Automorphism groups of certain Enriques surfaces

We calculate the automorphism group of certain Enriques surfaces. The Enriques surfaces that we investigate include very general $n$-nodal Enriques surfaces and very general cuspidal Enriques surfaces. We also describe the action of the automorphism group on the set of smooth rational curves and on the set of elliptic fibrations.


Introduction
A central theme in algebraic geometry is to study varieties using convex geometry. The cone of curves of a variety is the convex hull of the numerical equivalence classes of curves. Its dual is the cone of nef line bundles. Much of the birational geometry of a variety is encoded in these cones and their interplay with the canonical divisor. While for Fano varieties the nef cone is rational polyhedral [15,Theorem 3.7], in general the nef cone is not well understood. For instance it can have infinitely many faces or be round. The Morrison-Kawamata cone conjecture [20,12] gives a clear picture of the effective nef cone of a Calabi-Yau variety. It predicts that the action of the automorphism group on the effective nef cone admits a fundamental domain which is a rational polyhedral cone.
The conjecture is wide open in dimension three and beyond [18]. But it has been verified for K3 surfaces by Sterk [33], and for Enriques surfaces by Namikawa [21]. It follows that an Enriques surface admits up to the action of the automorphism group only finitely many smooth rational curves, finitely many elliptic fibrations, finitely many projective models of a given degree and its automorphism group is finitely generated and in fact finitely presented [19,Corollaries 4.15,4.16].
Naturally, enumerative questions arise: • Can one explicitly describe a fundamental domain?
• How many smooth rational curves, elliptic fibrations or projective models are there up to the action of the automorphism group? • Can one give generators for the automorphism group? Barth-Peters [2] noted that very general Enriques surfaces do not contain smooth rational curves. Hence their nef cone is round -it is the entire positive cone, and they proceed to answer the three questions for very general Enriques surfaces.
Enriques surfaces containing a smooth rational curve are called nodal. They form a subset of codimension one in the moduli space of Enriques surfaces. Very general nodal Enriques surfaces are treated by Cossec-Dolgachev [8] (see also the works of Allcock [1] and Peters-Sterk [25]).
When an Enriques surface is deformed to one containing more rational curves several phenomena working against each other occur. On the one hand the nef cone gets smaller and on the other hand the automorphism group may change drastically. p.395] write that they do not know whether one can control these effects. Albeit the behaviour of the nef cone and the automorphism group may be erratic, the cone conjecture promises that the fundamental domain on the nef cone stays of finite volume at least. Our first main result (Theorem 3.4) states that we can control the (change of) volume in a precise way under mild assumptions.
To generalize the aforementioned results of Barth, Peters, Cossec and Dolgachev to Enriques surfaces with more nodes, we introduce the notion of (τ,τ )-generic Enriques surfaces, which is closely related to the root invariant introduced by Nikulin [24]. See the next subsection for the precise definition. For instance the very general Enriques surface is (0, 0)-generic, a very general nodal Enriques surface is (A 1 , A 1 )-generic and if Y is an Enriques surface that is very general in the moduli of Enriques surfaces containing n disjoint smooth rational curves, then Y is (nA 1 , nA 1 )-generic. If Y is very general in the moduli of Enriques surfaces containing two smooth rational curves whose dual graph is ❝ ❝ (that is, Y is a very general cuspidal Enriques surface), then Y is (A 2 , A 2 )-generic.
Next we give algorithms to compute generators for the automorphism group Aut(Y ), a fundamental domain for Aut(Y ) on the nef and big cone Nef(Y ) and orbit representatives for its action on R(Y ) := the set of smooth rational curves on Y , E(Y ) := the set of elliptic fibrations Y → P 1 .
We apply Theorem 3.4 and the aforementioned algorithms to (τ,τ )-generic Enriques surfaces. This results in our second, series of main results: Theorem 1.18 expresses the volume of the fundamental domain of Aut(Y ) on the nef cone Nef(Y ) in terms of the Weyl group of τ , Theorem 1.19 relates the orbits of Aut(Y ) on the set of smooth rational curves R(Y ) to the connected components of the Dynkin diagram τ and Theorem 1.21 counts the Aut(Y )-orbits of the set of elliptic fibrations E(Y ) and their fiber types.
Our new idea is the lattice theoretic result obtained in [6] (see also Chapter 10]). For a lattice L with the intersection form −, − , let L(m) denote the lattice with the same underlying Z-module as L and with the intersection form m −, − . A lattice L of rank n > 1 is said to be hyperbolic if the signature is (1, n − 1). For a positive integer n with n mod 8 = 2, let L n denote an even unimodular hyperbolic lattice of rank n, which is unique up to isomorphism. Borcherds [4], [5] developed a method to calculate the orthogonal group of an even hyperbolic lattice S by embedding S primitively into L 26 and using the result of Conway [7]. This method has been applied to the study of automorphism groups of K3 surfaces by many authors. However, the method often requires impractically heavy computation (see, for example, [11] and [28]).
On the other hand, in [6], we have classified all primitive embeddings of L 10 (2) into L 26 and showed that they have a remarkable property (see Theorems 4.2 and 4.3) which enables us to calculate automorphism groups of Enriques surfaces efficiently and explicitly for the first time. The resulting speed up (roughly by a factor of 10 20 in the best situation see Remark 6.1) over a more direct approach, allows us to calculate the automorphism groups of the 184 families of (τ,τ )-generic Enriques surfaces.
1.1. Definition of (τ,τ )-generic Enriques surfaces. First, we define (τ,τ )generic Enriques surfaces. Let L be a lattice. We let the group O(L) of isometries of L act on L from the right, and write the action as v → v g for v ∈ L ⊗ R and g ∈ O(L). We have a natural identification O(L) = O(L(m)) for any non-zero integer m. A vector v of a lattice is called a k-vector if v, v = k. A (−2)-vector is called a root. Definition 1.1. An ADE-lattice is an even negative definite lattice generated by roots. An ADE-lattice R has a basis consisting of roots whose dual graph is a Dynkin diagram of an ADE-type. This ADE-type is denoted by τ (R).
A positive half-cone of a hyperbolic lattice L is one of the two connected components of { x ∈ L ⊗ R | x, x > 0 }. Let P be a positive half-cone of a hyperbolic lattice L. We put O P (L) := { g ∈ O(L) | P g = P }. In [29], we classified the ADE-sublattices of L 10 up to the action of O P (L 10 ). Let R be an ADE-sublattice of L 10 , and R the primitive closure of R in L 10 . It turned out that R is also an ADE-sublattice of L 10 . (2) The pair (τ,τ ) of ADE-types is equal to (τ (R), τ (R)) of an ADE-sublattice R of L 10 if and only if (τ,τ ) is one of the 184 pairs in Table 1.1, where the 3rd column being "−" means τ =τ .
Let R be an ADE-sublattice of L 10 . We denote by ι R : R ֒→ L 10 the inclusion. We define M R to be the Z-submodule of (L 10 (2) ⊕ R(2)) ⊗ Q generated by L 10 (2) and (ι R (v), ±v)/2 ∈ (L 10 ⊕ R) ⊗ Q, where v runs through R, and equip M R with an intersection form by extending the intersection form of L 10 (2)⊕R (2). By definition, M R is an even hyperbolic lattice with a chosen primitive embedding ̟ R : L 10 (2) ֒→ M R . If R ′ is another ADE-sublattice of L 10 such that (τ (R ′ ), τ (R ′ )) = (τ (R), τ (R)), then, by Proposition 1.2, we have an isometry g : L 10 ∼ − → L 10 that induces an isometry g| R : R ∼ − → R ′ , and hence we obtain an isometryg : M R ∼ − → M R ′ induced by g ⊕ g| R , which makes the following diagram commutative: By an explicit calculation, we obtain the following: Proposition 1.3. Let R be an ADE-sublattice of L 10 . Then the orthogonal complement of ̟ R : L 10 (2) ֒→ M R is isomorphic to R(2) for some ADE-lattice R. In the 4th column of Table 1.1, we give the ADE-type τ ( R) of R, where "−" means τ (R) = τ ( R).    Let Y be an Enriques surface. We denote by S Y the lattice of numerical equivalence classes of divisors of Y . It is well-known that S Y is isomorphic to L 10 . Let π : X → Y be the universal covering of Y , and let S X denote the lattice of numerical equivalence classes of divisors of the K3 surface X. Then theétale double covering π induces a primitive embedding π * : S Y (2) ֒→ S X . Definition 1.4. Let (τ,τ ) be one of the 184 pairs in Table 1.1. An Enriques surface Y is said to be (τ,τ )-generic if the following conditions are satisfied.
(i) Let T X be the transcendental lattice of X, and ω a non-zero holomorphic 2-form of X, so that we have Cω = H 2,0 (X) ⊂ T X ⊗ C. Then the group (ii) Let R be an ADE-sublattice of L 10 with (τ (R), τ (R)) = (τ,τ ). Then there exist isometries g : L 10 The numbering of the ADE-types in Table 1.1 of the present article is the same as the numbering in Table 1.1 of our previous paper [29], and hence the 1st-3rd columns of the two tables are identical. By definition, a (τ,τ )-generic Enriques surface exists if and only if the 4th column of the corresponding row of  [29], we have no S X (No. 86), or S X /M R is non-trivial ((Z/2Z) 2 for No. 87 and (Z/2Z) 3 for No. 88), that is, the inclusiong is not an isometry. Hence there exist no (τ,τ )generic Enriques surfaces with τ = 8A 1 , even though there exist surfaces with 8 ordinary nodes birational to Enriques surfaces.
Remark 1.7. The geometry of Enriques surfaces with O(T X , ω) = {±1} but with S X /M R being non-trivial and finite is left for future studies.
Let P Y (resp. P X ) be the positive half-cone of S Y (resp. S X ) containing an ample class. We regard P Y as a subspace of P X by the embedding π * ⊗ R. We put where [D] is the class of a divisor D. The following will be proved in Section 3.2. Proposition 1.8. Let Y and Y ′ be (τ,τ )-generic Enriques surfaces with the universal coverings π : X → Y and π ′ : X ′ → Y ′ , respectively. Then there exist isometries We denote by aut(Y ) the image of the natural representation Aut(Y ) → O P (S Y ). We embed the set R(Y ) of smooth rational curves C on Y into S Y by C → [C], and the set E(Y ) of elliptic fibrations φ : Y → P 1 into S Y by φ → [F ]/2, where F is a general fiber of φ. In Section 6, we will see that aut(Y ) and its actions on Nef Y , R(Y ), E(Y ) depend only on the data π * : S Y (2) ֒→ S X and Nef X . Therefore we obtain the following: Corollary 1.9. Let Y and Y ′ be as in Proposition 1.8. Then there exist an isomorphism aut(Y ) ∼ = aut(Y ′ ) and bijections R( Remark 1.10. The root invariant of a (τ,τ )-generic Enriques surface (defined by Nikulin [24]) is equal to (τ, Ker ξ), where ξ : R ⊗ F 2 → L 10 ⊗ F 2 is the linear homomorphism induced by the inclusion R ֒→ L 10 of the ADE-sublattice R of L 10 such that (τ,τ ) = (τ (R), τ (R)).

1.2.
Chambers. Before we state our geometric results, we define the notion of chambers of hyperbolic lattices, and recall the classical result of Vinberg [35].
A root r of an even lattice L defines the reflection s r : x → x + x, r r of L with respect to r. The Weyl group W (L) of L is the subgroup of O(L) generated by all the reflections s r with respect to the roots of L. Let L be an even hyperbolic lattice with a positive half-cone P. For v ∈ L ⊗ R with v, v < 0, let (v) ⊥ denote the hyperplane of P defined by x, v = 0. Then we have W (L) ⊂ O P (L), and the action of s r on P is the reflection into the mirror (r) ⊥ . A closed subset D of P is called a chamber if D contains a non-empty open subset of P and D is defined by inequalities x, v > 0 holds for one (and hence any) point x in the interior of D. We say that a closed subset A of P is tessellated by a set {D j } j∈J of chambers if A is the union of D j (j ∈ J) and the interiors of two distinct chambers D j and D j ′ in the family {D j } j∈J have no common points. Recall that L 10 is an even unimodular hyperbolic lattice of rank 10. Then L 10 has a basis e 1 , . . . , e 10 consisting of roots whose dual graph is given in Figure 1.1. Let P 10 be the positive half-cone of L 10 containing e ∨ 1 +· · ·+e ∨ 10 , where {e ∨ 1 , . . . , e ∨ 10 } is the basis of L ∨ 10 = L 10 dual to {e 1 , . . . , e 10 }. Theorem 1.13 (Vinberg [35]). The chamber D 0 in P 10 defined by x, e i ≥ 0 for i = 1, . . . , 10 is an L 10 -chamber, and {e 1 , . . . , e 10 } is the set of roots defining walls of D 0 . Definition 1.14. We call an L 10 -chamber a Vinberg chamber. 1.3. Main results. We investigate the geometry of a (τ,τ )-generic Enriques surface Y . In particular, we calculate a finite generating set of aut(Y ) and the action of aut(Y ) on Nef Y , R(Y ) and E(Y Let Y be an Enriques surface. Recall that aut(Y ) ⊂ O P (S Y ) is the image of the natural homomorphism Aut(Y ) → O P (S Y ). Since S Y is isomorphic to L 10 , we have Vinberg chambers in the positive half-cone P Y . Since Nef Y is bounded by ([C]) ⊥ , where C runs through R(Y ), and [C], [C] = −2, the cone Nef Y is tessellated by Vinberg chambers. We put V(Nef Y ) := the set of Vinberg chambers contained in Nef Y , on which aut(Y ) acts, and define vol(Nef Y /aut(Y )) := the number of orbits of the action of aut(Y ) on V(Nef Y ).
Our first main result is as follows. For an ADE-type τ , let W (R τ ) denote the Weyl group of the ADE-lattice R τ with τ (R τ ) = τ , that is, the finite Coxeter group defined by the Dynkin diagram of type τ . An automorphism of Y is called numerically trivial if it acts trivially on S Y . Theorem 1.18. Let Y be a (τ,τ )-generic Enriques surface. Then we have where c (τ,τ ) ∈ {1, 2} is the number of numerically trivial automorphisms of Y and is given in 6th column of Table 1.1.
In two non-geometric cases Nos. 142 and 170 (Remark 1.15), there exists a contribution to c (τ,τ ) not coming from a numerically trivial automorphism. (See Theorem 3.11 and Remark 3.12). Theorem 1.18 is in fact obtained from a more general result Theorem 3.4 on vol(Nef Y /aut(Y )). To obtain Theorem 3.4, we prove a result (Proposition 2.1) of the theory of discriminant forms in the spirit of Nikulin [22]. The proof of these theorems is conceptual. Nevertheless the ability to compute examples played a crucial role in finding the correct statements.
Next, we calculate explicitly a finite generating set of aut(Y ) and a complete set of representatives of the orbits of the action of aut(Y ) on Nef Y . The algorithms we use for this purpose are variations of a simple algorithm given in Section 4.1, which is an abstraction of the generalized Borcherds' method described in [28]. By means of these computational data, we analyze the action of aut(Y ) on R(Y ) and Our second main result is as follows.
(1) There exist smooth rational curves C 1 , . . . , C m on Y whose dual graph Γ is a Dynkin diagram of type τ . Under the action of aut(Y ), any smooth rational curve C on Y is in the same orbit as one of C 1 , . . . , C m .
(2) The size of R(Y )/aut(Y ) is given in the 7th column rat of Table 1.1. Except for the cases marked by × in this column, two curves C i and C j are in the same orbit if and only if the vertices of the dual graph Γ corresponding to C i and C j belong to the same connected component of Γ, and hence |R(Y )/aut(Y )| is equal to the number of connected components of the Dynkin diagram of type τ .
We calculate E(Y )/aut(Y ) for (τ,τ )-generic Enriques surfaces. Since the tables span 7 pages, we relegate a part of it to the ancillary files. Theorem 1.21. Let Y be a (τ,τ )-generic Enriques surface. Then the orbits of the action of aut(Y ) on the set E(Y ) of elliptic fibrations of Y are indicated in Section 6.5 for rank τ ≤ 7 and in the ancillary files [32] for rank τ ≥ 8.
1.4. The plan of the paper. This paper is organized as follows. In Section 2, we prepare basic notions about finite quadratic forms, discriminant forms, lattices and chambers. Proposition 2.1 in Section 2.1 plays a crucial role in the proof of the volume formula in the next section. The notion of L/M -chambers given in Section 2.4 is the main tool of our computation. In Section 3, we investigate the nef-and-big cone Nef Y of an Enriques surface Y from the point of view of L/Mchambers, and prove Proposition 1.8. Then, by means of Proposition 2.1, we prove a formula (Theorem 3.4) for the volume of Nef Y /aut(Y ), and in Section 3.4, we deduce Theorem 1.18 from Theorem 3.4.
In Section 4, we present a computational procedure on a graph (Procedure 4.1), which is an abstraction of the generalized Borcherds' method formulated in [28]. Then we recall the classification of primitive embeddings L 10 (2) ֒→ L 26 obtained in [6], and construct primitive embeddings S Y (2) ֒→ S X ֒→ L 26 for (τ,τ )-generic Enriques surfaces Y . In Section 5, we prepare some geometric algorithms used in the application of the generalized Borcherds' method to (τ,τ )-generic Enriques surfaces. In Section 6, we calculate aut(Y ) and Nef Y /aut(Y ), and prove Theorems 1.19 and 1.21. The table of elliptic fibrations is given in Section 6.5.
In Section 7, we exhibit some examples. In particular, we treat an (E 6 , E 6 )generic Enriques surface (No. 47 of Table 1.1) in detail, because we investigated this surface in [31]. Section 7.1 contains a correction of a wrong assertion made in [31].
In the second author's webpage and in the repository "zenodo" [32], we put a detailed computation data made by GAP [34].
Thanks are due to Professor Igor Dolgachev for his comments on the manuscript of this paper. The authors also thank the referees for many valuable comments.

Finite quadratic forms, lattices and chambers
We fix notions and terminologies about finite quadratic forms, discriminant forms, lattices and chambers.

Finite quadratic forms.
A finite quadratic form is a finite abelian group A with a quadratic form q A : A → Q/2Z. We say that a finite quadratic form is non-degenerate if the bilinear form The automorphism group of a finite quadratic form A is denoted by O(A), and we let it act on A from the right. For a subgroup D ⊂ A, let D ⊥ denote the orthogonal complement of D with respect to b A , and let The following proposition will play a crucial role in the proof of the volume formula (Theorem 3.4).
be the graph of φ, which is an isotropic subgroup with respect to q A ⊕ q B . We put C := Γ ⊥ /Γ. Then q A ⊕ q B induces a quadratic form q C on C, and we have a natural homomorphism We denote by K the kernel of this homomorphism. Then the homomorphism is injective, and the image of i A is equal to the kernel of the natural homomorphism In particular, for any β ∈ B, we have α ∈ A such that (α, β) ∈ Γ ⊥ .
Remark 2.2. Proposition 2.1 holds for non-degenerate finite bilinear forms (A, b A ) and (B, b B ) as well.
2.2. Discriminant forms and overlattices. Let L be an even lattice. We put the intersection form of L ′ is the extension of that of L. See Nikulin [22] for the details of the theory of discriminant forms and its application to the enumeration of even overlattices of a given even lattice.
To illustrate Proposition 2.1, we apply it to two known extreme cases. First suppose that L is unimodular. Then, by a result of Nikulin [22], We see that i A : K → O(A) is an isomorphism as predicted by Proposition 2.1. Let P and D be the closures of P and D in L ⊗ R, respectively. A half-line contained in (P \ P) ∩ D is called an isotropic ray of D.
Suppose that D has only finitely many walls, that they are defined by vectors in L ⊗ Q, and that the list of defining vectors of these walls in L ⊗ Q is available. Then we can make the list of faces of D by means of linear programming. For each isotropic ray R ≥0 v, we have a unique primitive vector v ∈ L that generates R ≥0 v, which we call a primitive isotropic ray of D. We can also make the list of primitive isotropic rays of D.

L/M -chambers.
Let (L, , L ) and (M, , M ) be even hyperbolic lattices with fixed positive half-cones P L and P M , respectively. Suppose that we have an embedding M ֒→ L that maps P M into P L . We regard P M as a subspace of P L by this embedding. The notion of L-chambers was introduced in Section 1.2. The following class of chambers plays an important role in this paper.
In particular, an L-chamber is an L/L-chamber.
Let pr : L → M ⊗ Q be the orthogonal projection. Then an L/M -chamber is the closure in P M of a connected component of where r runs through the set of roots r of L such that pr(r), pr(r) M < 0 holds, and (pr(r)) ⊥ = P M ∩ (r) ⊥ is the hyperplane of P M defined by pr(r). Hence, for Since a root of M is mapped to a root of L by the embedding M ֒→ L, an M -chamber is tessellated by L/M -chambers. More generally, we have the following proposition, which is easy to prove: Proposition 2.8. Suppose that M 1 ֒→ M 2 ֒→ L is a sequence of embeddings of even hyperbolic lattices that induces a sequence of embeddings P M1 ֒→ P M2 ֒→ P L of fixed positive half-cones. Then each M 2 /M 1 -chamber is tessellated by L/M 1chambers.
In general, two distinct L/M -chambers are not isomorphic to each other. See [11] and [28] for examples of K3 surfaces X with a primitive embedding S X ֒→ L 26 such that P X is tessellated by L 26 /S X -chambers of various shapes. The tessellation of P L by L/L-chambers is obviously reflexively simple.
3. The cone Nef Y Let Y be an Enriques surface with the universal covering π : X → Y . Let ε ∈ Aut(X) be the deck-transformation of π : X → Y , and we put Then S X+ is equal to the image of π * : S Y (2) ֒→ S X , and S X− is the orthogonal complement of S X+ . We regard P Y as a subspace of P X by π * ⊗ R.
More precisely, every wall of an S X /S Y (2)-chamber D Y is defined by a root r of S Y , and the reflection s r ∈ O P (S Y ) with respect to the root r is the restriction sr + sr − |S Y (2) of the product of two reflections with respect to rootsr + ,r − of S X .
Proof. Let −, − X and −, − Y be the intersection forms of S X and S Y , respectively. We denote by (u) ⊥ X the hyperplane of P X defined by u ∈ S X ⊗R, and by We first prove that r,r ε X = 0. Letr be written as , which is absurd. Let s and s ′ be the reflections with respect to the rootsr andr ε of S X , respectively. By r,r ε X = 0, we have ss ′ = s ′ s. Since s ′ = εsε, we see that ss ′ commutes with ε and hence ss ′ preserves P Y . The vector r :=r +r ε is contained in S Y . Moreover we have r, r Y = −2 and It is easy to confirm that the restriction of ss ′ to S Y is equal to the reflection with respect to the root r of S Y and therefore maps D 3.2. Proof of Proposition 1.8. We prove Proposition 1.8. By Proposition 1.2, we have isomorphisms ψ X and ψ Y that make the diagram (1.2) commutative. By 3.3. The volume of Nef Y /aut(Y ). In this subsection, we give a formula (Theorem 3.4) for vol(Nef Y /aut(Y )) under the assumption that We put (1) The action of G Y on P Y preserves the set of S X /S Y (2)-chambers, and aut(Y ) is equal to the stabilizer subgroup of Nef Y in G Y .
(2) The group W (R(Y )) is contained in G Y as a normal subgroup, and we have Proof. Since every g ∈ G Y lifts to an elementg of G X ⊂ O P (S X ), the action of G Y on P Y preserves the tessellation of P Y by S X /S Y (2)-chambers.
Let aut(X) be the image of the natural representation Aut(X) → O P (S X ). By the Torelli theorem for complex K3 surfaces ([3, Chapter VIII]), we have a natural embedding [22]). Therefore, by assumption (3.1), an isometryg ∈ O P (S X ) belongs to aut(X) if and only ifg preserves Nef X and acts on S ∨ X /S X as ±1. Let Aut(X, ε) denote the centralizer of ε in Aut(X). We have a natural identification Aut(Y ) ∼ = Aut(X, ε)/ ε . Suppose that g ∈ aut(Y ). We will show that g belongs to the stabilizer subgroup of Nef Y in G Y . It is obvious that g preserves Nef Y . Letγ be an element of Aut(X, ε) that induces g on S Y . We writeγ as (g, f ) by (3.3). Note that ε acts on T X as −1. Hence, replacingγ withγε if f = −1, we can assume f = 1. Then the actiong ∈ O P (S X ) ofγ on S X induces the trivial action on S ∨ X /S X , which meansg ∈ G X . Hence g =g|S Y belongs to G Y . Conversely, suppose that g is an element of the stabilizer subgroup of Nef Y in G Y . We will show that g ∈ aut(Y ). Letg be an element of G X such that g =g|S Y .
Since Nef Y contains an interior point of Nef X ,g preserves Nef X , and henceg belongs to aut(X). Letγ = (g, f ) be an element of Aut(X) that inducesg. Sincẽ g ∈ G X commutes with the action of ε on S X , the first factor of the commutator [γ, ε] ∈ Aut(X) is 1. Since O(T X , ω) = {±1} is abelian, the second factor of [γ, ε] is also 1. Henceγ ∈ Aut(X, ε), and therefore g is induced by an element of Aut(Y ). Thus assertion (1) is proved.
By Proposition 3.1, for each r ∈ R(Y ), the reflection s r = sr + sr − |S Y (2) belongs to G Y , because the reflections sr + and sr − act on S ∨ X /S X trivially and hence sr + sr − ∈ G X . Therefore we have W (R(Y )) ⊂ G Y . Moreover, by Proposition 3.1 again, we see that W (R(Y )) acts on the set of S X /S Y (2)-chambers transitively. If Therefore, by the standard method of geometric group theory (see, for example, Section 1.5 of [36]), we see that Nef Y is a standard fundamental domain of the action of W (R(Y )) on P Y , and W (R(Y )) acts on the set of S X /S Y (2)chambers simply-transitively. Recalling that aut(Y ) is the stabilizer subgroup of Moreover G Y is generated by the union of W (R(Y )) and aut(Y ). It remains to show that W (R(Y )) is a normal subgroup of G Y . Let r be a root in R(Y ) and g an arbitrary element of G Y . It is enough to show that g −1 s r g belongs to W (R(Y )). Note that g −1 s r g = s r g and r g defines a wall of the S X /S Y (2)-chamber D Y := Nef Y g . We have an element w ∈ W (R(Y )) such that D Y = Nef Y w . Then r ′ := r gw −1 defines a wall of Nef Y , and ws r g w −1 = s r ′ is an element of W (R(Y )).
Let (A + , q + ) and (A − , q − ) be the discriminant forms of S X+ = S Y (2) and S X− , respectively. We put and let D + ⊂ A + and D − ⊂ A − be the image of the projections of Γ X . Then Γ X is the graph of an isometry (D + , q + |D + ) ∼ = (D − , −q − |D − ), and the discriminant group of S X is canonically isomorphic to Γ ⊥ X /Γ X . We denote by G X+ and G X− the images of G X+ and G X− by the natural homomorphisms Then we have a commutative diagram where the two arrows below are isomorphisms by the first part of Proposition 2. Proof. There is an isomorphism of Aut nt (Y ) with Ker (G X → G X+ ) given by mapping a numerically trivial automorphism g to its liftg ∈ Aut(X) acting trivially on the 2-form ω and restricting to its action on S X . By the diagram (3.4) and Conversely any element of Ker(G X− → G X− ) can be extended to an element of Ker (G X → G X+ ) by complementing it with the trivial action of S X+ .
consisting of isometries g whose action on A − preserves D − . Then we have Proof. Recall that we have |G X | = |G X+ | = |G X− |. Let G BP be the kernel of the natural homomorphism O P (S X+ ) = O P (S Y (2)) → O(A + ). Then G BP is equal to aut(Y 0 ) by Theorem 1.17 and hence the index of G BP in O P (S X+ ) = O P (S Y ) is 1 BP . If g ∈ G BP , then (g, 1) ∈ O P (S X+ ) × O(S X− ) acts trivially on A + ⊕ A − , and hence preserves Γ X and acts on Γ ⊥ X /Γ X trivially. Therefore the action of (g, 1) on S X+ ⊕ S X− preserves the overlattice S X , and (g, 1)|S X is an element of Applying the second part of Proposition 2.1 to (A, B) = (A − , A + ), we see that . Hence the inclusion ⊂ in (3.5) is proved. Conversely, let f be an element of the right-hand side of (3.5), and denote byf ∈ O(A − ) the action of f on A − . By Proposition 2.1, we havef ∈ Im i A− and hence there exists a unique elementh ∈ K such that i A− (h) =f . We putḡ := i A+ (h). Since the natural homomorphism where the first equality follows from Proposition 3.2. From Lemma 3.3 we get the second equality of (3.6).
Since S X− is negative definite, O(S X− ) is a finite group and can be computed easily. Thus this formula enables us to calculate vol(Nef Y /aut(Y )).
Denote by π − : S X → S ∨ X− the orthogonal projection. Identify M R with S X viag. Then the following equalities hold: Note that we neglect the quadratic forms in (1)-(5) and just consider them as equalities of abelian groups.
Proof. The equality (1) is by the definition.
(2) Note that M R is spanned by Im ̟ R and {(i R (v) ± v)/2 | v ∈ R}. Hence π − (M R ) is spanned by 0 and 1 2 R. (3) As lattices we have R(2) = S X− , and ( R(2)) ∨ = 1 2 R ∨ yields the claim. (4) By definition, we have π − (S X )/S X− = D − . (5) Let x ∈ 1 2 R and y ∈ R ∨ . Then x, y MR = 2 x, y R ≡ 0 mod Z and x + R ∈ D − . This shows that . Let R be an ADE-lattice and Φ the set of its roots. We fix a subset Φ + ⊂ Φ of positive roots. There exists a unique Weyl-chamber C of R (see Definition 2.5) such that for all r ∈ Φ + and c ∈ C we have r, c > 0. We call C the fundamental chamber. The positive roots perpendicular to the walls of C are the so-called simple roots. The simple roots form a basis of R whose Dynkin diagram is of ADE-type τ (R). As  If τ (R) is irreducible, a case by case analysis shows that this map is injective: indeed for A 1 , E 7 and E 8 , Aut(τ (R)) = 1; for A k with k ≥ 1, D k with k > 4 and E 6 the group Aut(τ (R)) is of order two. A direct computation shows that it acts faithfully on the discriminant group. Suppose that the root system τ (R) is reducible. The decomposition of τ (R) into connected components corresponds to a decomposition of R into an orthogonal sum of irreducible ADE-lattices, which in turn induces a corresponding decomposition of the discriminant group R ∨ /R. The action of Aut(τ (R)) preserves the three decompositions. Hence the elements of Ker ψ must preserve the components which have a non-trivial discriminant group, that is, all components which are not of type E 8 . By the first part, they must act trivially on these components. Finally, since the E 8 diagram has no symmetry, the elements in the kernel act as a permutation of the connected components of τ (R) of type E 8 .
Lemma 3.7. Let R be an ADE-lattice of rank at most 10 and R an even overlattice. Consider the homomorphism If there is a component R j of R with τ ( R j ) = E 8 and τ ( R j ∩ R) = 2D 4 , then the kernel of (3.7) is W (R) ⋊ h where h ∈ Aut(τ (R), R) is an involution. Otherwise the kernel is just the Weyl group W (R).
Proof. Let Aut(τ (R), R) ≤ Aut(τ (R)) be the stabilizer of R. Since the elements of W (R) act trivially on R ∨ /R, they preserve R and The elements of W (R) act trivially on the domain of R ∨ /R ։ R ∨ / R, so they lie in the kernel of (3.7). Thus it suffices to compute the kernel of Indeed, the kernel of (3.7) is given by W (R) ⋊ Ker ϕ.
First we suppose that τ (R) is irreducible. If R = R, then W (R) = O 0 (R) by Lemma 3.6, and hence ϕ is injective. Otherwise (as rank R ≤ 10) the pair (τ (R), τ ( R)) ∈ {(A 7 , E 7 ), (A 8 , E 8 ), (D 8 , E 8 )}. Suppose we are in the case (A 7 , E 7 ). Then R ∨ /R ∼ = Z/8Z and R/R = 4(R ∨ /R). Then Aut(τ (R)) is of order two and acts as ±1 on R ∨ /R which is non-trivial in R ∨ / R ∼ = Z/4Z. A similar argument applies to (A 8 , E 8 ). Finally the symmetry of the D 8 diagram exchanges the two isotropic vectors of its discriminant. In particular it does not fix any non-trivial even overlattice which implies that Aut(τ (R), R) = 1 in the (D 8 , E 8 ) case. In any case ϕ is injective. Now suppose that R = R i has several irreducible components R i and let h ∈ Ker ϕ. Note that h preserves the decomposition R ∨ = R ∨ i . Let x ∈ R ∨ i be a non-zero element.
If x h lies in the same component R ∨ i as x, then h must preserve it. Hence we may restrict h to this component and the previous paragraph yields x h = x.
If x and x h lie in different components R ∨ i and R ∨ j , then these components are isomorphic and q( If y is any non-trivial element of R ∨ i , then x h and y h lie in the same connected component R ∨ j and the same reasoning applies. In particular ∀y ∈ R ∨ i : q(y) ≡ 0 mod Z which implies that R i is 2-elementary and q Ri has values in Z/2Z. Under the constraint rank R ≤ 10, this is possible only if τ (R i ) = τ (R j ) = D 4 . To sum up ϕ is injective, except possibly if τ (R) has two D 4 components. We analyse this case in detail.
We may assume that R = R 1 ⊕ R 2 is of type 2D 4 andR an overlattice of R. If R = R, then ϕ is injective by Lemma 3.6. Hence we may further assume that Then f h i = e σ(i) for some permutation σ ∈ S 4 with σ(4) = 4. Since h ∈ Ker ϕ, we have t i := e ∨ i − f ∨ i ∈ R for i ∈ {1, 2, 3}. Now the cosets of 0, t 1 , t 2 and t 3 constitute a maximal totally isotropic subspace of R ∨ /R contained in R/R. Since R/R is totally isotropic as well, the subspaces must be equal. We conclude that τ ( R) = E 8 . By the same reasoning we have f ∨ i − e ∨ σ(i) ∈ R. As R/R has only four elements, this is possible only if σ = 1. Hence h is an involution and uniquely determined by R/R. This shows that the kernel of ϕ is of order 2.
Lemma 3.8. Let R be an ADE-lattice and Φ + the set of its positive roots. Then the natural map Φ + → R/2 R is injective.
Proof. We may assume that R is irreducible. In what follows we explicitly compute η : Φ + → R/2 R for each case using classical constructions of the ADE-lattices (see e.g. [10, Theorem 1.2]).
Lemma 3.9. Let R = j∈J R j be an ADE-lattice with R j irreducible. Then the kernel of the natural homomorphism , is generated by the elements ⊕ j∈J g j with g j = ±1 Rj if R j is unimodular and g j = 1 Rj otherwise.
Proof. We identify 1 2 R ∨ / R and R ∨ /2 R. Let g ∈ Ker ψ. Since R ⊆ R ∨ , g acts trivially on R ∨ /2 R ∨ . The action of O( R) preserves the decomposition R = j∈J R j .
In particular g acts on the set J. As R ∨ /2 R ∨ = j∈J R ∨ j /2 R ∨ j and g is in Ker ψ we have j g = j. Hence g must fix each connected component of R and we may and will assume that R is irreducible.
We tensor the perfect pairing R ∨ × R → Z with F 2 , to obtain a perfect pairing R ∨ /2 R ∨ × R/2 R → F 2 . Since g acts trivially on the first factor, so does it on the second factor R/2 R. By Lemma 3.8 Φ( R)/{±1} ∼ = Φ + ( R) injects into R/2 R, which implies that g(r) = ±r for every root r ∈ Φ( R). As any simple root system of R is connected, the sign is the same for each simple root. Since the simple roots form a basis, g = ±1.
Set R ± = Ker(g ∓ 1) ⊂ R. We apply Proposition 2.1 to the primitive extension Since g acts trivially on the discriminant group 1 2 R ∨ / R of R(2), the implication (3.8) holds. By definition g| R− = −1 R− and then by the right hand side of (3.8), the lattice R − (2) must be 2-elementary, i.e. R − is unimodular. In particular R − is a direct summand of R. But we assumed the latter to be irreducible, so that R ∈ {0, R − }. Thus g = ±1 if R is unimodular and g = 1 else. Theorem 3.11. Let Y be a (τ,τ )-generic Enriques surface, and let R, R, R be as in Table 1.1. Let R = j R j be the decomposition into irreducible components. Then we have where d (τ,τ ) , e (τ,τ) are given as follows.

Proof. By Theorem 3.4 and Lemma 3.5, we have
which, by Lemma 3.7, is given by . By our dictionary in Lemma 3.5, we have G X− = ψ(G X− ). By Lemma 3.9, the kernel of ψ consists of those g = ⊕ j∈J g j with g j = ±1 Rj if R j is unimodular and g j = 1 Rj else. Further Ker ψ ∩ W (R) consists of those g with g j = ±1 if R j is unimodular and −1 ∈ W (R ∩ R j ), and g j = 1 else. Now Remark 3.10 yields the condition for e (τ,τ) . Since the g j = ±1 do not preserve any positive root system, the involution h is not in Ker ψ. This explains the presence of d (τ,τ ) . Finally, in the geometric situation, we have G X− = W (R) (see Remark 3.12 below), and hence Aut nt (Y ) ∼ = Ker G X− → G X− = Ker ψ ∩ W (R) gives e (τ,τ) = | Aut nt (Y )|, where the isomorphism follows from Lemma 3.3.
Remark 3.12. The factor d (τ,τ ) is nontrivial only for Nos. 142 and 170 which are not realized geometrically. This is explained by an extra "automorphism" of Y which exchanges two D 4 configurations of "smooth rational curves" and acts trivially on their orthogonal complement in S Y . This is not visible in the Weyl group. Thus in the geometric cases a nontrivial contribution of c (τ,τ ) = e (τ,τ) is indeed explained by the presence of a numerically trivial involution of Y .

Borcherds' method
4.1. An algorithm on a graph. The algorithms to prove our main results are variations of the following computational procedure.
Let (V, E) be a simple non-oriented connected graph, where V is the set of vertices and E is the set of edges, which is a set of non-ordered pairs of distinct elements of V . The set V may be infinite. Suppose that a group G acts on (V, E) from the right. We assume the following.
to v is finite and can be calculated effectively.
(VE-2) For any vertices v, v ′ ∈ V , we can determine effectively whether the set is empty or not, and when it is non-empty, we can calculate an element of v) of v in G is finitely generated, and a finite set of generators of T G (v, v) can be calculated effectively.
Suppose that V 0 is a non-empty finite subset of V with the following properties.
We fix an element v 0 ∈ V 0 .
Proposition 4.1. The natural mapping is a bijection, and the group G is generated by the union of T G (v 0 , v 0 ) and H.
Proof. Let H be the subgroup of G generated by H. First we prove that, for any v ∈ V , there exists an element h ∈ H such that v h ∈ V 0 . Let an element v ∈ V be fixed. A sequence is a path from V 0 to v H of length l − 1. Thus we obtain a path from V 0 to v H of length 0, which implies the claim. The injectivity of (4.2) follows from property (V 0 -1) of V 0 . The surjectivity follows from the claim above. Suppose that g ∈ G. By the claim, there exists an element h ∈ H such that v gh 0 ∈ V 0 . By property (V 0 -1) of V 0 , we have v 0 = v gh 0 and hence gh ∈ T G (v 0 , v 0 ). Therefore G is generated by the union of H and T G (v 0 , v 0 ).
To obtain V 0 and H, we employ Procedure 4.1. This procedure terminates if and only if |V /G| < ∞.
Replace flag by false. Break from the innermost for-loop. if flag = true then Append v ′ to the list V 0 as the last entry. Replace i by i + 1. Recall that L 26 is an even unimodular hyperbolic lattice of rank 26. The L 26 -chamber (that is, the standard fundamental domain of W (L 26 )) was studied by Conway [7]. He constructed a bijection between the set of walls of an L 26 -chamber D and the set of vectors of the Leech lattice, and showed that the automorphism group O(L 26 , D) of D is isomorphic to the group of affine isometries of the Leech lattice. Using this result, Borcherds [4], [5] developed a method to calculate the orthogonal group of an even hyperbolic lattice S by embedding S primitively into L 26 and investigating the tessellation of an S-chamber (that is, a standard fundamental domain of W (S)) by L 26 /S-chambers.
In [6], we apply this method to S = L 10 (2). We fix positive half-cones P 10 of L 10 and P 26 of L 26 . In [6], we have proved the following.  The explicit description of the 17 primitive embeddings and L 26 /L 10 (2)-chambers is given in [6] and [30]. From these data, we see the following. Let L 10 (2) ֒→ L 26 be a primitive embedding whose type is not infty, and D an L 26 /L 10 (2)-chamber. Let f be a face of D with codimension k. Then the defining roots of the walls of D containing f form a configuration whose dual graph is a Dynkin diagram of an ADE-type. The ADE-type of f is the ADE-type of this Dynkin diagram. The closure D of D in S X ⊗ R contains only a finite number of isotropic rays. Let v ∈ S X ∩ D be a primitive isotropic ray (see Section 2.3). Then the defining roots r of walls of D such that r, v = 0 form a configuration whose dual graph is a Dynkin diagram of an affine ADE-type. The affine ADE-type of the isotropic ray R >0 v is the affine ADE-type of this Dynkin diagram.

4.3.
Constructing S X . Let Y be an Enriques surface with the universal covering π : X → Y . We consider the following assumption: we have a primitive embedding S X ֒→ L 26 such that the composite S Y (2) ∼ = L 10 (2) ֒→ L 26 of π * : S Y (2) ֒→ S X and S X ֒→ L 26 is not of type infty, and we have the list of walls of an L 26 /S Y (2)-chamber D 0 that is contained in Nef Y .
Suppose that (4.4) holds. Then P Y has the following three tessellations, each of which is a refinement of the one below.
• by Vinberg chambers, • by L 26 /S Y (2)-chambers, each of which has only finite number of walls, and • by S X /S Y (2)-chambers, one of which is Nef Y . The tessellation of Nef Y by L 26 /S Y (2)-chambers is very useful in analyzing Nef Y .  1) and (4.4). Then the action of G Y on P Y preserves the tessellation of P Y by L 26 /S Y (2)-chambers. In particular, the action of aut(Y ) on Nef Y preserves the tessellation of Nef Y by L 26 /S Y (2)chambers.
Proof. It is enough to prove that the action ofg ∈ G X on P X preserves the tessellation of P X by L 26 /S X -chambers. Let id P be the identity of the orthogonal complement P of S X in L 26 . Since the action ofg on S ∨ X /S X is 1, the action of (g, id P ) on S X ⊕ P preserves the even unimodular overlattice L 26 of S X ⊕ P . Thus g extends to an isometry of L 26 , and hence its action on P X preserves the L 26 /S Xchambers. The second assertion follows from the fact that aut(Y ) is the stabilizer subgroup of Nef Y in G Y .
The purpose of this section is to construct a primitive embedding S X ֒→ L 26 for a (τ,τ )-generic Enriques surface Y , so that we can assume (4.4). We start from a primitive embedding ι : L 10 (2) ֒→ L 26 whose type is not infty and which has a fixed L 26 /L 10 (2)-chamber D 0 , and then proceed to the construction of S X between L 10 (2) ∼ = S Y (2) and L 26 such that the inclusion of L 10 (2) ∼ = S Y (2) into S X is the embedding π * , and that the fixed L 26 /L 10 (2)-chamber D 0 is contained in Nef Y .
Recall that, for a (τ,τ )-generic Enriques surface Y , the lattice S X is obtained from S Y (2) by adding roots of the form (r + v)/2, where r is a root of S Y and v is a (−4)-vector in S X− . To find roots in L 26 that yield an appropriate extension from S Y (2) to S X , we search for pairs α = (r, v) of a root r of L 10 defining a wall of D 0 and a (−4)-vector v of Q ι such that (r + v)/2 is in L 26 , where Q ι is the orthogonal complement of L 10 (2) in L 26 . For a finite set p = {α 1 , . . . , α m } of such pairs, we consider the sublattice M p of L 26 generated by L 10 (2) and the roots ( . . , α m } satisfies the following: (i) The dual graph of r 1 , . . . , r m is a Dynkin diagram of some ADE-type τ . By Proposition 1.2, the primitive closure R of the ADE-sublattice R of L 10 generated by r 1 , . . . , r m is also an ADE-sublattice of L 10 . Letτ denote the ADEtype of R. By Proposition 1.2, the embedding L 10 (2) ֒→ M p is isomorphic to L 10 (2) ֒→ M R , and hence, by Proposition 1.3, we see that L 10 (2) is a primitive sublattice of M p , and the orthogonal complement of L 10 (2) in M p contains no roots. We consider the following condition: (ii) M p can be embedded primitively into the K3 lattice (an even unimodular lattice of rank 22 with signature (3,19)). This condition is checked by calculating the discriminant form of M p and applying the theory of genera (see [22]). Suppose that M p satisfies condition (ii). Since 22 − rank M p = 12 − m > 2, the surjectivity of the period mapping of complex K3 surfaces ([3, Chapter VIII]) implies that there exists a K3 surface X with M p ∼ = S X such that O(T X , ω) = {±1}. Moreover, by [13], the K3 surface X has a fixed point free involution ε with the quotient morphism π : X → Y = X/ ε to the Enriques surface Y such that, under suitable choices of isometries M p ∼ = S X , the embedding L 10 (2) ֒→ M p is identified with π * : S Y (2) ֒→ S X . By the construction of M p , this Enriques surface Y is (τ,τ )generic. Thanks to Proposition 3.1, we can further assume that D 0 is contained in Nef Y by changing the isometry M p ∼ = S X .
Remark 4.7. Even when M p does not satisfy condition (ii), we can use M p as the Néron-Severi lattice S X of a "non-existing K3 surface" X and run the geometric algorithms below.

Geometric algorithms
We prepare some algorithms that will be used in the application of the generalized Borcherds' method to geometric situations.
Let Y be an Enriques surface with the universal covering π : X → Y . We assume (3.1) and (4.4). First we prepare the following computational data: (i) an integral interior point a Y 0 ∈ S Y of D 0 , which is an ample class of Y , (ii) the list of roots defining the walls of D 0 , Definition 5.1. Let L be an even hyperbolic lattice with a positive half-cone P, and let a 1 , a 2 be elements of P ∩ L. We say that a hyperplane (v) ⊥ of P separates a 1 and a 2 if v, a 1 and v, a 2 are non-zero and have different signs. We say that a vector v ∈ L ⊗ Q with v, v < 0 separates a 1 and a 2 if (v) ⊥ separates a 1 and a 2 .
By an algorithm given in [27], we can calculate, for any a 1 , a 2 ∈ P ∩ L, the set of roots of L that separate a 1 and a 2 .

Splitting roots.
Definition 5.2. We say that a root r of S Y splits in S X if there exists a rootr of S X such that π * (r) =r +r ε .
A root r of S Y splits in S X if and only if there exists a (−4)-vector v of S X− such that (π * (r) + v)/2 ∈ S X . Hence we can effectively determine whether a given root r of S Y splits in S X or not. Moreover, when r splits, we can calculate the rootsr = (π * (r) + v)/2 andr ε = (π * (r) − v)/2 of S X such that π * (r) =r +r ε .
Suppose that a root r of S Y satisfies that Nef Y ∩ (r) ⊥ contains a non-empty open subset of (r) ⊥ and that r, a Y > 0 for an ample class a Y of Y . Then the following are equivalent: • Nef Y ∩ (r) ⊥ is a wall of Nef Y (that is, the hyperplane (r) ⊥ is disjoint from the interior of Nef Y ), • r splits in S X , and • r is the class of a smooth rational curve C on Y . In this case, the rootsr andr ε of S X are the classes of the smooth rational curves C and C ε on X such that π −1 (C) = C + C ε .
and only if there exists an isometry h ∈ O(S X− ) such that the action of (g, h) on S X+ ⊕ S X− preserves the overlattice S X and thatg := (g, h)|S X acts on S ∨ X /S X trivially. Since we have the list of elements of the finite group O(S X− ), we can determine whether an element g ∈ O P (S Y ) belongs to G Y or not, and if g ∈ O P (S Y ), we can calculate a liftg ∈ G X of g.

5.4.
Membership criterion of aut(Y ) in G Y . Suppose that g ∈ G Y , and let g ∈ G X be a lift of g. Recall from Proposition 3.2 that g belongs to aut(Y ) if and only if g preserves Nef Y , or equivalentlyg preserves Nef X . Hence g ∈ aut(Y ) holds if and only if one of the following conditions that are mutually equivalent is satisfied: • For any ample classes a X and a ′ X of X, there exist no root of S X separating ag X and a ′ X . • For any ample classes a Y and a ′ Y of Y , any roots of S Y separating a g Y and a ′ Y does not split in S X . • There exist ample classes a X and a ′ X of X such that there exist no roots of S X separating ag X and a ′ X . • There exist ample classes a Y and a ′ Y of Y such that any root of S Y separating a g Y and a ′ Y does not split in S X . Thus we can determine effectively whether a given isometry g ∈ G Y belongs to aut(Y ) or not, because we have at least one ample class a Y 0 of Y .

Proofs of main theorems
We present algorithms that prove Theorems 1.19 and 1.21. Let Y be an Enriques surface with the universal covering π : X → Y . Suppose that Y is (τ,τ )-generic, where (τ,τ ) is not equal to No. 88 nor No. 146 in Table 1.1, so that we can assume (3.1) and (4.4).
6.1. Generators of aut(Y ) and representatives of Nef Y /aut(Y ). We calculate a finite generating set of aut(Y ) and a complete set of representatives of Nef Y /aut(Y ). This calculation affirms Theorem 1.18 computationally. Moreover the results will be used in the proofs of Theorems 1.19 and 1.21 below.
Let (V, E) be the graph where V is the set of L 26 /S Y (2)-chambers contained in Nef Y and E is defined by the adjacency relation of L 26 /S Y (2)-chambers. Let G be the group aut(Y ), and let v 0 ∈ V be the . Then we can calculate the set of roots defining the walls of D by mapping the set of roots defining the walls of D 0 by the isometry g. For each root r defining a wall of D, the chamber D sr = D gsr 0 adjacent to D across the wall D ∩ (r) ⊥ of D is contained in Nef Y if and only if r does not split in S X . Therefore we can determine D sr ⊂ Nef Y or not by the method in Section 5.2. Therefore condition (VE-1) in Section 4.1 is satisfied. Since we can calculate isom(Y, D g 0 , D g ′ 0 ) for any g, g ′ ∈ O P (S Y ) by Section 5.5, conditions (VE-2) and (VE-3) are also satisfied. Therefore we can apply Procedure 4.1 to the graph (V, E) and the group G, and obtain a complete set V 0 of representatives of orbits of the action of G on V , the stabilizer subgroups isom(Y, D, D) = aut(Y, D) of these representatives D ∈ V 0 , and a generating set of aut(Y ). Then we have Thus Theorem 1.18 is computationally affirmed.
Remark 6.1. The amount of computation of Procedure 4.1 grows quadratically as |V /G| becomes large, because we have to check T G (v, v ′ ) = ∅ for all pairs of distinct v, v ′ ∈ V 0 . We could calculate a finite generating set of aut(Y ) by using, naively, the graph (V ′ , E ′ ), where V ′ is the set of Vinberg chambers contained in Nef Y and E ′ is the adjacency relation of Vinberg chambers. However, the size of V ′ /aut(Y ) is approximately vol(D 0 ) times the size of V /aut(Y ). Thus, very roughly speaking, using the primitive embedding S Y (2) ֒→ L 26 of type 96C gives us computational advantage of multiplicative factor the square of vol(D 0 ) = 652758220800.
6.2. Calculating R temp , E temp and G X . From V 0 and G calculated above, we compute the following data, which will be used in Sections 6.3 and 6.4.
Since D ⊂ Nef Y , a root r defining a wall of D belongs to R(Y ) if and only if r splits in S X . Therefore we can calculate R(Y, D) by the method in Section 5.2. We put Then the mapping R temp ֒→ R(Y ) → → R(Y )/aut(Y ) is surjective. Via the generating set G, we can generate (pseudo-)random elements of aut(Y ) = G . For [C], [C ′ ] ∈ R temp , if we find g ∈ aut(Y ) such that [C] g = [C ′ ], then we remove [C ′ ] from R temp . Repeating this process many times, we obtain a smaller subset R ′ temp of R(Y ) that is mapped to R(Y )/aut(Y ) surjectively. Let φ : Y → P 1 be an elliptic fibration of Y , and F a general fiber of φ. Then f φ := [F ]/2 ∈ S Y is a primitive isotropic ray (see Section 2.3 for the definition) contained in the closure of Nef Y in P Y . For each D ∈ V 0 , let E(Y, D) be the set of primitive isotropic rays contained in the closure D of D in P Y . We put Then the mapping E temp ֒→ E(Y ) → → E(Y )/aut(Y ) is surjective. As above, from E temp and using G, we obtain a smaller subset E ′ temp of E(Y ) that is mapped to E(Y )/aut(Y ) surjectively.
Let Aut(X, ε) be the centralizer of ε ∈ Aut(X) in Aut(X), and let aut(X, ε) be the image of Aut(X, ε) in aut(X). We write an elementγ ∈ Aut(X) as (g, f ) by (3.3). Since O(T X , ω) = {±1} is abelian, we see thatγ commutes with ε ∈ Aut(X) if and only ifg commutes with ε ∈ aut(X). Hence aut(X, ε) is equal to the centralizer of ε ∈ aut(X) in aut(X). By the Torelli theorem (see the proof of Proposition 3.2), an elementg of O P (S X ) belongs to aut(X, ε) if and only ifg acts on S ∨ X /S X as ±1, preserves Nef X , and commutes with ε ∈ O P (S X ). Let aut(X, ε) 0 be the group consisting of elementsg ∈ aut(X, ε) that act on S ∨ X /S X as 1. We have aut(X, ε) 0 = aut(X, ε) ∩ G X .
The kernel K is naturally embedded into O(S X− ) byg →g|S X− . We put By definition K 0 acts trivially on S ∨ X+ /S X+ and by Proposition 2.1 it must act trivially on S ∨ X− /S X− as well. Hence, regarded as a subgroup of G X− ⊂ O(S X− ), K 0 is contained in the kernel of . Conversely the elements of Ker ψ can be extended by the identity on S X+ to elements of G X which trivially preserve Nef Y . Hence they are induced by automorphisms of Y and we have K 0 = Ker ψ. The kernel of ψ is explicitly computed in the proof of Theorem 3.11. Its order is given by e τ,τ ∈ {1, 2}. Suppose that e τ,τ = 2. If ε ∈ K 0 , then K = K 0 = ε . This is the case if in addition τ ( R) = E 8 . Otherwise For each g in the generating set G of aut(Y ), we calculate a liftg ∈ aut(X, ε) of g, and put G X := {g | g ∈ G } ∪ K. Then aut(X, ε) is generated by G X . 6.3. Rational curves on Y . We prove Theorem 1.19. By the construction of S X given in Section 4.3, we have a set of splitting roots that define some walls of D 0 ⊂ Nef Y and form the dual graph of ADE-type τ . Therefore the existence of C 1 , . . . , C m in assertion (1) is proved.
Let C be a smooth rational curve on Y , and r := [C] the class of C. Let V C be the set of L 26 /S Y (2)-chambers D such that D ∩ (r) ⊥ is a wall of D and that D is located on the same side of (r) ⊥ as Nef Y . Let D be an element of V C , and suppose that F := D ∩ (r) ⊥ ∩ (r ′ ) ⊥ is a face of codimension 2 of D that is a boundary of the wall D ∩ (r) ⊥ , where r ′ is a root of S Y defining a wall of D. Then there exists a unique element D ′ of V C such that D ∩ D ′ = F holds. We say that this chamber D ′ is adjacent in V C to D across F . This L 26 /S Y (2)-chamber D ′ is calculated as follows. As is seen from the set of faces of L 26 /S Y (2)-chambers (see [30]), we have r, r ′ = 0 or r, r ′ = 1. Let s and s ′ be the reflections with respect to the roots r = [C] and r ′ , respectively. Then Suppose that D is contained in Nef Y . Then D ′ is contained in Nef Y if and only if r ′ is not the class of a smooth rational curve on Y , or equivalently, r ′ does not split in S X . We consider the graph (V C , E C ), where V C is the set of L 26 /S Y (2)-chambers D ∈ V C contained in Nef Y , and E C is the restriction to V C ⊂ V C of the adjacency relation on V C defined above. Then the stabilizer subgroup of C in aut(Y ) acts on (V C , E C ). For D, D ′ ∈ V C , we have where T G (D, D ′ ) ⊂ G C is defined by (4.1), and isoms(Y, D, D ′ ) is defined in Section 5.5. Therefore (V C , E C ) and G C satisfy conditions (VE-1), . . . , (VE-3) in Section 4.1. We apply Procedure 4.1 to every C ∈ R ′ temp and obtain a complete set V C,0 of representatives of orbits of the action of G C on V C .
Two elements C and C ′ of R ′ temp are contained in the same orbit under the action of aut(Y ) on R(Y ) if and only if we have one of the following conditions that are mutually equivalent.
• Let D be an arbitrary element of V C,0 . Then there exists an L 26 /S Y (2)chamber D ′ in V C ′ ,0 such that isoms(Y, D, D ′ ) contains an isometry g such that [C] g = [C ′ ]. • There exist a pair of L 26 /S Y (2)-chambers D ∈ V C,0 and D ′ ∈ V C ′ ,0 and an isometry g ∈ isoms(Y, D, D ′ ) such that [C] g = [C ′ ]. Applying this method to all pairs C, C ′ of distinct elements of R ′ temp , we obtain a complete set of representatives C ′ 1 , . . . , C ′ k of orbits of the action of aut(Y ) on R(Y ). We then apply this method to the representatives C ′ 1 , . . . , C ′ k and the smooth rational curves C 1 , . . . , C m in assertion (1), and complete the proof of Theorem 1.19.
The algorithm given above is a priori guaranteed to work. A posteriori, Theorem 1.19 can be verified by the following simple strategy. Let aut(X, ε)|S X− be the image of the homomorphism given byg →g|S X− . Since we have calculated a finite generating set G X of aut(X, ε), we can calculate the elements of the finite group aut(X, ε)|S X− . Let C, C ′ be elements of R(Y ). If the orbit of {±v C } ⊂ S X− by aut(X, ε)|S X− and that of {±v C ′ } are disjoint, then the orbits of C and C ′ by aut(Y ) are disjoint. Even though the converse does not necessarily hold, we know a posteriori that once the size of R ′ temp is small enough, this separates the orbits of R ′ temp .
6.4. Elliptic fibrations of Y . Let φ : Y → P 1 be an elliptic fibration of Y . We consider the following graph (V φ , E φ ). We define V φ to be the set of L 26 /S Y (2)chambers D contained in Nef Y such that the closure D of D in S Y ⊗ R contains the primitive isotropic ray f φ = [F ]/2, where F is a general fiber of φ, and E φ to be the set of pairs of adjacent L 26 /S Y (2)-chambers in V φ . The stabilizer subgroup . Then condition (VE-1) is satisfied. Indeed, the set of L 26 /S Y (2)-chambers in V φ adjacent to D ∈ V φ is the set of all D sr , where r runs through the set of non-splitting roots of S Y defining walls of D such that r, f φ = 0. For D, D ′ ∈ V φ , the subset T G (D, D ′ ) of G φ is the set of isometries belonging to isoms(Y, D, D ′ ) that fixes f φ . Therefore (VE-2) and (VE-3) are also satisfied. We apply Procedure 4.1 to every φ ∈ E ′ temp and obtain a complete set V φ,0 of representatives of orbits of the action of G φ on V φ . We also obtain a finite generating set G φ of the stabilizer subgroup aut(Y, φ).
The set Σ φ of classes of smooth rational curves C contained in some fiber of φ is calculated as follows. Let a Y be an ample class of Y . Every class [C] ∈ Σ φ satisfies [C], f φ = 0 and 0 < [C], a Y < 2 f φ , a Y . We calculate the set Σ ′ of all roots r of S Y satisfying r, f φ = 0 and 0 < r, a Y < 2 f φ , a Y . Then r ∈ Σ ′ belongs to Σ φ if and only if r splits in S X (see Section 5.2) and there exist no roots r ′ ∈ Σ φ such that r ′ , a Y < r, a Y and r, r ′ < 0. Therefore we can calculate Σ φ by sorting the elements r of Σ ′ according to r, a Y and applying the above criterion to r ∈ Σ ′ in this order.
Each connected component of the dual graph of roots in Σ φ corresponds to a reducible fiber of φ, and is the Dynkin diagram of an affine ADE-type. Let Γ be a connected component. The weighted sum of roots in Γ with appropriate weights according to the ADE-type of Γ (see, for example, [26,Theorem 5.12]) is either f φ or 2f φ . The former case occurs when the corresponding reducible fiber is a multiple fiber, while the latter occurs when the fiber is non-multiple.
We can construct a sequence of primitive embeddings S Y (2) ֒→ S X ֒→ L 26 from the primitive embeddings L 10 (2) ֒→ L 26 of type 20E. We see that D 0 is a fundamental domain of the action of aut(Y ) on Nef Y , and hence vol(Nef Y /aut(Y )) = vol(D 0 ) = 1 BP 51840 = 1 BP |W (R E6 )| .
In fact, the L 26 /S Y (2)-chamber D 0 is equal to the chamber D Y in [31]. We then obtain the same result as  [31] says that there exist 10 orbits of the action of aut(Y ) on R(Y ). In fact, the argument in Section 7.6 of [31] for the calculation of the number of aut(Y )-orbits of RDPconfigurations is wrong, and Table 1.2 of [31] should be replaced by Table 7.1 below.
Here we present a correct method for the calculation of aut(Y )-orbits of RDPconfigurations. Let ψ : Y → Y be a birational morphism to a surface Y that has only rational double points as its singularities, and let h ψ be an ample class of Y . Since the L 26 /S Y (2)-chamber D 0 is a fundamental domain of the action of aut(Y ) on Nef Y , we can assume that ψ * (h ψ ) ∈ S Y belongs to D 0 by composing ψ with an ADE-type number  For a given face f of D 0 , we calculate the set of roots r of S Y such that f ⊂ (r) ⊥ . From this set, we can calculate Γ(f ) by using the ample class a Y and the set of (−4)-vectors in S X− . We calculate Γ(f ) for all faces f of D 0 , and obtain 750 RDPconfigurations of smooth rational curves. Every RDP-configuration on Y is equal to one of them modulo the action of aut(Y ). Let Γ be one of the 750 RDP-configurations. We put µ := |Γ|, that is, µ is the total Milnor number of the singularities of the surface Y corresponding to Γ. The sublattice Γ of S Y generated by the classes in Γ is negative definite of rank µ, and its orthogonal complement Γ ⊥ is hyperbolic of rank 10 − µ. Let P Γ ⊥ be the positive half-cone of Γ ⊥ contained in P Y . Composing the primitive embedding Γ ⊥ ֒→ S Y with the primitive embedding S Y (2) ֒→ L 26 of type 20E, we have L 26 / Γ ⊥ (2)-chambers of P Γ ⊥ . The intersection f 0 := P Γ ⊥ ∩ D 0 is one of the L 26 / Γ ⊥ (2)-chambers, and it is the maximal face of D 0 among all the faces f of D 0 such that Γ(f ) = Γ. Let (V Γ , E Γ ) be the graph where V Γ is the set of L 26 / Γ ⊥ (2)chambers on P Γ ⊥ contained in P Γ ⊥ ∩Nef Y and E Γ is the usual adjacency relation of chambers. Then D → P Γ ⊥ ∩ D gives a bijection to the set V Γ of vertices from the set of L 26 /S Y (2)-chambers D contained in Nef Y such that P Γ ⊥ ∩ D is a face of D of dimension 10 − µ, or equivalently, such that P Γ ⊥ ∩D contains a non-empty open subset of P Γ ⊥ . The group G Γ := { g ∈ aut(Y ) | Γ g = Γ } acts on the graph (V Γ , E Γ ). We apply Procedure 4.1 to (V Γ , E Γ ) and G Γ , and obtain a complete set V Γ,0 of representatives of V Γ /G Γ . Let Γ ′ be one of the 750 RDP-configurations with the same ADE-type as Γ. Let V Γ ′ ,0 be a complete set of representatives of V Γ ′ /G Γ ′ . Then the RDP-configurations Γ and Γ ′ are in the same orbit under the action of aut(Y ) if and only if there exists an L 26 / Γ ′ ⊥ (2)-chamber f ′ = P Γ ′ ⊥ ∩ D ′ ∈ V Γ ′ ,0 with D ′ ⊂ Nef Y such that isoms(Y, D 0 , D ′ ) contains an element g satisfying Γ g = Γ ′ . Since |V Γ ′ ,0 | is finite, we can determine whether Γ and Γ ′ are in the same orbit or not. Applying this method to all pairs Γ and Γ ′ with the same ADE-type, we obtain a complete set of representatives of RDP-configurations modulo aut(Y ).  Table 1.1). We construct a sequence S Y (2) ֒→ S X ֒→ L 26 from the primitive embedding L 10 (2) ֒→ L 26 of type 96C. The complete set V 0 of representatives of orbits of the action of aut(Y ) on the set of L 26 /S Y (2)-chambers contained in Nef Y consists of 5 elements with the orders of stabilizer subgroups 1, 1, 1, 2, 1. Since vol(D 0 ) = 1 BP /72, we have vol(Nef Y /aut(Y )) = vol(D 0 ) 1 1 The set R temp is of size 56 and the set E temp is of size 6270. We also construct S Y (2) ֒→ S X ֒→ L 26 for a (4A 1 , D 4 )-generic Enriques surface (No. 8 of Table 1.1) from the primitive embedding of type 96C. The set V 0 consists of 18 elements with the orders of stabilizer subgroups 4, . . . , 4. We have |R temp | = 154 and |E temp | = 21452. 7.3. A (D 5 , D 5 )-generic Enriques surface. We have to use the primitive embedding of type 40A to construct S Y (2) ֒→ S X ֒→ L 26 for a (D 5 , D 5 )-generic Enriques surface (No. 24 of Table 1.1). The set V 0 consists of 6 elements with the orders of stabilizer subgroups 2, . . . , 2. In this case, we have vol(D 0 ) = 1 BP /5760 and vol(Nef Y /aut(Y )) = vol(D 0 )  Fig. 1.4]. The chamber Nef Y is isomorphic to an L 26 /L 10 (2)chamber D 0 of the primitive embedding L 10 (2) ֒→ L 26 of type 12A, and hence vol(D 0 ) = 1 BP /174182400 (see [6]). Therefore vol(Nef Y /aut(Y )) = vol(D 0 ) 4 = 1 BP 2 14 The group aut(Y ) decomposes R(Y ) as 2 + 2 + 2 + 2 + 4. For a very general Enriques surface Y with finite automorphism group of type II, the chamber Nef Y is isomorphic to an L 26 /L 10 (2)-chamber D 0 of the primitive embedding L 10 (2) ֒→ L 26 of type 12B. We have vol(D 0 ) = 1 BP /3870720. Note that 3870720 · |S 4 | = |W (R D9 )|. The Enriques surface Y is (D 9 , D 9 )-generic (No. 184 of Table 1.1), and we have Aut(Y ) ∼ = aut(Y ) ∼ = S 4 . The group aut(Y ) decomposes R(Y ) as 6 + 6.