Quartic surfaces up to volume preserving equivalence

We study log Calabi–Yau pairs of the form (P3,Δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {P}^3,\Delta )$$\end{document}, where Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document} is a quartic surface, and classify all such pairs of coregularity less than or equal to one, up to volume preserving equivalence. In particular, if (P3,Δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathbb {P}^3,\Delta )$$\end{document} is a maximal log Calabi–Yau pair then we show that it has a toric model.


Log Calabi-Yau pairs
One topic of much contemporary interest is the geometry of log Calabi-Yau pairs. 1 In part, this is because the interior of a maximal log Calabi-Yau pair is expected to have remarkable properties predicted from mirror symmetry (see e.g.[8, §1]).It is therefore important to understand the classification of log Calabi-Yau pairs up to volume preserving equivalence.
The coregularity.The most important volume preserving invariant of a log Calabi-Yau pair (X, ∆ X ) is an integer 0 ≤ coreg(X, ∆ X ) ≤ dim X, called the coregularity of (X, ∆ X ), which is the dimension of the smallest log canonical centre on a dlt modification of (X, ∆ X ) (see Definition 2.6).At one end of the spectrum are the pairs with coreg(X, ∆ X ) = dim X.These are necessarily of the form (X, 0), where X is a variety with trivial canonical class K X ∼ 0, and hence this case reduces to the study of (strict) Calabi-Yau varieties.At the opposite end are the pairs satisfying coreg(X, ∆ X ) = 0, which are also known as maximal pairs.These form the next most important case to understand, particularly given the role that maximal pairs play in mirror symmetry via the Gross-Siebert program.They are characterised by the property that the dual complex D(X, ∆ X ) has the largest possible dimension.

Toric models.
The simplest examples of maximal log Calabi-Yau pairs are toric pairs, and these lie in a single volume preserving equivalence class.We say that (X, ∆ X ) has a toric model if it also belongs to the same volume preserving equivalence class as a toric pair.A toric model for (X, ∆ X ) is a particular choice of volume preserving map ϕ : (X, ∆ X ) (T, ∆ T ) onto a toric pair (T, ∆ T ).
Remark 1.1.We note three immediate consequences for a d-dimensional maximal log Calabi-Yau pair (X, ∆ X ) with a toric model.
1. X is rational, since it is birational to a toric variety.
2. Every irreducible component D ⊂ ∆ X is rational.This is because, after choosing a suitable toric model ϕ : (X, ∆ X ) (T, ∆ T ), D maps birationally onto a component of ∆ T .

Main result
In this paper we consider log Calabi-Yau pairs of the form (P 3 , ∆) where ∆ is a quartic surface.
The behaviour of (P 3 , ∆) depends upon the trichotomy coreg(P 3 , ∆) = 2, 1 or 0, which is equivalent to the condition that a general pencil of quartic surfaces passing though ∆ defines a type I, type II or type III degeneration of K3 surfaces respectively.For the cases with coreg(P 3 , ∆) ≤ 1, we prove the following result.

Outline of the proof.
A log Calabi-Yau pair (P 3 , ∆) has coreg(P 3 , ∆) ≤ 1 if and only if ∆ has a singularity which is strictly (semi-)log canonical.Thus to prove Theorem 1.2, we start by consulting the extensive literature on the classification of singular quartic surfaces ( [20,21,22,23,24] etc.) and organise all such pairs into eleven different deformation families of pairs (P 3 , ∆) depending on the singularities of ∆.These are We then construct ten explicit volume preserving maps (i)-(x), as shown in Figure 1, which link the different families together.
Figure 1: The volume preserving maps that link the eleven different families.
Ultimately this shows that every pair admits a volume preserving map onto a pair from the family (C.4) which, by definition (cf.§3.4), consists of all pairs (P 3 , ∆) whose boundary divisor ∆ = D 1 + D 2 is the union of a plane D 1 and the cone over a plane cubic curve D 2 .At this point the proof of Theorem 1.2 follows easily (see §6).

The two-dimensional cases
As a toy example, and because it also illustrates the basic process of our proof, we describe the 2-dimensional analogue of Theorem 1.2.
Classification of two-dimensional log Calabi-Yau pairs.If (X, ∆ X ) is a two-dimensional log Calabi-Yau pair then, after replacing (X, ∆ X ) by a minimal resolution of singularities and consulting the classification of surfaces, it follows that (X, ∆ X ) is given by one of the following.
1.If coreg(X, ∆ X ) = 2 then X is either an abelian surface or a K3 surface and ∆ X = 0.

If coreg(X, ∆
and  It follows from the existence of toric models that the three maximal cases are all volume preserving equivalent and, indeed, it is simple to construct explicit volume preserving maps that relate them.Recall that a quadratic transformation ϕ : P 2 P 2 is determined by a linear system |O P 2 (2)− p 1 − p 2 − p 3 | of conics that pass through three non-collinear (but possibly infinitely near) basepoints p 1 , p 2 , p 3 ∈ P 2 .We can define a volume preserving quadratic transformations (P 2 , ∆ a ) ) by picking basepoints as illustrated in Figure 2 (which also shows the base- Volume preserving maps between the three maximal pairs of the form (P 2 , ∆).
In other words, the basepoints of ϕ 1 are p 1 , p 2 , p 3 ∈ ∆ a , where p 1 is the node of ∆ a and p 2 , p 3 are general points.Similarly, the basepoints of ϕ 2 are p 1 , p 2 , p 3 ∈ ∆ b , where p 1 is one of the nodes of ∆ b and p 2 , p 3 are general points on the conic component of ∆ b .In each case, for ϕ i to be volume preserving the basepoints are required to belong to ∆, and in order to pull out a new irreducible component we let one of the basepoints coincide with a node of ∆ (i.e. a minimal log canonical centre of (P 2 , ∆)).
Our proof of Theorem 1.2 proceeds in a similar (but more involved) manner.For a given pair (P 3 , ∆), we find a collection of points and curves contained in the log canonical centres of (P 3 , ∆) which form the baselocus for volume preserving map ϕ : (P 3 , ∆) (P 3 , ∆ ′ ) such that ∆ ′ is 'simpler' than ∆ (where 'simpler' is to be interpreted in accordance with the structure of the graph in Figure 1).

Characterising maximal pairs with a toric model
Finding criteria which characterise maximal log Calabi-Yau pairs with a toric model is a difficult problem which originated in work of Shokurov.Theorem 1.2(2) is a special case of the following conjecture.
Conjecture 1.4.Suppose that (X, ∆ X ) is a maximal log Calabi-Yau pair and X is a rational 3-fold.Then (X, ∆ X ) has a toric model.
Unfortunately the simple and appealing statement of Conjecture 1.4 fails miserably as soon as one tries to relax any of the given assumptions.
Remark 1.5.In light of the three consequences of Remark 1.1, we note that the following conditions in the statement of Conjecture 1.4 are essential.
1.It is necessary to assume that X is rational, since there exist examples of non-rational maximal log Calabi-Yau 3-fold pairs constructed by Kaloghiros [10] and Svaldi [10, Example 5].
(This is in contrast to the 2-dimensional setting, in which maximal pairs are always rational.) 2. It is necessary to assume that dim X = 3, since the examples of Kaloghiros can easily be used to produce a maximal log Calabi-Yau pair of the form (P 4 , ∆) where ∆ contains an irreducible component which is a non-rational quartic 3-fold.

Cremona equivalence of rational quartic surfaces with a plane
Mella [15] has shown that every rational quartic surface ∆ ⊂ P 3 is Cremona equivalent to a hyperplane H ⊂ P 3 .That is to say that there exists a birational map ϕ : P 3 P 3 which maps ∆ birationally onto H.Our theorem strengthens this result of Mella (at least in the case that (P 3 , ∆) is log canonical) by showing that a rational quartic ∆ ⊂ P 3 can be mapped onto a hyperplane by a volume preserving map for the pair (P 3 , ∆).Most of the maps that Mella constructs do not extend to volume preserving maps of (P 3 , ∆), and thus we need to proceed rather more carefully.Roughly speaking, in order for ϕ to be volume preserving we need to ensure that the k-dimensional components of the baselocus of ϕ are contained in (k + 1)-dimensional log canonical centres of (P 3 , ∆).

Volume preserving subgroups of Bir(P n )
A log Calabi-Yau pair (P n , ∆) (up to volume preserving equivalence) determines a subgroup Bir vp (P n , ∆) ⊆ Bir(P n ) (up to conjugation), where Bir vp (P n , ∆) is the subgroup consisting of volume preserving birational self-maps of (P n , ∆).It is an interesting question to know how big (or small) this subgroup can be, depending on the geometry of (P n , ∆), and whether one can describe a set of maps that generate it.
A complete picture is known in the case of P2 .For the pairs (P 2 , ∆) of coregularity one, any map ϕ : (P 2 , ∆) (P 2 , ∆) induces a birational map ϕ| ∆ : ∆ ∆ which is necessarily an isomorphism.Thus Bir vp (P 2 , ∆) coincides with the decomposition group of the smooth plane cubic curve ∆, which has been studied by Pan [18].For the pairs of coregularity zero, Blanc [2] has given a very explicit description of the group Bir vp (P 2 , ∆) when ∆ = V(xyz) is the triangle of coordinate lines.In dimension 3, Araujo, Corti & Massarenti [1] consider the case of a very general quartic surface ∆ ⊂ P 3 (in particular, ∆ is smooth and has Picard rank 1), and show that Bir vp (P 3 , ∆) consists only of those automorphisms of P 3 that preserve ∆.Moreover, they also give an explicit description of Bir vp (P 3 , ∆) in the case that ∆ is a general quartic surface with a single ordinary double point.
1.4.4Pairs (P 3 , ∆) of coregularity two Theorem 1.2 only treats the case of pairs (P 3 , ∆) of coregularity at most one.The remaining case coreg(P 3 , ∆) = 2 occurs if and only if ∆ is an irreducible quartic surface with at worst Du Val singularities.Aside from the results of [1] mentioned above, giving an explicit classification of all such pairs up to volume preserving equivalence will be difficult, and significantly more involved than simply classifying quartic surfaces up to birational equivalence.For example, Oguiso [17] has given an example of two smooth isomorphic quartic surfaces ∆ 1 , ∆ 2 ⊂ P 3 for which there is no map ϕ ∈ Bir(P 3 ) (let alone a volume preserving one) that maps ∆ 1 birationally onto ∆ 2 .

Notation
We use dP d to denote a del Pezzo surface of degree d, possibly with Du Val singularities.We often need to consider curves which are either smooth elliptic curves, or reduced nodal curves of arithmetic genus 1.Since repeating this each time we want to use it is a bit of a mouthful we call such a curve an ordinary curve.

Acknowledgements
I would like to thank Anne-Sophie Kaloghiros for some very helpful correspondence and comments on the topic of this paper.

Log Calabi-Yau pairs
We begin with some useful results concerning the geometry of log Calabi-Yau pairs.Definition 2.1.A log Calabi-Yau pair (X, ∆ X ) is a log canonical pair consisting of a proper variety X over C and a reduced effective integral Weil divisor 2 and which is uniquely determined up to scalar multiplication.

Volume preserving maps
The natural notion of birational equivalence between log Calabi-Yau pairs is that of volume preserving equivalence (cf.[12,Definition 2.23]).

Definition 2.2. A proper birational morphism of pairs
where f and g are crepant birational morphisms.
In the context of log Calabi-Yau pairs (X, ∆ X ) and (Y, ∆ Y ), crepant birational maps are also known as volume preserving maps, 3 since ϕ * ω ∆ X = ω ∆ Y for an appropriate rescaling of the naturally defined volume form on each side [3,Remark 5].
Remark 2.3.An easy consequence of the definition is that a volume preserving map preserves discrepancies, i.e. that a E (X, ∆ X ) = a E (Y, ∆ Y ) for any exceptional divisor E over both X and Y , where a E (X, ∆ X ) ∈ Q denotes the discrepancy of E over (X, ∆ X ).Moreover a composition of volume preserving maps is volume preserving.

Dlt modifications
The main problem with considering log canonical pairs (X, ∆ X ) in general is that they can exhibit rather complicated singularities.Life becomes easier if we focus on pairs with divisorial log terminal (dlt) singularities.This is always possible by passing to a dlt modification.
One of the most pleasing consequences of working with a dlt pair (X, ∆ X ) is that it is easy to understand the log canonical centres of (X, ∆ X ) and they satisfy some very pleasing forms of adjunction.

is a volume preserving map which restricts to a birational map of log canonical centres ϕ| Z
Thus the boundary divisor ∆ X of a dlt log Calabi-Yau pair can be thought of as a collection of log Calabi-Yau pairs of dimension d − 1, glued together along their boundary components.One can make a similar study of log canonical log Calabi-Yau pairs, but in general the picture is significantly more complicated (see [12, §4] for details).

The coregularity
Since volume preserving maps preserve discrepancies, they map log canonical centres onto log canonical centres.In particular, one can use this to show that the dimension of a minimal log canonical centre on a dlt modification is a volume preserving invariant5 of (X, ∆ X ).
Definition 2.6.The coregularity coreg(X, ∆ X ) is defined to be the dimension of a minimal log canonical centre in a dlt modification ϕ : Given a log canonical centre Z ⊂ X of a dlt pair (X, ∆ X ) then coreg(X, ∆ X ) = coreg(Z, ∆ Z ), since any smaller log canonical centre Z ′ ⊂ Z ⊂ X restricts to a log canonical centre of (Z, ∆ Z ) by [12,Theorem 4.19(3)].

The dual complex D(X, ∆ X )
Although we will not use it, we briefly recall the dual complex D(X, ∆ X ) of a log Calabi-Yau pair (X, ∆ X ) since it was mentioned in the introduction.This is a simplicial complex which encodes the geometry of the log canonical centres of (X, ∆ X ) obtained by associating a (k − 1)-dimensional simplex σ Z to each k-codimensional log canonical centre Z X, which are then glued together according to inclusion.Thus D(X, ∆ X ) has dimension dim D(X, ∆ X ) = dim X −coreg(X, ∆ X )−1, and this is of maximal possible dimension if coreg(X, ∆ X ) = 0 (which is one explanation for the terminology 'maximal pair').A key theorem of Kollár & Xu relates volume preserving maps of pairs to homeomorphisms of their dual complexes.

A rough classification of quartic surfaces
We now recall some results on the classification of quartic surfaces.Quartic surfaces can have one of many thousands of different singularity types [4], but they have been well-studied and the study of the classification of singular quartic surfaces goes back to Jessop [9].Moreover, since then other authors have also given very precise descriptions of the type of singularities that a quartic surface can have, e.g.[ We divide log Calabi-Yau pairs (P 3 , ∆) of coregularity ≤ 1 into eleven different families according to the singularities of ∆, as described in §1.2.Each family is taken to be closed under degeneration and they are not supposed to be mutually exclusive.Moreover, every such pair belongs to one of these eleven families.Clearly every reducible quartic surface ∆ is either the union of a plane and cubic surface (C.1) or two quadrics (C.2), or a degeneration of one of these two cases.The fact that every log canonical pair with irreducible boundary divisor belongs to one of the other seven families follows from Proposition 3.1 and Proposition 3.3.

Two-dimensional semi-log canonical singularities
In Table 1 we present the classification of two-dimensional strictly (semi-)log canonical hypersurface singularities V(f (x, y, z)) ⊂ A 3 x,y,z , up to local analytic isomorphism, cf.[14].When we refer to p ∈ ∆ as an ' E k singularity' we implicitly take that to include the possibility that p ∈ ∆ is a degeneration of an E k singularity.For example, the cusp singularities T pqr with 3 ≤ p, q, r ≤ ∞ are E 6 singularities, the cusp singularities T 2qr with 4 ≤ q, r ≤ ∞ are E 7 singularities, and the cusp singularities T 23r with 6 ≤ r ≤ ∞ are E 8 singularities.
Coregularity of (P 3 , ∆).Now let (X, ∆ X ) be a 3-fold pair and p ∈ ∆ X ⊂ X is a point at which X is smooth, but ∆ X has an E 6 , E 7 or E 8 singularity.We consider the weighted blowup with weights {a, b, c} = {1, 1, 1}, {2, 1, 1} or {3, 2, 1} given to the local coordinates x, y, z for the normal form presented in Table 1, so that deg f (x, y, z) ≥ 3, 4 or 6 in each case respectively.In all cases E is a log canonical centre of (X, ∆ X ) and, setting ∆ * ∆ X is also a log canonical centre of (X, ∆ X ), and is a smooth elliptic curve if p ∈ ∆ X is a simple elliptic singularity, and a reduced nodal curve otherwise.
In particular if coreg(P 3 , ∆) = 1 then ∆ can only have either simple elliptic singularities or a double curve with a finite number of pinch points.Similarly, coreg(P 3 , ∆) = 0 if and only if ∆ has a cusp singularity or a degenerate cusp singularity.

Irreducible quartic surfaces with isolated singularities
There are four distinct ways in which an irreducible quartic surface can have an isolated strictly log canonical singularity.Singularities of type E 6 and E 7 each appear in an essentially unique way, but singularities of type E 8 can appear in one of two different ways (cf.[9,16,20,24]).Proposition 3.1.Suppose that ∆ = V(F (x, y, z, t)) ⊂ P 3 is a reduced irreducible quartic surface such that ∆ has at least one isolated simple elliptic (or cusp) singularity and possibly some additional Du Val singularities.Then, up to projective equivalence, one of the following occurs.Moreover, in order for p ∈ ∆ to be log canonical, in each case the appropriate weighted tangent cone must define the cone over an ordinary curve.
Remark 3.2.The two different types of E 8 singularity can be distinguished by the fact that the quartics in family (A.3) contain a line V(y, z) which through P ∈ ∆, whereas the generic member of family (A.4) does not contain any line.From the point of view of GIT stability, that quartics in family (A.3) are unstable but those in (A.4) are stable [20,24].
Parameterisation of the rational cases.The study of rational quartics with these four types of simple elliptic singularities goes back to Noether [16].The rational parameterisation of quartics with a triple point (A.1) is straightforward.In each of the other cases (A.2-4) Noether produced a rational parameterisation which we now describe.First note that (∆, 0) is a log Calabi-Yau pair by adjunction, and we consider the volume preserving minimal resolution of log Calabi-Yau pairs µ : ( ∆, D) → (∆, 0) where D ⊂ ∆ the reduced exceptional curve (or reduced exceptional cycle) over the simple elliptic (or cusp) singularity p ∈ ∆.By the classification of two-dimensional log Calabi-Yau pairs §1.3, ∆ is a rational surface.Let ( ∆ 0 , D 0 ) := ( ∆, D) and, for i = 1, . . ., k, let f i : ( ∆ i−1 , D i−1 ) → ( ∆ i , D i ) be a sequence of volume preserving blowdowns (i.e.obtained by setting By choosing a sequence of contractions carefully it is possible to show that we can always find a sequence ending with ∆ k = P 2 and therefore D k ⊂ P 2 is either a smooth or nodal cubic curve.2. The ruled elliptic cases.The remaining ruled elliptic cases (A.2*) and (A.3*) were described by Umezu [22,Theorems 1& 2].She gives a similar construction of them, by taking a minimal volume preserving resolution of singularities and blowing down to a minimal ruled elliptic surface.

Irreducible quartic surfaces with non-isolated singularities
The classification of non-normal quartic surfaces is contained in Jessop's book [9], but it has also been considered in more modern times by Urabe [23].We follow Urabe's treatment and his subdivision into eight classes.Only two families in Urabe's classification have a general member which has worse than semi-log canonical singularities: (I) corresponding to the cone over a plane quartic curve, and (II-2) corresponding to a ruled elliptic surface with a line of cuspidal singularities.We recall the remaining cases.For the case (B.1), Urabe shows that it is always possible to find a sequence of nine contractions ending in D 9 = P 2 , and for which A = 4h − 2e 1 − 9 i=2 e i .In the three remaining cases, we have that µ * |O ∆ (1)| |A| is a strict linear subsystem of |A|.In these cases the normalisation ∆ can be realised as ν : (∆ ⊂ P h 0 (A)−1 ) (∆ ⊂ P 3 ), where ν is the projection from a general linear subspace of P h 0 (A)−1 which is disjoint from ∆.In particular, one of the following cases occurs.We summarise these results in Table 3.

Reducible quartic surfaces
The remaining families correspond to pairs with a reducible boundary divisor and we divide them up into the following four families.The subdivision into these four particular cases may look somewhat artificial or arbitrary.Our only reason for considering it is that it corresponds to the logical structure of our proof of Theorem 1.2. 4 Low degree maps in Bir(P 3 ) The bidegree of a 3-dimensional birational map ϕ ∈ Bir(P 3 ) is given by (deg ϕ, deg ϕ −1 ) ∈ Z 2 ≥1 .Maps with low bidegree are well-understood and there are some very detailed classification results.For example, Pan, Ronga & Vust [19] show that quadratic maps can have bidegree (2, d) for d = 2, 3, 4, and these three types of map comprise three irreducible families F (2,d) of dimensions 29, 28, 26 respectively.Moreover they give a complete description of the strata of F (2,d) , by exhibiting all possible ways in which the baselocus Bs(ϕ) can degenerate.Deserti & Han [5] provide a similar analysis for a large part of the landscape of maps of degree 3.

Strategy
In this section we exhibit a few examples of maps ϕ ∈ Bir(P 3 ) of low bidegree that we will use to construct some of the links between in our families in §5.For each map ϕ we construct a resolution of the following form.X Then, to simplify a given pair (P 3 , ∆), we will find boundary divisors ∆ X ⊂ X and ∆ ′ ⊂ P 3 such that (P 3 , ∆ ′ ) belongs to a simpler family (according to Figure 1), and (X, ∆ X ) is a diagram of volume preserving maps of Calabi-Yau pairs, as in Definition 2.2.
Throughout the following calculations we let H denote the hyperplane class on the lefthand copy of P 3 (i.e. the domain of ϕ), and H ′ the hyperplane class on the righthand copy (i.e. the range of ϕ).By abuse of notation we refer to the strict transform of a subvariety (whenever it makes sense) by the same name as for the original.

The generic map of bidegree (2, 2)
The generic map ϕ : P 3 P 3 of bidegree (2, 2) is defined by |2H − C − p|, the linear system of quadrics passing through a plane conic C ⊂ P 3 and a general point p ∈ P 3 .Let E ′ be the plane containing C and let F ′ be the quadric cone through C with vertex at p.The map ϕ is resolved by a symmetric diagram of the form where 1. ψ blows up p with exceptional divisor E ∼ = P 2 and C with exceptional divisor and we have the following relations between divisor classes.

The generic map of bidegree (3, 2)
Consider three pairwise skew lines ℓ 1 , ℓ 2 , ℓ 3 ⊂ P 3 and a fourth line ℓ 0 which meets each of the first three.The generic map ϕ : P 3 P 3 of bidegree (3, 2) is defined by the linear system |3H − 2ℓ 0 − ℓ 1 − ℓ 2 − ℓ 3 |.Let F ′ ⊂ P 3 be the unique quadric surface containing all four lines ℓ 0 , . . ., ℓ 3 ⊂ F ′ and let E ′ i ⊂ P 3 be the plane containing ℓ 0 and ℓ i for i = 1, 2, 3. Then ϕ can resolved by a diagram of the form F ′ and we have the following relations between divisor classes.
and is the generic map of bidegree (2, 3).(3,3) 4.4.1 The generic map of bidegree (3,3) The generic map of bidegree (3,3) is the classical cubo-cubic Cremona transformation and has been studied by many authors, e.g [11].Let C ⊂ P 3 be a smooth curve of degree 6 and genus 3 defined by the 3 × 3-minors of a 3 × 4-matrix with linear entries.The trisecant lines to C span a ruled surface E ′ ⊂ P 3 of degree 8 with multiplicity 3 along C. Then the birational map ϕ : P 3 P 3 defined by the linear system |3H − C| can be resolved by a symmetric diagram of the form

Maps of bidegree
where ψ is the blowup of C with exceptional divisor E and

A special map of bidegree (3, 3)
We consider a degenerate case of the previous example in which C = Γ∪ℓ 1 ∪ℓ 2 ∪ℓ 3 is the union of a twisted cubic curve Γ and three lines ℓ 1 , ℓ 2 , ℓ 3 which are secant lines to Γ.This kind of cubo-cubic Cremona transformation was considered by Mella [15, Proof of Proposition 2.2] and we briefly recall the description.
. The birational map ϕ : P 3 P 3 defined by the linear system |3H − Γ − ℓ 1 − ℓ 2 − ℓ 3 | can be resolved by a symmetric diagram of the form and the following relations between divisor classes.

Connecting the eleven families
We now constructing the ten types of volume preserving map (i)-(x), appearing in Figure 1, that connect our eleven families.

The map (x) between (A.1) and (C.4)
Although it appears last in Figure 1, we begin by explaining map (x) since it is by far the easiest case to deal with.Suppose that ∆ has an E 6 singularity, or in other words a triple point.Thus we may write ∆ = V(a 3 t + b 4 ) for some polynomials a 3 , b 4 ∈ C[x, y, z].We let ∆ ′ = V(a 3 t) and consider the birational map ϕ : (P 3 , ∆) which is volume preserving.To see this we can restrict to the affine patch of P 3 where z = 1 (on both sides of ϕ) and then we compute that Lastly, note that ∆ ′ = V(t) + V(a 3 ) is the union of a plane and the cone over a plane cubic curve, and thus (P 3 , ∆ ′ ) is a member of family (C.3).
The map ϕ 1 .We start with a quartic ∆ ⊂ P 3 in the family (A.3).By Proposition 3.1 it is defined by an equation F 4 (t, x, y, z) of the form for some homogeneous polynomials a 0 , . . ., h 4 ∈ C[y, z].
The map ϕ 2 .The construction of ϕ 2 is very similar to that of ϕ 1 .By Proposition 3. and, if we let u = y −1 z, then it is easy to check that this produces a volume preserving map of pairs, where the boundary divisor ∆ ′ = V(tG 4 ) ⊂ P(1, 1, 1, 2) is given by the union of a plane and a (possibly degenerate) dP 2 defined by the equation The map ϕ 4 .The construction of ϕ 4 is very similar to that of ϕ 3 .Indeed if t, x, y, z are coordinates on P(1, 1, 1, 2) and ∆ = V(tF 4 ), then, arguing as before, we may assume that the component D 2 = V(F 4 ) contains a line ℓ = V(y, z).Then the map ϕ 4 : P(1, 1, 1, 2) P 3 ϕ(t, x, y, z) = (t, x, y, y −1 z) is easily seen to produce a volume preserving map, for a boundary divisor ∆ ′ ⊂ P 3 given by the union of the plane V(t) and the cubic surface obtained by contracting ℓ ⊂ D 2 .

Reduction to (B.1).
To reduce to a simpler case, we apply the quadratic map given by the linear system of all quadrics passing through C and the infinitely near point p ∈ C.This induces a volume preserving map ϕ : (P 3 , ∆) (P 3 , ∆ ′ ) where ∆ ′ is the quartic defined by the equation which is singular along the line ℓ = V(y, z).

The map (iv) between (B.1) and (C.1)
Throughout this subsection we assume that ∆ ⊂ P 3 has a line of double points along ℓ 0 = V(y, z), so that the equation of ∆ can be written in the form a 2 t 2 + b 2 tx + c 2 x 2 + d 3 t + e 3 x + f 4 (5.1) for homogeneous polynomials a 2 , . . ., f 4 ∈ C[y, z].We deal with these surfaces in three steps.
Singularities away from ℓ 0 .First we consider the cases in which ∆ is singular at a point outside of the line ℓ 0 .
Thus we may assume that ∆ \ ℓ 0 is smooth, or else proceed via the (A.2) case.
2. The map f is the projection from p ∈ X, which contracts D 1 onto C and maps D 2 birationally onto ∆.
3. The map g is the projection from a general point q ∈ Γ.Since g maps D 1 birationally onto a plane and D 2 birationally onto a cubic surface, we set ∆ ′ := g(∆ X ).

The map (vii) between (B.3) and (C.2)
We suppose that ∆ ⊂ P 3 has double points along a twisted cubic curve Σ.If Σ is a degenerate twisted cubic curve then it splits into the union of a line and a (possibly reducible) conic.In particular we can treat (P 3 , ∆) as a special case of family (B.1).We now follow the argument of Mella [15] very closely.Since we can assume Σ is smooth, without loss of generality it is given by the equations and thus we can write the blowup of Σ as σ : X ⊂ P 3 × P 2 ξ 1 ,ξ 2 ,ξ 3 → P 3 where X is the complete intersection of codimension two cut out by the two equations (5.3).

(A. 1 - 4 )(B. 1 - 3 )(C. 1 - 4 )
the first four families described in Proposition 3.1, corresponding to irreducible quartic surfaces with a simple elliptic (or cusp) singularity, the first three families described in Proposition 3.3, corresponding to irreducible nonnormal quartic surfaces, the four families described in §3.4,corresponding to reducible quartic surfaces.
(a) ∆ a := ∆ is an irreducible nodal cubic curve, (b) ∆ b := ∆ is the sum of a conic and (non-tangent) line, or (c) ∆ c := ∆ is a triangle of lines.
Let f : ∆ → P 2 be the composition of the f i and consider NS( ∆) = Z h, e 1 , . . .e k given with its standard basis, where h = f * O P 2 (1) and e i is the total transform of class of the exceptional divisor of f i .Then we have D ∼ 3h − e 1 − . . .− e k and the map µ : ∆ → ∆ ⊂ P 3 is induced by a nef divisor class A = µ * O ∆ (1) satisfying h 0 ( ∆, A) = 4, A 2 = 4 and A • D = 0.The possibilities for A are given in Table

Proposition 3 . 3 (Remark 3 . 4 .
[23]).Suppose that ∆ ⊂ P 3 is a reduced irreducible non-normal quartic surface with semi-log canonical singularities.Then ∆ has double points along a curve Σ ⊂ ∆ and possibly some Du Val singularities outside of Σ. Moreover Σ is (possibly a degeneration of) one of the following cases.1.∆ is a rational surface and the curve Σ is (B.1) a line [23, (III-C)], (B.2) a plane conic [23, (III-B)], (B.3) a twisted cubic [23, (III-A-2) & (III-A-3)], (B.4) the union of three concurrent lines [23, (III-A-1)], or 2. ∆ is a elliptic ruled surface and the curve Σ is (B.1*) a pair of skew lines [23, (II-1)].As one may see, cases (B.4) and (B.1*) do not appear in the list of families considered in §1.2.The generic member of family (B.4) is isomorphic to Steiner's Roman surface.Given that this surface necessarily has a triple point at the intersection of the three lines, we treat it as a special case of (A.1).We treat case (B.1*) as a special case of (B.1).Parameterisation of the rational cases.Urabe also provides an explicit construction for all cases of his classification, which is analogous to the results of Noether and Umezu discussed in isolated singularity case above.We recall the description for the rational cases.The normalisation ν : (∆, D) → (∆, 0) is a volume preserving map of log Calabi-Yau pairs, where D = ν −1 (Σ) is the preimage of the double curve.Let µ : ( ∆, D) → (∆, D) → (∆, 0) be the volume preserving minimal resolution of singularities which factors through ν.As in (3.1) above, we consider a sequence of volume preserving blowdowns f= f k • • • • • f 1 from ( ∆ 0 , D 0 ) := ( ∆,D) to a minimal pair ( ∆ k , D k ) and we keep the same notation for NS( ∆) = Z h, e 1 , . . ., e k and the divisor class A = µ * O ∆ (1).

(B. 2 )
∆ ∼ = Bl 5 P 2 ⊂ P 4 is an anticanonically embedded dP 4 and ∆ → ∆ is at worst the crepant resolution of some Du Val singularities.(B.3) ∆ = ∆ ⊂ P 5 is either P 1 × P 1 embedded by O P 1 ×P 1 (2, 1), or F 2 embedded by |s + f | wheres is a (positive) section of F 2 and f is the class of a fibre.By blowing up one more point on ∆ we can treat these both as one case, where ∆ = Bl 2 P 2 is a dP 7 .(The difference between the two cases is then whether these two points are infinitely near or not.)(B.4) ∆ = ∆ ∼ = P 2 ⊂ P 5 is embedded by O P 2 (2) (i.e. the second Veronese embedding of P 2 ).

(C. 1 )(C. 2 )(C. 3 )(C. 4 )
∆ is the union of a plane and a cubic surface, ∆ is the union of two quadrics, ∆ is the union of a plane and a singular cubic surface, ∆ is the union of a plane and the cone over a cubic curve.

Table 1 :
Two-dimensional strictly semi-log canonical hypersurface singularities [24,[24, Theorem 8.1(iii)]) ∆ is a rational surface with exactly one such singularity p ∈ ∆.The type of singularity p ∈ ∆ and the form of the equation F (x, y, z, t) are given by one of the following four cases.

Table 2 :
Irreducible rational quartic surfaces with an isolated log canonical singularity.