ 1, then F ∼ xy + f (z, t), where f (z, t) has multiplicity greater than or equal to n + 1. Since In all cases, since µ(F = 0) is finite and f has no repeated factor, C[[z, t]]/J f has dimension deg ∂f ∂z · deg ∂f ∂z ≥ n(N − 1). Now assume that (F = 0) is a cD m singularity with m > 4. Then, as in [AGZV85, I, Section 12], if F 0 is the quasihomogeneous part of F then µ(F = 0) = µ(F 0 = 0), and µ(F 0 = 0) is given by the formula [AGZV85, I, Corollary 3, p. 200]. In the notation of Theorem 3.4, using the methods of [AGZV85, I, Section 12], we obtain that: 3.2. Bounding the local analytic type of singularities on a terminal factorial Fano Mori fibre space. We bound the local analytic type of singularities on a terminal factorial Fano 3-fold with Picard rank 1.
[Nam97] Let X be a Fano 3-fold with terminal Gorenstein singularities. Then X has a smoothing, i.e. there is a one parameter flat deformation: Xt . The existence of a smoothing X ֒→ X allows us to bound the Milnor numbers of singularities on X.
When X is a terminal and factorial Fano 3-fold, the second and fourth Betti numbers of X are equal, that is b 2 (X) = b 4 (X), so that: The third Betti numbers of non-singular Fano 3-folds with Picard rank 1 are known (see [IP99, Table 12.2]), and we obtain a bound on the sum of Milnor numbers of singular points on X that only depends on −K 3 X . When −K 3 X = 4, we have the following. Proposition 3.9. Let X ⊂ P 4 be a terminal factorial quartic hypersurface. If (P ∈ X) is a singular point of type cA n , then n is at most 7.
If (P ∈ X) is a singular point of type cD m , then m is at most 8.
Proof. Let X ֒→ X be a smoothing, then for all t = 0, X t is a nonsingular quartic hypersurface and b 3 (X t ) = 60 (see [IP99, Table 12.2]). By Theorem 3.8, we have µ(X, P ) is bounded above by 60, and the result follows immediately from the lower bounds obtained in Lemma 3.6.
Remark 3.10. The bound on the local analytic type of cA points is sharp, Example 4.9 is an example of a terminal factorial quartic hypersurface with a cA 7 singular point. We do not believe that the bound on the local analytic type of cD points is optimal, as we have not been able to write down examples attaining it. We give examples of terminal factorial quartic hypersurfaces with isolated singular points of type cD 4 , cD 5 and cE 6 , cE 7 and cE 8 in Section 4.3.
Remark 3.11. By the classification of non-singular Fano 3-folds, the bounds on the local analytic type of singularities lying on a terminal factorial Fano 3-fold with Picard rank 1 and anticanonical degree −K 3 > 4 are even more restrictive than in the case of a quartic hypersurface.
Remark 3.12. We can use the same methods to bound the local analytic type of singularities on an arbitrary terminal Gorenstein Fano 3-fold X with ρ = 1. Indeed, by Theorem 3.8: where σ(X) = b 4 (X) − 1 is the defect of X. The defect of terminal Gorenstein Fano 3-folds with ρ = 1 is bounded in [Kal11]. For example, if X is a (not necessarily factorial) terminal quartic hypersurface, then Our main interest in this article is in non-rigid quartic 3-folds; in particular, we have not tried to write down examples of non-factorial quartic hypersurfaces with a singular point of type cA 8 . We believe that such an example would be found with extra work.
3.3. Divisorial extractions with centre at a cA n point. Kawakita classifies the germs of divisorial extractions f : Z → X with centre at a cA n point. We recall this classification here, as we will use it in Section 4.
Remark 3.14. Let X ⊂ P 4 be a terminal factorial quartic hypersurface. Then, the discrepancies of possible divisorial extractions f : Z → X with centre at a cA n point (P ∈ X) can be bounded in the same way as in Proposition 3.9. Indeed, by Lemma 3.6, if there is an extraction of general type in Theorem 3.13, the Milnor number µ(X, P ) satisfies µ(X, P ) ≥ n(a(n + 1) − 1), but by Theorem 3.8, µ(X, P ) ≤ 60. It follows that the discrepancy of f satisfies: n 6, 7 5 4 3 2 a 1 ≤ 2 ≤ 3 ≤ 5 ≤ 11

Examples of non-rigid terminal quartics
In this section, we present examples of non-rigid terminal factorial quartic hypersurfaces X ⊂ P 4 . Each of these examples has a Sarkisov link initiated by an extraction f : Z → X that contracts a divisor to a singular point. In most of our examples, the singular point is of type cA n , so that the germ of f is a weighted blowup as in Kawakita's classification (Theorem 3.13). Our examples are obtained by globalising these germs: we can write down an explicit description of f in projective coordinates.
Let X ⊂ P 4 be a terminal factorial quartic hypersurface, and assume that P = (1:0:0:0:0) ∈ X is a cA n point. Up to projective change of coordinates, the equation of X can be written: , where ϕ 4 is a homogeneous polynomial of degree 4 in the variables x 0 , · · · , x 4 and ψ 3 and θ 4 are homogeneous polynomials in the variables x 1 , · · · , x 4 . The first step in our construction is to look for examples of hypersurfaces X ⊂ P 4 that can be embedded in a larger weighted projective space P in such a way that the restriction of a suitable weighted blowup F : F → P is a divisorial contraction f : Z → X. We take X to be a complete intersection of the form: for homogeneous forms α, β of degree 2. The equation of the hypersurface X ⊂ P 4 is recovered by substituting α, β in the third equation. Explicitly, we want the germ of f : Z → X to be a weighted blowup of general type in the classification of Theorem 3.13. This means that up to local analytic identification, denoting by a the discrepancy of f , we have (6) (P ∈ X) ≃ 0 ∈ {xy + g(z, t) = 0} ⊂ C 4 , where g(z, t) = z n+1 + g ≥a(n+1) (z, t) and g has weighted degree a(n + 1). We choose ϕ 4 so that for suitable α, β, setting x 0 = 1 in the third equation gives: αβ + g(x 3 , x 4 ) + (higher weighted order terms) = 0.
In other words, the restrictions of α, β to {x 0 = 1} define the local analytic isomorphisms that bring the equation of X ∩{x 0 = 1} into the desired form (6). The divisorial contraction f : Z → X we construct are restrictions of weighted blowups F : F → P(1 5 , 2 2 ), where the weights assigned to the variables α, β, x 3 , x 4 are as in Theorem 3.13. The second step in our construction is to show that some of these divisorial contractions initiate Sarkisov links. Since F is a Mori dream space (it is toric), it has a 2-ray configuration as in (1). We check directly that the 2-ray configuration or that we can find another embedding Z ⊂ F ′ via unprojection such that the 2-ray configuration on F ′ restricts to a Sarkisov link. Note that we do not make any assumption on the singularities of F, in particular, F needs not be terminal and Q-factorial. We check the following: 1. The map Φ |Z is an isomorphim in codimension 1 and Z + is terminal. In our examples, Φ |Z is the composition of finitely many elementary maps that are isomorphisms or antiflips, flops and flips (in that order). If Z i ϕ i Z i+1 is Knonpositive (e.g. a flip or flop) and Z i is terminal and Q-factorial, then so is Z i+1 . We need to check directly that antiflips preserve the terminal condition; we do this by identifying the antiflips as inverses of flips appearing in [Bro99]. 2. The restriction of F + → P + is the contraction of a K-negative extremal ray Z + → X + .
Remark 4.1. We do not give details of how to check that each of our examples is factorial. This relies on ad hoc methods and the general scheme is as follows. As X ⊂ P is a Picard rank 1 hypersurface, X is (Q-)factorial precisely when the rank of the divisor class group satisfies rkCl(X) = ρ(X) = rkCl(P).
Since Sing X has codimension 3, we have an isomorphism Cl(X) ≃ Cl(X Sing(X)).
Let π : P → P be a map from a smooth variety P that restricts to a resolution X → X with exceptional locus E X . We have natural isomorphisms and similarly Cl(P) ≃ Pic( P Exc π).
Now, X is factorial precisely when Pic( P Exc π) ≃ Pic( X E X ). As the classical Grothendieck-Lefschetz theorem guarantees that Pic( X) ≃ Pic( P), the result follows by comparison of the kernels of the surjective maps r 1 : Pic( P) → Pic( P Exc π) and r 2 : Pic( X) → Pic( X E X ).
These kernels are isomorphic to the free abelian groups on irreducible divisorial components of Exc π and of E X respectively, and can be worked out in each case. Note that we have 2-ray configuration (7), and therefore all varieties in (7) are Q-factorial if and only if one of them is. These varieties are complete intersections in P, F, F + and P + and their Picard ranks are 1, 2, 2 and 1 by construction, and the method above can be applied to determine the divisor class group of any of them. In some cases, it can be easier to determine Q-factoriality of another variety in (7) than X (one needs to keep track of Gorenstein indices).
Since G is a semi quasi-homogeneous polynomial of degree 1 with respect to w, and since no element of its Jacobian algebra has degree strictly greater than 1, by [AGZV85, I, Section 12], G ∼ xy + z n+1 + t n+1 and (P ∈ X) is a cA n point. The quartic hypersurfaces X i,j are terminal (Sing X i,j = {P }) and factorial, and hence are Mori fibre spaces.
Remark 4.2. Taking (i, j) = (4, 0) gives a terminal quartic hypersurface with a cA 7 point is not factorial as f = q 1 q ′ 1 + q 2 q ′ 2 , where q i , q ′ i are quadric polynomials. Notation 4.3. We embed X i,j as a complete intersection in a scroll whose coordinates are those of P 4 on the one hand, and projectivisations of the (non-linear) components of ϕ ∈ Aut C[[x, y, z, t]], the automorphism we used to transform the equation of X ∩ U 0 into G. Here, ϕ 3 (x, y, z, t) = z and ϕ 4 (x, y, z, t) = t, so we only need to introduce the coordinates: .
With the grading defined in (9), taking the ample model of a divisor whose class is in 0 k is the morphism given by: This is precisely the weighted blowup F : F → P(1 5 , 2 2 ) we are after.
In what follows, we always denote by L the pullback of O P (1)-so that L ∈ 0 1 -and by The form of the expression (8) imposes that the discrepancy of f is 1. We have −K Z = f * (−K X ) − E, where E = Exc f , so that a = 1 and r 1 + r 2 = n + 1, where (n, (i, j)) are as in Table 1. Set r 1 = r, r 2 = n + 1 − r, and assume as we may that r ≤ n + 1 − r. We have: Lemma 4.4. The weighted blowup F : F → P restricts to a divisorial contraction f : Z → X with discrepancy 1 if the weights r, w 1 , w 2 of F in (9) are one of: 1. i = 0 (n = 2 or 3) and w 1 = w 2 = 1, r = 1 or 2, 2. w 1 = w 2 = 2 and r ≥ 2 (n ≥ 3).
Proof. The 3-fold Z is a general complete intersection of 3 hypersurfaces of degrees determined by r, w 1 and w 2 . Once these degrees are known, we use adjunction to write the anticanonical class of Z. Since −K X ∈ O(1) and a = 1, −K Z ∼ L − E and this yields the possible values for r, w 1 and w 2 . For example, if w 1 , w 2 , r ≥ 2, Z is the proper transform of X under F and it is a complete intersection: and this forces w 1 = w 2 = 2. Other cases are entirely similar.
We check that, with one exception labelled "bad link", for all weights in Lemma 4.4, the 2-ray configuration on F induces a 2-ray configuration on T V (I, A) that induces a Sarkisov link for X i,j . In the case of the bad link, the second birational contraction g : Z Y has relatively trivial canonical class, so that Y is not terminal and the 2-ray configuration is not a Sarkisov link. Table 2 gives details of the construction of each Sarkisov link.
Theorem 4.5. There are examples of non-rigid terminal factorial quartic hypersurfaces in P 4 with an isolated cA n point for all 2 ≤ n ≤ 6. For each combination (i, j), n in Table 1, X i,j ⊂ P 4 is a non-rigid terminal factorial quartic 3-fold with an isolated singular point (P ∈ X) of type cA n . Table 2 lists Sarkisov links initiated by a divisorial contraction f : Z → X i,j with centre at (P ∈ X) and discrepancy 1. Each entry specifies the weights of (α, β, x 3 , x 4 ) for the germ of f in the notation of Theorem 3.13, and gives the explicit construction of the link.
Proof. Each case is treated individually. To illustrate the computations involved, we treat the cA 6 case in detail. We then say a few words about the cA 2 case, where we recover the example of a non-rigid quartic constructed in [CM04].
Remark 4.6. The case labelled as (⋆) in Table 2 is a quadratic involution in the language of [CPR00]. In particular, the link does not produce a new Mori fibre space; it is just a birational selfmap of X 1,2 that is not an isomorphism.
Non-rigid quartic with a cA 6 singular point. Consider the terminal, factorial quartic hypersurface: 1 + x 4 2 = 0} ⊂ P 4 . As above, we embed X as a complete intersection in P = P(1 5 , 2 2 ), where the variables of weight 2 are α, β: We construct Sarkisov links initiated by a divisorial contraction f : Z → X, which is the restriction of a weighted blowup F : F → P. We assume that the weights assigned to the variables (α, β, x 3 , x 4 ) are those in Theorem 3.13. By Theorem 3.13 and Lemma 4.4, F is the Picard rank 2 toric variety T V (I, A), where A is of the form   (5, 2, 1, 1).

Case 1. The germ of f is a blowup with weights
We re-order the coordinates of P and write the action A as follows: In this case, Z is the complete intersection: Taking the difference of the first two equations shows that the variable β is redundant and: where we now denote by F the toric variety T V (I, A), for The 2-ray configuration on F is: where Φ is a small map that is the ample model for L − (1 + ε)E and G : F + → P + is a divisorial contraction, where P + ≃ Proj(F, n(L − 2E)) for suitable n > > 1. The only pure monomials in u, x 0 , x 3 , x 4 in the equation of Z are in the expression x 2 3 + x 2 4 in the first equation, so that the restriction of Φ 1 to Z is a flop in 2 lines (a copy of P 1 u,x 0 above each of the two points {x 2 3 + x 2 4 = 0} ⊂ P 1 x 3 ,x 4 ), and since α doesn't divide the equations of Z, G does restrict to a divisorial contraction.
To determine P + , we find a suitable change of basis in which to express the action A. In practice, we look for a matrix M in Sl 2 (Z) such that The matrix M = 1 3 −2 −5 transforms the action A into: so that P + = P(1 2 , 2, 3 2 , 5), with coordinates x 1 , x 2 (degree 1), uα (degree 2), x 3 α, x 4 α (degree 3) and x 0 α 2 (degree 5). Writing the equations of the proper transform of Z + shows that Y ⊂ P + is so that Y is the complete intersection of two hypersurfaces of degree 6 in P + . Since x 3 α is a section of g * O P + (3), the 3-fold Y has Fano index 3, and by construction of f , its basket consists of a single [5, 2] singular point at P α ; Y is the Fano variety number 41920 in [GRD].
We re-order the coordinates of P, and write the action A: The 3-fold Z is the complete intersection: The 2-ray configuration on F is: where Φ 1 is a small map and F 1 the ample model of L − (1 + ε)E, Φ 2 is a small map and F 2 the ample model of 2L − (3 + ε)E, and G is a divisorial contraction to P + ≃ Proj(F, n(L − 2E)) for suitable n > > 1. Since x 2 3 , x 2 4 appear in two of the equations defining Z, Z does not contain any curve contracted by Φ 1 and Φ 1|Z is an isomorphism. We still denote by Z its image under Φ 1 .
We show that Φ 2|Z is a flip. We study the behaviour of Z near P β = (0:0:0:0:1:0:0:0). The restriction of Z to U β = {β = 1} is a hypersurface: we may use the second and third equation to eliminate u and α, so that Z ∩ U β is the hypersurface defined by the first equation. As above, under the change of coordinates associated to 1 1 2 3 ∈ SL 2 (Z), the action becomes: so that once the variables u, α and the second and third equations defining Z have been eliminated, we are left with a flip of the hypersurface defined by the first equation of Z in C x 0 ,x 3 ,x 4 ,x 1 ,x 2 , which is x 0 (x 1 + x 2 ) + x 2 3 + x 2 4 + · · · = 0, that is, in the notation of [Bro99], of the form (3, 1, 1, −1, −1; 2). There are thus 2 flipped curves, and while Z has a cA/3 singularity over P β , Z 2 is Gorenstein over P β . The map G is a fibration over P(α, x 1 , x 2 ) = P (1, 1, 2), and the equations of Z show that the restriction Z 2 → P (1, 1, 2) is a conic bundle. Non-rigid quartic with a cA 2 point. Consider the terminal, factorial quartic hypersurface By Lemma 4.4, after re-ordering the coordinates of P, F can only be the Picard rank 2 toric variety T V (I, A) and Z is given by the equations: 2 ) = 0. The first equation shows that we may eliminate the variable α. The first step of the 2-ray configuration of F is a small map but introduces a new divisor on Z, hence is not a step in the 2-ray game of Z. Note that the second equation is in the ideal (u, x 0 ), we will re-embed Z by unprojection into a toric variety of Picard rank 2 whose 2-ray configuration restricts to suitable maps on Z. To do so, we introduce an unprojection variable s and replace the second equation with: From the two equations that define s, we eliminate the variable β and x 1 so that the (isomorphic) image of Z under the unprojection is: Again, we see that the equation defining Z is in the ideal (u, x 0 ). If we denote by f, g (non-unique) polynomials such that Z = {x 0 f + ug = 0}, so that we have f = x 3 3 + x 3 4 + · · · and g = x 4 1 + (us) 4 + · · · , and introduce a second unprojection variable then we see that Z is the complete intersection The 2-ray configuration of F is where Φ is a small map and F + the ample model of L − (1 + ε)E and G is a divisorial contraction. The restriction Φ |Z : Z Z + is a flop in 12 lines that are copies of P 1 u,x 0 lying over the 12 points {x 4 1 + x 4 3 − x 4 4 = x 3 2 + x 3 3 + · · · = 0} ⊂ P x 1 ,x 3 ,x 4 .
Since Z ⊂ {s = 0}, the restriction G |Z + : Z + → Y is a divisorial contraction. As above, applying the coordinate change for the action associated to and we see that P + = P(1 4 , 2 2 ), with coordinates su, x 1 , x 3 , x 4 (degree 1), sx 0 , st (degree 2). The proper transform of Z is i.e. Y is the complete intersection of a cubic and a quartic hypersurface in P + , a Fano 3-fold of codimension 2 and genus 2. The map Z + → Y is a Kawamata blowup of one of the two 1/2(1, 1, 1) points on Y . This link is constructed in [CM04].
Remark 4.7. In several cases, one or both of the variables α, β are redundant. This means that f : Z → X is the restriction of a weighted blowup of some P ′ with P 4 ⊂ P ′ ⊂ P: the construction could have been obtained with a "smaller embedding". For example, this is the case in our treatment of a terminal factorial quartic hypersurface with a cA 2 point: [CM04] construct the same link without introducing α, β. In a given example, it is usually clear how many (if any) variables need to be introduced. We have chosen to always introduce two variables (and then eliminate redundant ones) in order to present our results in a unified way.
Remark 4.8. It is crucial to understand that we make no claim about the existence of Sarkisov links initiated by divisorial contractions f : Z → X whose germs have weights different from those in Lemma 4.4, where X is one of the hypersurfaces X i,j . Such divisorial contractions may occur, but they are not restrictions of weighted blowups of P = P(1 5 , 2 2 ). We expect that in some cases, one may construct such contractions by considering a different embedding of X ⊂ P ′ and looking at restrictions of weighted blowups of P ′ . For instance, we do not know wether X 3,1 admits a Sarkisov link initiated by a divisorial contraction whose germ is a weighted blowup (6, 1, 1, 1).

4.2.
Other examples with cA singularities. In this section, we use similar techniques to give an example of a non-rigid terminal factorial quartic hypersurface with a cA 7 singular point. These can also be used to construct Sarkisov links initiated by divisorial contractions with discrepancy a > 1 and centre at a cA n point for n ≥ 2, where the possible values a, n are determined in Proposition 3.9. We give an example with n = 2 and a = 2.
Since G is a semi quasi-homogeneous polynomial of degree 1 with respect to w, and since no element of its Jacobian algebra has degree strictly greater than 1 with respect to these weights, and (P ∈ X) is a cA 7 point. The hypersurface X ⊂ P 4 is a terminal and factorial quartic hypersurface, and hence is a Mori fibre space. As in Section 4.1, we embed X as the complete intersection We consider the weighted blowup F : F → P, where F is the Picard rank 2 toric variety T V (I, A), where I = (u, x 0 ) ∩ (x 1 , · · · , x 4 , α, β) is the irrelevant ideal and A is the action of C * × C * with weights: and we check that the restriction of F to Z is indeed a divisorial contraction f : Z → X with discrepancy a = 1. The 2-ray configuration on F is: where Φ is a small map and F + is the ample model of L − (1 + ε)E and G is a fibration morphism and P + = P(1, 1, 2, 2) = P x 1 ,x 2 ,α,β is the ample model Proj(n(L − 2E)) for suitable n > > 1. The restriction of Φ to Z is an isomorphism because the monomials x 2 3 , x 2 4 appear in the first two equations, we still denote by Z its image. The restriction of G to Z is a conic bundle over the quartic surface S 4 ⊂ P(1, 1, 2, 2) defined by the third equation of Z. Each of the Sarkisov links we have constructed so far is initiated by a divisorial contraction with centre at a cA n point and discrepancy a = 1. We now construct an example with higher discrepancy.
Example 4.11. We construct a Sarkisov link initiated by a divisorial contraction with centre at a cA 2 point. Unlike the previous examples, which only involved flips and flops, the Sarkisov link in this example involves an antiflip.
Consider the terminal, factorial quartic hypersurface Then Sing X = {P, P 4 }, and, setting x 0 = 1 in the expression above, we see that: (P ∈ X) ∼ 0 ∈ {xy + z 3 + t 6 + (higher order terms) = 0}, so that P is a cA 2 point and a divisorial contraction with centre at (P ∈ X) has discrepancy 1 or 2 by Theorem 3.13. We embed X as If we consider a divisorial contraction with discrepancy 1, we obtain a link of the same form as above. We now consider the case when f is a divisorial contraction with discrepancy 2. Then, by Theorem 3.13, the weights of (α, β, x 3 , x 4 ) are either (1, 5, 2, 1) or (3, 3, 2, 1). We consider the second case, and as in Lemma 4.4, we show that F is a toric variety T V (I, A), where  , for (p, q) = (2, 2), (2, 3) or (3, 2).
We consider the case when (p, q) = (2, 2), so that, after re-ordering, and the proper transform of X is given by the equations: As in the proof of the cA 2 case in Theorem 4.5, the first equation is in the ideal (u, x 0 ), we re-embed Z so that it follows the ambient 2-ray configuration: we introduce an unprojection variable s such that: We then see that the variables x 1 and α are redundant so that the expression of Z is x 0 (sβ + x 3 3 + x 3 2 ) + u 2 ((su) 4 + x 4 2 + x 4 3 ) = 0.
As above, since the second equation is in the ideal (u 2 , x 0 ), we need to introduce a second unprojection variable η such that We now get that Z is the complete intersection where F denotes the toric variety T V (I, A), for The 2-ray configuration on F is: -Φ 1 : F F 1 is a small map, and F 1 is the ample model of L − (1 + ε)E, -Φ 2 : F 1 F 2 is a small map, and F 2 is the ample model of 2L − (3 + ε)E, -Φ 3 : F 2 F 3 is a small map, where F 3 is the ample model of L − (2 + ε)E, -G : F 3 → P + is a divisorial contraction, and P + is the ample model of 8L − 3E.
We study the restriction of this 2-ray configuration to Z. Since the monomial x 2 4 appears in one of the equations of Z, the restriction Φ 1|Z is an isomorphism. We still denote by Z its image.
We now prove that Φ 2|Z : Z Z 2 is a small birational map. Since it is a K-positive contraction, we also need to prove that Z 2 has terminal singularities. The exceptional locus of Φ 2|Z is at most 1-dimensional, as the only pure monomial in u, x 0 , x 4 that appears in the equations of Z is x 2 4 .The exceptional locus of (Φ 2|Z ) −1 is also at most 1-dimensional, as pure monomials in x 2 , x 3 , η, s appear in two of the three equations defining Z.
In order to study Φ 2|Z , we localise near P β . Setting β = 1, we use the first two equations to eliminate the variables u and s, so that X ∩ {β = 1} is the hypersurface defined by the third equation: {ηx 0 = x 4 2 + x 2 3 + (ηu 2 − x 3 3 − x 3 2 ) 4 (x 0 x 2 − x 2 4 ) 4 = 0} As above, under the change of coordinates associated to 1 −1 −2 3 ∈ SL 2 (Z), the action becomes: so that once the variables u, s and the second and third equations defining Z have been eliminated, we are left with the inverse of a flip of the hypersurface defined by the first equation of Z in C η,x 2 ,x 3 ,x 0 ,x 4 , which is, in the notation of [Bro99], of the form (7, 1, 1, −3, −1; 4). The map Φ 2|Z is thus an antiflip between 3-folds with terminal singularities. The exceptional locus of Φ 2|Z is empty because the equation of X ∩ U β has no pure monomial in x 0 , x 4 ; the exceptional locus of (Φ 2|Z ) −1 consists of 4 lines {x 4 2 + x 4 3 = 0} ⊂ P(1, 1, 7) = P x 2 ,x 3 ,η . By the construction of f , the basket of Z consists of two [3, 1] singularities, one of which lies over P β . From [Bro99], the basket of Z 2 consists of one [7, 1] singular point and one [3, 1] singular point.
In summary, we have constructed a Sarkisov link from a terminal factorial quartic hypersurface X with a cA 2 point (P ∈ X) to a complete intersection Y 4,4 ⊂ P(1 4 , 2, 3) with a cE 7 point (Q ∈ Y ), which is of the form: -f is a discrepancy 2 divisorial contraction with centre at P , -ϕ = ϕ 3 • ϕ 2 • ϕ 1 , with ϕ 1 , ϕ 3 isomorphisms and ϕ 2 an antiflip, -g is a discrepancy 2 divisorial contraction with centre at Q.

4.3.
Examples with cD and cE singularities. We now give examples of non-rigid factorial quartic hypersurfaces with singular points that are not of type cA. The study of the pliability of quartics with cD m and cE 6,7,8 singular points is complicated by the fact that, unlike in the cA n case, there is no classification of the germs of divisorial extractions f : Z → X with centre at a cD or cE point. We only know the germs of a few explicit divisorial extractions with these centers: those that are weighted blowups with discrepancy 1.
The following examples are non-rigid quartic hypersurfaces with a cD or cE singular point.
Note the similarity of this link to the link between a quartic X ′ ⊂ P 4 with a cA 2 point and a general Y 3,4 ⊂ P(1 4 , 2 2 ) studied in [CM04]. In our case, Y is a special quasi-smooth model in its family, and we conjecture that as in [CM04] X is birigid, i.e. P(X) = {[X], [Y ]}.