{'fraction_non_alphanumeric': 0.0667471576617366, 'fraction_numerical': 0.029037126471476004, 'mean_word_length': 3.05086403651777, 'pattern_counts': {'":': 0, '<': 32, '<?xml version=': 0, '>': 31, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 70, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Groups Acting Freely on Calabi-Yau Threefolds Embedded in a Product of del Pezzo Surfaces

In this paper, we investigate quotients of Calabi-Yau manifolds Y embedded in Fano varieties X which are products of two del Pezzo surfaces - with respect to groups G that act freely on Y. In particular, we revisit some known examples and we obtain some new Calabi-Yau varieties with small Hodge numbers. The groups G are subgroups of the automorphism group of X, which is described in terms of the automorphism group of the two del Pezzo surfaces.


Introduction
In [13] and [14] Tian and Yau discover a new Calabi-Yau manifold with Euler characteristic equal to -6. Let us briefly explain their seminal example. To begin with, they consider the product X of two cubic Fermat surfaces in P 3 C . Next, they pick a smooth hyperplane section Y in X, which is invariant with respect to a group G isomorphic to the cyclic group of order three. By adjunction and by Lefschetz's Hyperplane Theorem, Y turns out to be a smooth Calabi-Yau threefold, i.e., a smooth compact Kähler threefold with trivial canonical bundle and no holomorphic p-forms for p = 1, 2. The Euler characteristic of Y is −18 and the two significant Hodge numbers h 1,1 (Y ) and h 1,2 (Y ) are 14 and 23, respectively. To reduce to Euler characteristic and the Hodge numbers, Tian and Yau take the quotient of Y with respect to G that turns out to act freely on it. The quotient manifold Y /G is a Calabi-Yau variety with Hodge numbers h 1,1 = 6 and h 1,2 = 9.
In recent years, physicists have focused on Calabi-Yau manifolds with small Hodge numbers: see, for instance, [2], [4], [3], [6] and [9]. In fact, imagine to plot the distribution of Calabi-Yau varieties on a diagram with variables the Euler characteristic χ(Y ) (on the horizontal axis) and the height h(Y ) := h 11 (Y ) + h 12 (Y ) (on the vertical axis). Fix a pair (χ 0 , h 0 ) of positive integers such that χ 0 is even and −2h 0 ≤ χ 0 ≤ 2h 0 . For h 0 ≤ 30, it turns out that there are still a lot of missing examples of Calabi-Yau varieties with Euler characteristic χ 0 and height h 0 . The example in [13] is even more significant because the Euler characteristic is −6. In general, special attention is given to those Calabi-Yau manifolds that have Euler characteristic 6 in absolute value since they correspond to three-generation families (see, for instance, [3]).
Remarkably, the example in [13] can be generalized in the following way. The two cubic Fermat surfaces are examples of degree three del Pezzo surfaces, i.e., smooth surfaces with ample anticanonical divisor which can be obtained as the blow-up of P 2 C at six points in general position. A first generalization in this direction was given by Braun, Candelas and Davies in [3]. In that paper, they discover a new Calabi-Yau manifold with Euler characteristic −6 and small Hodge numbers. They replace the two Fermat surfaces in P 3 C by two del Pezzo surfaces of degree six and come up with a group of order twelve that acts freely on a suitable hyperplane section of the product.
In this paper we generalize the examples mentioned above even further and we put them in a more general context. Indeed, let us consider two suitable smooth del Pezzo surfaces S 1 and S 2 . The product X is a smooth Fano fourfold, i.e., −K X is ample. In X we pick a smooth threefold Y which is in | − K X |. As pointed out by the example in the Introduction in [11] this requires some work: in fact, for some choice of the two del Pezzo surfaces it is not even possible. Moreover, we pick a finite group G in Aut(S 1 × S 2 ) that acts freely on Y so that the quotient variety is a Calabi-Yau manifold. Since the Euler characteristic χ(Y ) is negative, it is easy to verify that the height of Y /G is less than the height of Y for any non-trivial group G. Within this set-up, we obtain the two examples mentioned above; further, we find new Calabi-Yau manifolds with small Hodge numbers. The smoothness and the free action of G on a suitable Y are proved as follows. We pick a group G that has only finitely many fixed points on X. We decompose the representation of G on H 0 (X, −K X ) as a direct sum ⊕V i of irreducible subrepresentation. We consider a subspace W such that for every g ∈ G and every s ∈ W , g * (s) = λ g s for some λ g ∈ C * , i.e. for every g ∈ G, W is an eigenspace for g * .We pick a section s ∈ W , if there are some, so that the corresponding zero locus does not intersect the fixed locus of G. Next, we look at the base points of the subsystem W ≤ H 0 (X, −K X ). In case there are some, we take a generic section and prove that the base points are smooth. This is done by direct computation with MAPLE. A Bertini-type argument yields the existence of a smooth threefold Y in X on which G acts freely.
In Section 5 we present the examples we obtain case by case. Except for the last subsection of that Section, all the examples have height less than 20. Unfortunately, we do not obtain any new three-generation manifolds, i.e., a manifold with |χ(Y )| = 6. Moreover, in Section 8, you may find all the examples of quotients of Calabi-Yau threefolds Y embedded in S 1 × S 2 by groups which are of maximal order. In other words, we take the quotient by a group H ≤ Aut(S 1 × S 2 ) such that the restriction to Y yields a free action and H can not have order greater than the groups used. Finally, we investigate the height of the quotient variety. In several cases, we are able to say that the height for the quotient threefold is the least possible within this framework.
The following picture represents the tip of the distribution of the Calabi-Yau manifold with respect to the Hodge numbers. The diagonal axis are h 1,1 (Y ) and h 1,2 (Y ) whereas the horizontal and the vertical axis are χ(Y ) and h(Y ), respectively. We plot only the known manifolds with height less or equal than 31. The solid dots correspond to quotients found in this paper. The blue rings represent the ones known until now (with respect to the data collected in [2], [4], [3] and [6]). The black rings are quotients by groups whose order is maximal. From the picture below, we can summarize our results as follows. The dots (3,5), (2,7) and (5,13) represent NEW Calabi-Yau threefolds. There exists a Calabi-Yau manifold corresponding to the pair (1, 5) with non-abelian fundamental group: see [4]. Our example in Section 5.1 has abelian fundamental group isomorphic to the product of the cyclic group of order two and that of order eight. Moreover, we come up with a Calabi-Yau manifold with Hodge numbers (2, 11) (cf. (5.3)), which are the same as those described in [4]. Finally, we construct other varieties with greater height (see Section 5.6) but they correspond to existing dots in the picture below. In all the cases where other Calabi-Yau manifolds already exist, it would be interesting to know whether our examples are isomorphic to those or not.
In some cases, it is not possible to consider non-trivial quotients with our method. In fact, we prove, for instance, that there does not exist a Calabi-Yau variety which is the quotient by a group of order seven of a smooth anticanonical section Y in a product of two del Pezzo surfaces of degree two. This type of results is collected in Section 6. To prove them, we use the following theorem which is proved in Section 7. For this purpose, we first use some Mori theorem of Fano fourfolds which are products of two Fano varieties. Second, we also recall that for low degree del Pezzo surfaces are toric varieties. Thus, we apply a theorem due to Demazure (later generalized by D. Cox in [5]) on the structure of the automorphism group of toric varieties. More specifically, the following holds (see Section 7).
Theorem. Let S 1 and S 2 be two del Pezzo surfaces. Then where S 4 is the symmetric group with 24 elements.

Preliminaries
We say that a complex surface S is a del Pezzo surface if it is projective, smooth, simply-connected and the anticanonical divisor −K S is ample. Examples of del Pezzo surfaces are blow-ups of the projective plane in a finite set ∆ of 0 ≤ n < 9 points in general position and P 1 × P 1 . As proved in [7], this list is exhaustive. We often write dP d to mean a del Pezzo surface that is obtained by blowing up 9 − d points of P 2 that are in general position. Let S = Bl ∆ P 2 . We can identify H 0 (S, −K S ) with the vector space of the homogeneous polynomials of degree 3 with variables {x 0 , x 1 , x 2 } such that f (P ) = 0 for all P ∈ ∆. It is easy to show where H is the hyperplane divisor on the projective plane and the E i 's are the exceptional divisors. Thus, K 2 S = 9 − k = d. For d ≥ 3 we have that −K S is very ample. For d = 2 the anticanonical system | − K S | gives a 2 : 1 map of S in P 2 branched along a smooth quartic. For d = 1 the anticanonical model of S is a finite cover of degree two of a quadratic cone Q ramified over a curve B in the linear system |O Q (3)|.
Suppose that Y is a Calabi-Yau threefold and that G is a group that acts freely on Y . Then it is well known that the quotient Y /G has a canonical complex structure such that the projection on the quotient is holomorphic. Furthermore, the quotient map is a local isomorphism.
Theorem 2.1. If the action of G is free then Y /G is also a Calabi-Yau threefold. Moreover, the quotient is projective.
Proof. Take g ∈ G \ {Id}. The manifold Y is a Calabi-Yau threefold, so There exists ω ∈ H 3,0 (Y ) such that ω P ≡ 0 for all P ∈ Y (this is equivalent to K Y ≡ 0). We want to show that g * ω = ω. The maps are zero for p = 1, 2 whereas for p = 0, g * is the identity. We apply the Holomorphic Lefschetz Fixed Point formula, which in this case reads as follows: Since h 3,0 (Y ) = 1 (Y is a Calabi-Yau manifold) we get g * = Id for p = 3 and for all g ∈ G. Thus the action of G on H 3,0 (Y ) is trivial. We have the following isomorphism ([1], p. 198): and there exists a holomorphic 3−formω on Y /G such that π * ω = ω and, as π is a local isomorphism, ω P = 0 for all P ∈ Y /G. This is equivalent to K Y /G ≡ 0. Finally, using h p,0 (Y /G) = h p,0 (Y ) G one has h 1,0 (Y /G) = h 2,0 (Y /G) = 0 and this concludes the proof. As for the projectivity of Y /G, see, for example, [10], p. 127.
We will adopt the following framework. We will take two del Pezzo surfaces S 1 and S 2 , their product X = S 1 × S 2 , which is a Fano fourfold, and a smooth element Y of | − K X |. First of all, we will define a number M (S 1 , S 2 ) which bounds the maximum order of a finite group acting freely on Y and which depends only on the degree of S 1 and S 2 .
Notice that if Y ⊂ S 1 × S 2 is a Calabi-Yau threefold and G is a finite group that acts freely on Y , then |G| divides M (S 1 , S 2 ).
With the definition of M (S 1 , S 2 ) in mind, we will search for a group G with the following properties: Note that if Fix(G) ⊂ X contains a curve L, by the Nakai-Moishezon criterion of ampleness, −K X · L > 0, and since Y = −K X we will have some fixed points on Y . Hence it's necessary to choose groups whose action on X has at most a finite number of fixed points. We anticipate that there are cases in which M (S 1 , S 2 ) > 1 but the only group with these requests is the trivial group (that is m(S 1 , S 2 , Y ) = 1 for all Y ).

Necessary Conditions
Assume that S 1 and S 2 are smooth projective surfaces and Y is a Calabi-Yau threefold embedded in X = S 1 × S 2 . Then the following result holds: Proof. By the exact sequence of vector bundles and, as Y is a Calabi-Yau manifold ( which implies c 1 (Y ) = 0), we have: and, in particular, Using the fact that X is a product of surfaces we have and Hence, by the identification H 6 (Y, Z) Z, we have Now, assume Y is asmooth ample divisor in X. Thus, the following isomorphisms hold: For any divisor class D ∈ H 2 (Y /G, Z) denote by D 1 and D 2 divisors classes such that π * (D) = D 1 + D 2 , where π is the projection of Y onto the quotient. Finally, we denote by K i the divisor classes such that K X = K 1 + K 2 . Then the following holds.
Theorem 3.2. Let G be a group that acts freely on Y . Then for any D ∈ H 2 (Y /G, Z) and D 1 , D 2 as above, we have Proof. We recall that the Riemann-Roch formula for the Calabi-Yau threefold Y /G is The action of G is free, hence This yields . In a similar way, we obtain Merging these results and using Nöther formula 1 , we complete the proof.
We focus our attention on a particular divisor on the quotient: a divisor D such that π * D = −ι * K X =. Such a divisor always exists because the canonical divisor is G-invariiant for any gruoup of automorphisms G. We can specialize the previous formula for nD obtaining Hence |G| has to divide χ(−nι * K X ) for all 2 n. We can obtain a similar condition using Theorem 3.1: the Euler characteristic of the quotient Y /G of Y by a finite group G that acts freely is the Euler characteristic of Y divided by the order of the group. Moreover, it is known that a Calabi-Yau threefold has even Euler number so we obtain that |G| must divide χ(Y )/2. This gives a motivation to Definition 2.2.
The following table gives the values of M (S 1 , S 2 ) for every distinct values of degrees of S 1 and S 2 , with S 1 and S 2 del Pezzo surfaces -distinguishing the case dP 8 and P 1 × P 1 .
The self-intersection of −K S , where S is a del Pezzo surface, is positive and is equal to its degree and this, using Theorem 3.1, means that χ(Y ) < 0 regardless of the choice of which surfaces we are using. Therefore, by recalling that the action of G is free, we have that the height h := h 1,1 + h 1,2 of Y and that of 2 One could easily check that Y /G satisfy the following inequality: By finding a group whose order is maximal -and such that the dimension h 1,1 (Y ) G of the invariant part of H 1,1 (Y ) is the smallest possible -we obtain the least possible height for the quotient.
In the following sections we give some examples (both known and new) and some results of non-existence.

Known Examples
With the following examples we revisit some known examples in the framework presented. The first one is due to Braun, Candelas and Davies and can be found in [3]. The second one is due to Tian and Yau and is presented in [14] and [13].
4.1 dP 6 × dP 6 with maximal order 12 There is a unique del Pezzo surface of degree 6 and this surface can be obtained as the complete intersection of two global sections of O P 2 ×P 2 (1, 1). Explicitly we can take S to be the surface in P 2 × P 2 given by the equations where x ij is the j-th coordinate on the i-th copy of P 2 . In this way, S is the surface obtained by blowing up the points P 0 = (1 : 0 : 0), P 1 = (0 : 1 : 0) and P 2 = (0 : 0 : 1) of P 2 and the exceptional divisors E i are given by We define S 1 = S 2 = S and embed X = S × S in (P 2 ) 4 using x i0 , x i1 and x i2 as projective coordinates of the i−th P 2 for i = 1, 2, 3, 4. Let P be the point Consider the automorphism of X defined by It is easy to check that g 3 3 = g 4 4 = Id and g 4 g 3 = g 3 g 2 4 hence G =< g 3 , g 4 > Z 4 Z 3 :=: Dic 3 , which is called the dicyclic group of order 12. The set of the fixed points Fix of G is given by the union of We are looking for a global section s of O X (−K X ) that is G−invariant and whose zero-locus V (s) is smooth and doesn't intersect Fix. We have an exact sequence with the surjection given by the inclusion ι : S → P 2 × P 2 . Hence, we have a surjection with kernel given by The representation of Dic 3 in H 0 (X, −K X ) C 49 has an invariant space H 0 (X, −K X ) G of dimension 5. By direct inspection, we have checked that the generic invariant section s doesn't intersect Fix and is smooth. Then Y = V (s) is a Calabi-Yau threefold with a free action of Dic 3 .

If we call R the representation of Dic
where we used the base where h(Y /G) = 5. Note that, because h 1,1 (Y /G) = 1, this example achieves the minimum of the height for the quotient Y /G, where G is isomorphic to the Dic 3 and Y is as above. It is interesting to note that taking the group G spanned by g 4 and g 3 is cyclic of order 12 and a generator is g 3 g 4 := g 12 . Following the same argument as the previous case it can be shown that exist a Calabi-Yau Y such that G acts on Y freely. The quotient Y /G is hence again a Calabi-Yau and has the same Hodge diamond as Y /G. However these two manifolds aren't even diffeomorphic because

dP 3 × dP 3 with maximal order 3
Suppose S 1 and S 2 del Pezzo surfaces of degree 3. Then −K Si is very ample and gives an embedding in P 3 . The surface obtained is a cubic (is called anticanonical model of S i ) and all smooth cubic surfaces in P 3 can be obtained in this way.
Set f 1 := x 3 0 + x 3 1 + x 3 2 + x 3 3 and f 2 := y 3 0 + y 3 1 + y 3 2 + y 3 3 and consider the Fermat surfaces S i := V (f i ). Denote, as usual, X = S 1 × S 2 ⊂ P 3 × P 3 and consider the automorphism given by There is an isomorphism so we have to study the polynomial of bidegree (1, 1). The action of Z 3 on X gives a representation of Z 3 in H 0 (X, −K X ) and a basis for the invariant space By direct computation, one can check that the generic section s doesn't intersect Fix(Z 3 ) hence the action of G restricted to V (s) is free. For example, taking s to be G 4 + G 5 = x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 gives a section whose zero locus Y is smooth and Y ∩ Fix(Z 3 ) is empty.
where π i : X → S i is the projection onto the i−th factor of the product X. In fact, by Lefschet's Hyperplane Theorem, the group H 1,1 (Y ) is isomorphic to H 2 (X). The dimension of the space of invariants with respect to G is equal to the traces of the homomorphisms induced on the second cohomology group of X = S 1 × S 2 by the elements of G. By linear algebra and the Künneth formula, the traces on the cohomology groups of the product X is the sum of the traces on the cohomology on the factors H 2 (S i ) for i = 1, 2. These traces can be computed via the Lefschetz fixed-point Theorem. In this case we obtain h 1,1 (Y /G) = 6. The same number could be obtained by studying the invariant space of H 2 (X, C) with respect to the representation of Z 3 given by By Theorem 3. As shown in [7], up to isomorphism of P 3 , there are 3 possible pairs (f, G) where f is a homogeneous polinomial of degree 3 and G is a group fixing f of order 3. One can show that Fix(f ) is either one of the following: 3 points or 6 points, or one line. Thus, the least value that can be assumed by χ(Fix f ) is 3 if we exclude the case with one line of fixed points. Hence, the example presented here achieves the minumum for h(Y /G).

New Examples
We present some new examples.
5.1 (P 1 × P 1 ) × (P 1 × P 1 ) with maximal order 16 Take S 1 = S 2 = P 1 × P 1 and define X to be S 1 × S 2 . We begin to search for a group H ≤ (Aut(P 1 )) 4 S 4 ≤ Aut(X) such that |H| = 8 and | Fix(H)| < ∞. Moreover, we want a section s that is an eigenvector for the action of H on H 0 (X, −K X ) and does not intersect Fix(H). After that, we try to extend H to a group of order 16 with the same properties.
Let g ∈ (Aut(P 1 )) 4 S 4 be an element of finite order. Without loss of generality, we can take g of the form where σ ∈ S 4 and a i ∈ C * for i = 1, 2, 3, 4.
If the order o(g) of g is 2, we can choose σ ∈ {Id, (12), (12)(34)}. An easy check shows that ((x : y), (x : a 2 y), (1 : 0), (1 : 0)) is a line of fixed points if σ = (12) or σ = (12)(34); so we must take σ = Id. The only possible case is a j = −1 for which If o(g) = 4, we can take σ ∈ {Id, (12), (12)(34), (1234)}. The automorphism σ cannot be a permutation of order 4. In fact, in this case g 2 would have a fixed line, as previously showed. Then, we have Fix(g) ⊂ Fix(g 2 ) = Fix(g 2 ). Suppose σ = Id or σ = (12) and consider an eigenvector s ∈ H 0 (X, −K X ) = O X (2, 2, 2, 2)(X). The condition o(g) = 4 is then equivalent to a 4 j = 1 for σ = Id and a 2 1 a 2 2 = a 4 3 = a 4 4 = 1 for σ = (12). Necessarily g satisfies g 2 = g 2 and this implies respectively a 2 j = −1 and a 1 a 2 = a 2 3 = a 2 4 = −1. One can see that for all P ∈ Fix(g 2 ) there exists a unique element e i of the usual basis of O X (2, 2, 2, 2)(X) such that e i (P ) = 0. For example, we have Then s has to be an element of the eigenspace of both x 2 1 x 2 2 x 2 3 x 2 4 and x 2 1 x 2 2 x 2 3 y 2 4 , but these have different eigenvalues ( 1 and a 2 4 = −1 respectively) so s = 0. Suppose that σ = (12)(34). The conditions o(g) = 4 and g 2 = g 2 show that g has to be of the form Finally, take g to be an automorphism of order 8. Then σ has to be a permutation of order 3 4. For example, pick σ = (1324) (that gives the following conditions on the a i 's: a 1 a 2 a 3 a 4 = −1) and let a 1 = a 2 = a 3 = −a 4 = 1. A basis for H 0 (X, −K X ) is given by {e 1 , . . . e 11 }, where Now, we try to extend the group H. Define h to be the involution of (P 1 ) 4 such that An easy check shows that gh = hg and that the following hold: . This means that, defining G to be the group generated by g and h, Fix(G) is a finite set composed of 48 points and G Z 8 × Z 2 .
If we take C i e i and impose both s(P ) = 1 for all P ∈ Fix(g) and h * (s) = s, we have the following conditions on the C i 's: By evaluating at the other fixed points, we obtain 4 different non identicallyzero linear-combinations of the C i 's; so the generic invariant section does not intersect Fix(G). For example, the section obtained by taking C 1 = 1 and C 6 = 2 fulfills all our requests. Moreover, it is smooth, so there exists a group of order 16 = M (P 1 × P 1 , P 1 × P 1 ) that acts freely on a Calabi-Yau threefold embedded in (P 1 ) 4 .
The representation of G on H 2 (Y, C) is given by This action has then a unique fixed class in H 2 (Y, C) (the sum of the four P 1 's). By 5.2 dP 4 × dP 4 with maximal order 8 As proved, for instance, in [7], every del Pezzo surface of degree 4 can be obtained as a complete intersection of two quadrics of P 4 . Moreover, one can choose the equations to be of the form where a i = a j ∈ C for i = j. We choose Let r be the automorphism which sends (x, y) to the point ((x 0 : x 1 : −x 2 : x 3 : −x 4 ), (y 0 : y 1 : −y 2 : y 3 : −y 4 )) .

Consider the groups
as a basis of the space of sections of the anticanonical bundle. It is easy to see that the vector space V spanned by is such that for all h ∈ H and for all s ∈ V , h * (s) = λs for some λ ∈ C * . By taking the generic section s ∈ V and imposing r * s = t * s = s (so that for every automorphism g of G, V is an eigenspace with respect to g * ), we obtain where A i ∈ C. Let a and b be fixed roots of 2z 2 + 1 + i and 2z 2 + 1 − i, respectively. Then To look for the fixed points of G it suffices to know the fixed points of r, t 2 and rt 2 . In fact, the following holds: ).
An easy check shows that for generic values of A 1 , A 3 , A 4 and A 7 , the section s does not intersect Fix(G).
We can check directly that the section corresponding to A 1 = 1, A 3 = −2, A 4 = 3 and A 7 = 1 is smooth and doesn't intersect Fix(G); so there exists a Calabi-Yau threefold Y embedded in S 1 × S 2 with Z 4 × Z 2 acting freely on Y .
We don't have an explicit description of a basis for Pic(Y ) = Pic(S 1 )⊕Pic(S 2 ) Z 12 , but we can use the Lefschetz Fixed Point fomula to get the traces we need to compute h 1,1 (Y ) G . For example, notice that r = r 1 × r 2 with r i ∈ Aut(S i ); so the trace of r * : H 2 (S 1 × S 2 , C) → H 2 (S 1 × S 2 , C) is equal to the sum of the traces of r * i : H 2 (S i , C) → H 2 (S i , C). By recalling that 16 = χ(Fix(r)) = χ(Fix(r 1 × r 2 )) = χ(Fix(r i )) 2 and by Lefschetz Fixed Point formula, we have With the same method we obtain Tr((t * ) 2 ) = Tr(r * (t * ) 2 ) = 4. We can write t as (t 1 × t 2 ) • σ where σ is the the permutation of the two copies of S. Hence t * will swap H 2 (S 1 ) and H 2 (S 2 ) in the sum H 2 (S 1 ) ⊕ H 2 (S 2 ) and this means that its trace is zero. In the same way we obtain Tr((t * ) 3 ) = Tr(r * t * ) = Tr(r * (t * ) 3 ) = 0. Merging these results and recalling that χ(Y ) = −32, we obtain 5.3 P 2 × P 2 with maximal order 9 Let (x 0 : x 1 : x 2 ) and (y 0 : y 1 : y 2 ) be the projective coordinates on the two copies of P 2 and set a = e 2πi/3 . Consider the automorphism of P 2 × P 2 := X defined by g := (x 0 : ax 1 : a 2 x 2 ) × (y 0 : ay 1 : a 2 y 2 ) := g 1 × g 2 and h := (x 1 : It is easy to show that the group G generated by g and h is isomorphic to Moreover, it is easy to see that where Then g * (e i,j ) = a j e i,j , h * (e i,j ) = a i e i,j , g * (e 0 ) = h * (e 0 ) = e 0 ; hence is a basis of H 0 (P 2 , O P 2 (3)) composed of eigenvectors of both g * and h * . Since a basis for the space of invariant sections is given by By direct computation, we can show that the generic invariant section doesn't intersect Fix(G). Moreover, the system |H 0 (X, −K X ) G | is base-point free. By Bertini's Theorem, the generic section is smooth. Hence there exists a Calabi-Yau threefold Y embedded in P 2 × P 2 equipped with a free action of G.
The space H 2 (X, Z) is free of rank two and is generated by π * 1 H and π * 2 H where < H >= H 2 (P 2 , Z). Every automorphism of P 2 fixes H, so H 2 (X, C) G = H 2 (X, C). This implies that the following is the Hodge diamond of Y /G : Its height is 13. An element g ∈ Aut(P 2 × P 2 ) = (Aut(P 2 ) × Aut(P 2 )) Z 2 of order 3 has to be of the form g = g 1 × g 2 with g i ∈ Aut(P 2 ). This means that H 2 (X, Z) <g> = H 2 (X, Z) and thus the minimum for h(Y /G) is achieved by this example.
5.4 dP 5 × dP 5 with maximal order 5 Fix P 1 = (1 : 0 : 0), P 2 := (0 : 1 : 0), P 3 := (0 : 0 : 1) and P 4 := (1 : 1 : 1) in P 2 . Let S be the unique del Pezzo surface of degree five. It is well known that the automorphism group of S is isomorphic to the symmetric group of order 120. The sections of O S (−K S ) are the cubics through the points P i for i = 1, 2, 3, 4. Hence a basis of H 0 (dP 5 , −K dP5 ) can be taken to be where x 0 , x 1 , x 2 is a system of homogeneous coordinates on P 2 . Consider the following transformation T on the projective plane, namely: It is easy to check that T acts as automorphism of S and its action on H 0 (S, −K S ) is determined by y 1 → y 1 y 2 → y 1 + y 5 − y 2 y 3 → y 2 y 4 → y 2 + y 3 − y 1 y 5 → −y 6 + y 5 − y 2 y 6 → −y 4 − y 1 + y 3 .
It's easy to see that the order of T is five thus G :=< T > is isomorphic to Z 5 . Let us now consider the action of G diagonally on X = S × S. We will use x 0 , x 1 , x 2 and z 0 , z 1 , z 2 as projective coordinates on the two P 2 's we blow up to obtain the two copies of S. There is an action of G on H 0 (X, −K X ). Let ω be a primitive fifth root of unity. The space of invariants under this action is generated by the following polynomials, namely: where we set: and k i = h i (z 0 , z 1 , z 2 ). It is easy to check that h i are eigenvectors with eigenvalue ω i with respect to the action of T on H 0 (S, −K S ). Let where ρ satisfies the degree two equation ρ 2 + ρ − 1 = 0 and thus there are four fixed points on X. Note that, by the Lefschetz Formula this is the least number of fixed points one could obtain. By a suitable choice of the A i 's, we restrict to the locus Σ such that the sections s in (1) do not pass through the fixed points above. It is easy to see that Σ is not empty. For s ∈ Σ the set of zeroes Y = V (s) is thus invariant with respect to the free action of G on it. Now, we look for base points of the system above. First, we look for solutions on P 2 × P 2 of the equations Next, we recall that S is obtained from P 2 by blowing-up the points P i . After some computation we show that there are 20 base points.
Analogously to the previous cases, we can conclude that there exists a smooth Calabi-Yau threefold Y ⊂ X and a group G Z 2 ⊕ Z 2 acting freely on it. The quotient has the following Hodge diamond Hence the height of the quotient is 18.

Other similar examples
For brevity we don't treat explicitly some examples. These are some threefolds in P 2 × dP 6 , P 2 × dP 3 , (P 1 × P 1 ) × dP 6 and dP 6 × dP 4 . The threefolds in P 2 × dP 6 and in P 2 × dP 3 admit a free action of

Results of non-existence
In this section we present some results of non-existence. In particular, we show that there are cases for which M (S 1 , S 2 ) > 1 but a group G that fulfills our requests doesn't exist.
Lemma 6.1. If S is a del Pezzo surface and g ∈ Aut(S) is such that o(g) = p is prime, then g has a fixed point.
Proof. Every del Pezzo surface S is a rational surface. Suppose that the fixed locus of g is empty. Recall that p is prime. Let G :=< g > be the group generated by g. Then Fix(G) is empty. In fact, for every n ≡ p 0 there exists m such that nm ≡ p 1; this implies Fix(g n ) ⊂ Fix((g n ) m ) = Fix(g).
Therefore R := S/G is a smooth surface and R is rational. In particular Π 1 (R) = {Id}. But this is not possible because S is simply connected, so Π 1 (R) G {Id}. Hence, g must have at least one fixed point. By [7], every automorphism of a del Pezzo surface S of degree 8 comes from an automorphism of P 2 that fixes the point R such that S = Bl {R} P 2 . Suppose S = dP 8 . Then we search for a group G ≤ Aut(dP 8 )×Aut(S). We are interested in the cases S ∈ P 1 × P 1 , dP 6 , dP 4 , dP 2 for which M (dP 8 , S) is respectively 16, 2, 4 and 2. It is then enough to show that there are not groups of order 2 whose action is free on Y . Let g = (g 1 , g 2 ) be an involution. By Corollary 6.2 there exists a fixed point P of g 2 . The automorphism g 1 comes from an involution of P 2 , hence it has a line L of fixed points, therefore L × {P } is a line of fixed points for g.
If S = dP 8 , then Aut(dP ×2 8 ) = Aut(dP 8 ) ×2 Z 2 . Let G =< g > where g = (g 1 , g 2 ). Using the same result as above, we will have a surface of fixed points. Then, it suffices to analyze the case g = (g 1 , g 2 ) • τ , where τ is the involution that switches the two copies of dP 8 . Then, by changing projective coordinates, we can assume that for some a, b ∈ C * . It is easy to see that ((ax : by : 0), (x : y : 0)) is a line of fixed points.
In conclusion, we have shown that m(dP 8 , S, Y ) = 1 for a del Pezzo surface S (here we have checked all the cases for which M (dP 8 , S) = 1) and for all Y Calabi-Yau embedded in dP 8 × S.
6.2 dP 7 × dP 7 with estimated maximal order 7 There is only one del Pezzo surface S of degree 7. It is given as the blow-up of P 2 in P 0 = (1 : 0 : 0) and P 1 = (0 : 1 : 0). We will show that there does not exist a section s of −K S×S such that g * s = cs for some c ∈ C * and g ∈ Aut(S × S) of order 7 which doesn't intersect the fixed locus of < g >.
By [7], every automorphism of a del Pezzo surface of degree 7 comes from an element of P GL(3) fixing the set {P 0 , P 1 }. Thus, we have Recall that Aut(S × S) = Aut(S) ×2 Z 2 . Since we need g of order 7, we have to choose an element of the form g = (g 1 , g 2 ), where g i ∈ Aut(S) and After a change of projective coordinates that fixes the points P 0 and P 1 , we may assume b i = c i = 0 so that g i is in diagonal form. The condition o(g) = 7 gives a 7 i = d 7 i = 1. Since we need a finite number of fixed points, we must impose a i = 1 = d i and a i = d i .
In conclusion, we can take g of the form where λ = e 2πi/7 and 0 = n i , m i and n i = m i .
The fixed points of g i as an automorphism of P 2 are P 0 , P 1 and P 2 , whereas the fixed points of g i as an automorphism of S are Here, for example, with ((0 : 1 : 0), (1 : 0)) we mean the point (1 : 0) on the exceptional divisor E 1 = π −1 (P 1 ), where we use the standard local description of S in a neighbourhood of E 1 as the surface of C 2 × P 1 such that um = vl with {((0, 0), (l : m))} = E 1 . Hence, in total, G :=< g > has 25 fixed points.
We blow up P 2 in P 0 and P 1 . Then, the following isomorphism holds: The correspondence is given by taking the strict transform of a polynomial see as a global section of O P 2 ×P 2 (3, 3). We call e i the elements of the base on the first del Pezzo surface and f i the elements of the base on the second one so that, by the Künneth formula, we obtain Suppose that s is an eigenvector of H 0 (S × S, −K S×S ) and that s(P ) = 0 for all P fixed points of G. Then, for example, if and only if s is in the eigenspace of x 2 0 x 1 y 2 0 y 2 . But these two eigenvectors have corresponding eigenvalues λ m1+m2 and λ m1+n2 and these numbers are different if and only if m 2 = n 2 , which it is true by hypotesis. This means that s must be zero and we have a contradiction.
Albeit M (dP 7 , dP 7 ) = 7, this shows that an automorphism of S × S with finite order cannot act freely on a smooth section of −K S×S . 6.3 dP 6 × dP 3 with estimated maximal order 9 In this case recall that M (dP 3 , dP 6 ) = 9. Nonetheless, the maximum order of G to have a free action on a Calabi-Yau threefold Y embedded in X is 3. We will also give an example for which m(dP 6 , dP 3 , Y ) = 3.
Suppose that G ≤ Aut(dP 6 ) × Aut(dP 3 ) has order 9. Then either G Z 9 or G Z 3 × Z 3 . First, we will show that if G Z 9 then G must have a fixed curve and so it can't satisfy our assumption on G. Next, we will deal with the case G Z 3 × Z 3 . We'll first find all the groups whose fixed locus is finite. Essentially, this will be done by projecting G on Aut(dP 6 ) and Aut(dP 3 ) so that the projections G 1 and G 2 satisfy G 1 G 2 G Z 3 × Z 3 . There is only one useful choice for G 2 =< g 2 , h 2 > whereas there are infinitely many possibilities for G 1 , which are parametrized by (C * ) 2 . Once we fix G 1 :=< u, v >, we will consider all the possible G s such that the projection of G on Aut(dP 3 ) and Aut(dP 6 ) are G 1 and G 2 , respectively. This will be done by choosing all the possible pairs (g 1 , h 1 ), not necessarily equal to (u, v), that generate G 1 . We thus consider the group G :=< g, h >, where g = g 1 × g 2 and h = h 1 × h 2 . For every case we have checked that all the sections of H 0 (X, −K X ) that are eigenvectors of both g * and h * are zero on a fixed point of the group G (we will show an explicit calculation for one of the cases).
Suppose that G Z 9 and consider its projection G 1 on Aut(dP 3 ). Necessarily, G 1 G. On the contrary, if G =< g 1 × g 2 > with g 3 1 = Id, G would have infinitely many fixed points. Hence G 1 has to be a group isomorphic to Z 9 in Aut(dP 3 ). If S is a smooth cubic surface in P 3 and if g 1 ∈ Aut(S) has order 9 then, by [7], there exist a projective automorphism of P 3 such that where a satisfies a 3 = 1 = a 9 . On the other hand, we have Hence Fix(< g 1 >) contains a curve C. This means that, by Corollary 6.2, we have a fixed curve in Fix(G), which contradicts our assumptions.
Suppose, now, that G Z 3 × Z 3 ≤ Aut(dP 6 ) × Aut(dP 3 ) and consider the projection G 2 on Aut(dP 3 ) so that G 2 G. Fix two generators g 2 , h 2 of G 2 and consider dP 3 = V (f ) ⊂ P 3 . By [7], if V (f ) is a smooth cubic andG Z 3 × Z 3 ≤ Aut(V (f )), we can change coordinates to obtain f = y 3 i . In this case Aut(V (f )) Z 3 3 S 4 , where each Z 3 acts as multiplication of a variable by a k (we write the elements in Z 3 3 as (1, a k1 , a k2 , a k3 )) and S 4 = Sym(0, 1, 2, 3) is generated by the permutation of the variables. By requiring | Fix(G 2 )| < ∞ we obtain G 2 ≤ Z 3 3 . There is only one group isomorphic to G 2 in Z 3 3 that has a finite number of fixed points on V (f ) and it is < g 2 , h 2 > where g 2 = (1, 1, a, a 2 ) and h 2 = (1, a, a 2 , a 2 ). We call V (2) i,j the maximal subspace of H 0 (dP 3 , −K dP3 ) such that g * 2 (s) = a i s and h * 2 (s) = a j s for every s ∈ V i,j . This vector space is the intersection of the eigenspaces Λ a i of g 2 and Λ aj of h 2 relative to a j . The following table summarizes the situation providing generators for these spaces.
Now, consider the projection G 1 of G on Aut(dP 6 ) = (S 3 × Z 2 ) (C * ) 2 . Any element of order 3 can be written in the form diag(1, b, c) • (123) k for some fixed b, c ∈ C * and k = 0, 1, 2. Easy arguments show that G 1 cannot satisfy G 1 ≤ (C * ) 2 (if it happens, one has | Fix(G 1 )| = ∞) and that G 1 has exactly two non-trivial elements in (C * ) 2 . These are diag(1, a, a 2 ) and its inverse. Moreover, these two elements commute with every element of the form (1, b, c) • (123) k , thus every subgroup of Aut(dP 6 ) isomorphic to Z 3 ×Z 3 and with a finite number of fixed points can be written in the form < u, v > where for some fixed b, c ∈ C * . We define d to be a fixed third root of bc. Set Then F j is an eigenvector of both u and v and the corresponding eigenvalues are the ones in the following table: This shows that {F j } form a base for H 0 (dP 6 , −K dP6 ). The following are the Fixed points (k = 0, 1, 2) g 2 , g  are fixed points of g = g 1 × g 2 . Suppose that is a section such that g * (s) = a k1 s and that s(P j ) = 0. Then This means that at least one between x i F j with i = 0, 1 and j = 2, 4, 6 has a non zero coefficient and the same is true for x i F j with i = 0, 1 and j = 1, 3, 5. But, Then each eigenvector of g is zero if evaluated in P 1 or in P 2 .
The same result is true for every other case: we have checked that, for every b, c ∈ (C * ), for every choice of g 1 , h 1 generators of G 1 =< u, v >, every section of H 0 (X, −K X ) that is an eigenvector of both g and h where g = g 1 × g 2 and h = h 1 × h 2 is zero on at least one fixed point of G =< g, h >. In conclusion the restriction of the action of a group G ≤ Aut(dP 6 ) × Aut(dP 3 ) of order 9 to a Calabi-Yau threefold Y ⊂ dP 6 × dP 3 cannot be free. Hence m(dP 6 , dP 3 , Y ) < M (S 1 , S 2 ) = 9 for every Y .
We have obtained m(dP 6 , dP 3 , Y ) ≤ 3 for all Y . We now give an example such that m(dP 6 , dP 3 , Y ) = 3. Take dP 3 to be the Fermat surface in P 3 . Call g 1 the automorphism of dP 6 such that x i,j → x i,j+1 and g 2 the authomorphism of dP 3 . Notice that the minimum for the number of fixed points for an automorphism of order 3 in Aut(dP 6 ) × Aut(dP 3 ) is achieved by g = g 1 × g 2 . The dimension of H 0 (X, −K X ) G , where G =< g >, is 10. It can be shown that the base locus for |H 0 (X, −K X ) G | has only 9 points and that these are 7 On the Relation between Aut(S 1 )×Aut(S 2 ) and Aut(S 1 × S 2 ) Let X be a projective complex manifold. We will denote by NE(X) the cone of effective curves of X. An extremal subcone V of NE(X) is a closed convex cone such that for every v, w ∈ NE(X) if v + w ∈ V then v, w ∈ V . An extremal ray is an extremal subcone of dimension 1. For every D divisor on X a subcone V ⊂ NE(X) is said to be D−negative if for every v ∈ V one has v · D < 0. The Contraction Theorem says that for every extremal K X -negative subcone V of NE(X) the contraction c V of V is well defined, that is to say, a morphism c V : X → W with connected fibers such that W is a normal variety. Moreover, a curve in X is contracted if and only if is numerically equivalent to a curve in V and the Picard number ρ(W ) is equal to ρ(X) − dim(< V >). For a morphism f we recall that NE(f ) is given by the intersection NE(X) ∩ ker(f * ), where f * is the map induced by f on the vector space spanned by NE(X).
If φ ∈ Aut(S 1 × S 2 ) we will write φ(x, y) = (φ 1 (x, y), φ 2 (x, y)) where φ i = π i • φ where π i is the projection of S 1 × S 2 on S i . Lemma 7.1. Let S 1 and S 2 be two del Pezzo surfaces and let φ ∈ Aut(S 1 × S 2 ) Let π i be the projection from S 1 × S 2 onto the i-th factor S i for i = 1, 2.
Proof. Assume φ * (NE(π i )) = NE(π i ). Fix x 1 , x 2 ∈ S 1 and take two distinct irreducible curves C 1 and C 2 on S 1 whose intersection is non empty and such that x i ∈ C i . We have φ(C i × y) = D i × y i because the image of C i × y is a curve that is numerically equivalent to a curve in NE(π 2 ). But C 1 × y and C 2 × y are two curves with nonempty intersection so their images have nonempty intersection. In particular y 1 = y 2 and this implies that φ 2 (x, y) = φ 2 (y). The same argument works with the first component (φ 1 (x, y) = φ 1 (x)) and with φ −1 meaning that φ i is an automorphism of S i .
With the same method, if φ * switches the two cones, one has and that φ i are biholomorphism thus S 1 = S 2 .
In particular, every extremal ray of X is generated by a curve of the type P 1 ×E 2 or E 1 × P 2 , where E i is a (−1)−curve on S i . Observe that the image V of an extremal subcone V by an automorphism φ is again an extremal subcone. In This implies that v and w are in V , so V also is extremal. This implies that φ induces a permutation of the extremal rays of X.
Suppose that there exists an extremal curve E × P 1 such that φ * (E 1 × P 2 ) = P 1 × E 2 . Then φ * maps the extremal ray V := [E 1 × P 2 ] to the extremal ray V := [P 1 × E 2 ]. The contractions c V and c V associated to the extremal subcones V and V are respectively p 1 × Id and Id ×p 2 , where p i : S i →Ŝ i are the blow up with exceptional divisor E i . Observe thatŜ i is smooth and that the fibers of c V and c V have dimension 0 or 1 and are connected. By construction a curve C is contracted by c V if and only φ * C is contracted by c V . These two facts imply that the map f : Then these two are sent to two disjoint curves of the form T i × E 2 by φ and, at last, contracted to the same point by c V . This implies that the fiber of this point with respect to c V contains two disjoint curves and, being connected, has to be at least of dimension 2. But we have seen that every fiber has dimension at most 1, so necessarily R 1 = R 2 . By construction f is also surjective and so it is a bijective map.
Suppose, now, that ρ(S 1 ) = ρ(S 2 ) ≥ 3. Fix a blow-up model for S i . Then the (−1)−curves on S i are either E ij , and are contracted to points by the model, or are sent to curves (lines, conics (ρ(S i ) ≥ 5) and cubics (ρ(S i ) ≥ 7)). If, for all j, the image of E 1j × P belongs to [Q × E] for some (−1)−curve E that depends on j, then the same holds true for the other exceptional curves of the same type: φ(E 1 × P ) ∈ [Q × E] for some E depending on E 1 . Thus, saying that there exist two exceptional curves E i × P such that φ(E 1 × P ) ∈ [Q × E] and φ(E 1 × P ) ∈ [E × Q] is equivalent to requiring that there are two indices (for examples j = 1 and j = 2) such that Suppose, then, that this could happen. Then, as in the previous case, we can construct a commutative diagram where c V = r × Id and c V = p 1 × p 2 where r : S 1 →S 1 is the contraction of two E 11 = r −1 (R 1 ) and E 12 = r −1 (R 2 ) whereas p 1 and p 2 are the blow-up with exceptional divisor respectively E 1 and E 2 . Note that the cone V spanned by E 11 × P and E 12 × P is an extremal subcone because for a >> 0, L : This implies that its image V is extremal. Again, the construction of f make sense because c V contracts a curve if and only if c V contracts its preimage and because all the fibers of c V are connected and have at most dimension one.
The fibers E 11 × Q i are mapped to two disjoint curves of the formQ i ×E 2 and then contracted to the same point. Then the fiber S of this point has dimension at least 2 (exactly 2 by construction) and is then an effective curve that is contracted to a point by c V so its preimage D intersects E 11 × Q i and is contracted by c V . Hence Q 1 = Q 2 . In a similar way we dealt with the other cases and prove that f is injective. By construction, f is also surjective and hence bijective.
Again f is a map that is holomorphic outisde two disjoint smooth subvariety of S 1 × S 2 whose codimension is 2. Thus, by Hartogs' Theorem, f is everywhere holomorphic. Then f is an isomorphism but checking the equality of the Euler numbers one obtain 2 + ρ(S 2 ) = 2 + ρ(S 1 ) = χ(S 1 ) = χ(S 2 ) + 1 = 3 + ρ(S 2 ) and then again a contradiction. Hence the two types of extremal rays cannot be mixed by φ. There are two cases: the first corresponding to the case for which ∀φ ∈ Aut(S), φ * NE(π i ) = NE(π i ) and the second where there exists φ ∈ Aut(X) that switches the two cones. By Lemma 7.1, in the first case Aut(S 1 × S 2 ) = Aut(S 1 ) × Aut(S 2 ) and S 1 = S 2 whereas, in the second, we have S 1 = S 2 and Aut(S 1 × S 2 ) = Aut(S 1 ) ×2 Z 2 .
If S 1 = P 2 and φ ∈ Aut(X), fix a point s ∈ S and consider the map obtained as composition of the inclusion P 2 P 2 ×{s} ⊂ P 2 ×S 2 , φ and the projection on S. The resulting map β s cannot be a dominant morphism because, in this case, P 2 would have divisors with negative self-intersection 4 . Moreover its image cannot have dimension greater than 0; in fact, every surjective map P 2 → C induces a surjective map P 2 → P 1 but this cannot exist. Hence β s (P 2 ) is a point, or equivalently, doesn't depend on P . Hence φ(P, s) = (α(P, s), β(s)) and the same holds true for φ −1 so β ∈ Aut(S 2 ) and, by a composition with Id ×β −1 , we can restrict to the case β = Id. Consider now for a fixed s ∈ S 2 the morphism α s : P 2 → P 2 . As before, its image cannot have dimension 1. If dim(α s (P 2 )) = 0 then φ(P 2 ×{s}) ⊂ P t×S 2 , and because φ is an automorphism, we would obtain an isomorphism between P 2 and a del Pezzo surface of Picard number strictly greater than 1 which is impossible. Hence α s is a dominant map. Suppose α s (P ) = α s (Q). Then φ(P, s) = (α s (P ), s) = (α s (Q), s) = φ(Q, s) but φ is injective so P = Q and α s is also injective. This shows that α s is an automorphism for every s and in particular we have a map f : s ∈ S 2 → α s ∈ P GL(3) = SL(3)/Z 3 . Then f lifts to a map from S 2 to SL(3) that is affine and then f doesn't depend on s. So Aut(P 2 × S) = Aut(P 2 ) × Aut(S 2 ).
If S 1 = P 1 × P 1 then the extremal rays of X = S 1 × S 2 are of the form [(P 1 × In particular, because extremal rays are permuted by every automorphism and because the intersection numbers are preserved, we have φ * (NE(π i )) = NE(π i ) and then Aut(S 1 × S 2 ) = Aut(S 1 ) × Aut(S 2 ).
We call ∆ Si ⊂ Z 2 =: N i the fan of S i and denote with ∆ Si (1) = {e 0 , . . . , e ri } the set of the rays of the fan.
It is possible to associate a divisor D i to each e i ∈ ∆(1) and we say than e i ∼ e j iff D i and D j are linearly equivalent. Call {∆ i } the partition of ∆(1) obtained by taking the quotient with respect to ∼. Call S ∆i the pemutation group over ∆ i . It is easy to see that this partition doesn't mix rays coming from different factors of the product so we can write S 1 ∆i or S 2 ∆i to mean a permutation group that acts on the first or on the second factor. Call H the quotient of Aut(N, ∆) with respect to ΠS ∆i = ΠS 1 ∆i × ΠS 2 ∆i . Then Here a small summary of these groups.
S Aut(N S , ∆ S ) S ∆i Aut(N S , ∆ S )/ S ∆i P 2 Sym(e 1 , e 2 , e 3 ) Sym(e 1 , e 2 , e 3 ) Id P 1 × P 1 < (13), (1234) > < (13), (24) > Z 2 dP 8 < (24) > < (24) > Id dP 7 < (12)(34) > Id Z 2 dP 6 Sym(e 1 , e 2 , e 5 )× < − Id > Id To see that the sequence splits, consider, for example, the case X = dP 8 × dP 7 for which H = Id ×Z 2 =< σ >. This group is generated by the automorphism of the fan of dP 7 that switches the rays associated to the two exceptional divisors of dP 7 , thus a section of Aut(X) → H is given by σ → A where A is an automorphism of P 2 that switches the two points that are blown-up to obtain dP 7 . All the other cases can be described in a similar way.
Aut 0 (X) is the connected component of the identity in Aut(X) and now we will show that Aut 0 (X) = Aut 0 (S 1 ) × Aut 0 (S 2 ). By a result of Cox (see again [5]) Aut 0 (X) Aut g (S) Hom Z (Pic(X), C * ) where Aut g (S) is the group of the automorphisms of the homogeneous coordinate ring S of X, regarded as graded C−algebra. This group is spanned by (C * ) |∆(1)| = (C * ) |∆ S 1 (1)|+|∆ S 2 (1)| and by the elements y m (λ) where λ ∈ C and m ∈ R(N, ∆) (the elements of R(N, ∆) are the roots of Aut(X)). We show that each y m (λ) can be written in a unique way as the product of f i ∈ Aut g (R i ) where R i is the coordinate ring of S i . This shows that Aut g (S) Aut g (R 1 ) × Aut g (R 2 ). The group Hom Z (Pic(X), C * ) splits as Hom Z (Pic(S 1 ), C * ) × Hom Z (Pic(S 2 ), C * ) because Pic(X) = Pic(S 1 ) ⊕ Pic(S 2 ). Then, the quotient can be viewed as a product of the quotient giving Aut 0 (X) = Aut 0 (S 1 ) × Aut 0 (S 2 ).

List of the Threefolds Obtained
In the previous sections we constructed examples of quotients of Calabi-Yau threefolds Y embedded in S 1 × S 2 by groups that are of maximal order in the sense that a group H ≤ Aut(S 1 × S 2 ) such that the restriction to Y gives a free action, cannot have greater order than the ones used. If Y is a Calabi-Yau threefold and G is a group acting freely on Y the same holds true each H ≤ G. Moreover Y /H → Y /G is anétale covering. In the following table we summarize all the quotients analyzed and all theétale coverings obtained by taking quotient with respect to subgroups. Also the known examples are shown. The column m(|G|)/M represents the ratio of the maximal order of the existing group action freely on Y and the estimated (M = M (S 1 , S 2 )). In the column Π 1 (Y /G) the fundamental group of the quotient is written. When for two isomorphic subgroups H 1 and H 2 of G we obtain h 11 (Y /H 1 ) = h 11 (Y /H 2 ) and h 12 (Y /H 1 ) = h 12 (Y /H 2 ) we represent them in the table in one row indicating that multiple subgroups give the same result by their number between round brackets. For example, taking S 1 = S 2 = P 2 and G Z 3 ⊕ Z 3 there are 4 subgroups of order 3 and each of them gives a manifold with Hodge numbers (2,29). In the table this is summarized by writing Z 3 (4) in the column of Π 1 (Y /H). In the last column a "Y " means that the height obtained for the quotient threefold is the least possible, a "N " means the opposite and a "?" means that we don't know if this is the case or not. The pairs (S 1 , S 2 ) for which M (S 1 , S 2 ) = 1 are omitted.