Exceptional collections, and the Neron-Severi lattice for surfaces

We work out properties of smooth projective varieties over a (not necessarily algebraically closed) field that admit collections of objects in the bounded derived category of coherent sheaves that are either full exceptional, or numerically exceptional of maximal length. Our main result gives a necessary and sufficient condition on the Neron-Severi lattice for a smooth projective surface with holomorphic Euler characteristic 1 to admit a numerically exceptional collection of maximal length, consisting of line-bundles. As a consequence we determine exactly which complex surfaces with vanishing irregularity and geometric genus admit a numerically exceptional collection of maximal length. Another consequence is that a minimal geometrically rational surface with a numerically exceptional collection of maximal length is rational.


Introduction
Recently, a substantial amount of work [1,7,8,16] was carried out in order to exhibit exceptional collections of line-bundles of maximal length on complex surfaces of general type with p g = q = 0, motivated by the will to exhibit geometric (quasi)-phantom triangulated categories, i.e., categories with trivial or torsion Grothendieck group K 0 ; see also [17]. Kuznetsov's recent ICM address [29] sets the notion of exceptional collection into the wider picture of semiorthogonal decompositions for bounded derived categories of smooth projective varieties.
The purpose of this work is twofold : we give arithmetic, and geometric, constraints for the existence of exceptional collections of maximal length on smooth projective surfaces over a field. For instance, on the geometric side, we show that there are no numerical obstructions to the existence of exceptional collections of maximal length on complex surfaces of general type with p g = q = 0 (in fact we give in Theorem 3.10 a complete classification of complex surfaces with p g = q = 0 that admit a numerically exceptional collection of maximal length). On the arithmetic side, we show that a minimal geometrically rational surface over a perfect field k that admits a numerically exceptional collection of maximal length is rational (Theorem 3.7). We also show that a numerically exceptional collection of maximal length, consisting of linebundles, on a surface S defined over an arbitrary field k remains of maximal length after any field extension (Theorem 3.3). These results are deduced from a general criterion (the main Theorem 3.1) that gives, for a smooth projective surface S defined over an arbitrary field k, a necessary and sufficient condition on the Néron-Severi lattice of S for S to admit a numerically exceptional collection of maximal length, consisting of line-bundles.
Along the way, we find that if a surface admits a full exceptional collection, then its integral Chow motive is a direct sum of Lefschetz motives (Theorem 2.7). On a slightly unrelated note, we also provide a new characterization of projective space (Theorem 1.2).
Derived category of coherent sheaves and exceptional objects. Let k be a field and let T be a k-linear triangulated category. The typical example of triangulated category that we have in mind is given by the bounded derived category D b (X) of coherent sheaves on a smooth projective variety X defined over k. Given a morphism f : T 1 → T 2 between two objects T 1 and  (ii) For all T ∈ T , there exist T i ∈ T and a sequence 0 = T n → T n−1 → . . . → T 1 → T 0 = T such that cone(T i → T i−1 ) ∈ A i for all i.
The simplest triangulated category is probably the bounded derived category D b (k−vs) of k-vector spaces. Given a triangulated category T , it is natural to be willing to split off copies of D b (k−vs) inside T , in the sense of semi-orthogonal decompositions. Consider then a functor D b (k−vs) → T . Such a functor is determined by the image E ∈ T of the one-dimensional k-vector space placed in degree 0. Let us denote this functor ϕ E ; then, for any complex An exceptional collection is a collection of exceptional objects E 1 , . . . , E n ∈ T such that Hom(E i , E j [l]) = 0 for all i > j and all l ∈ Z.
An important feature of exceptional objects is that strictly full triangulated subcategories generated by exceptional collections are admissible, meaning that the inclusion functor admits a left and a right adjoint. Consequently, given an exceptional collection (E 1 , . . . , E n ) of objects in T , we have a semi-orthogonal decomposition where A = {T ∈ T : Hom(T, E i ) = 0 for all 1 ≤ i ≤ n}, and where, by abuse, we have denoted E i the subcategory generated by E i . When A = 0, the semi-orthogonal decomposition T = E 1 , . . . , E n is said to be a full exceptional collection.
Let us now give some examples of smooth projective varieties X whose bounded derived category of coherent sheaves admits a full exceptional collection.
• X = P n : we have the Beilinson sequence [4] (this is perhaps the most famous example of a full exceptional collection) • X = P 2 the blow-up of P 2 at a point : if E denotes the exceptional divisor, then is a full exceptional collection. More generally, there is a blow-up formula due to Orlov [38]. Note that, by mutating O E (1) to the right, one obtains a full exceptional collection consisting of line-bundles : (2) .
• X = Q n ⊂ P n+1 C : a smooth complex quadric. Kapranov [22] showed that Other examples of varieties admitting exceptional collections include complex Grassmannians (Kapranov loc. cit.), and several other complex rational homogeneous spaces. In fact, it is expected that if G is a semi-simple algebraic group over an algebraically closed field of characteristic zero and P ⊂ G is a parabolic subgroup, then there is a full exceptional collection of vector bundles in D b (G/P ) ; see [30,Conjecture 1.1] and the results in that direction therein. The projective space P n admits a full exceptional collection consisting of n + 1 line-bundles. At least in characteristic zero, this property characterizes completely the projective space among n-dimensional smooth projective varieties : Theorem 1 (Theorem 1.2). Let X be a smooth projective variety of dimension n over a field k of characteristic zero. Assume that L 0 , . . . , L n is a full exceptional collection of D b (X) for some line-bundles L 0 , . . . , L n . Then X is isomorphic to the projective space P n .
As is apparent, the class of varieties admitting full exceptional collections is rather restricted. Perhaps the simplest constraint for a smooth projective variety to admit a full exceptional collection is the following. If we have a semi-orthogonal decomposition T = A, B , then K 0 (T ) = K 0 (A) ⊕ K 0 (B). Since K 0 (D b (k−vs)) = Z, we see that if D b (X) = E 1 , . . . , E r is a full exceptional collection, then K 0 (X) = Z r . As will be explained in the introduction of Section 2, this implies that the Chow motive of X with rational coefficients is a direct sum of Lefschetz motives. Such a constraint was originally obtained via the theory of non-commutative motives by Marcolli and Tabuada [34].
Chow motives and Lefschetz motives. Let R be a ring. The category of Chow motives with R-coefficients over k is constructed as follows. First one linearizes the category of smooth projective varieties over k by declaring that Hom(X, Y ) = CH d (X × Y ) ⊗ Z R, that is, by declaring that the morphism between X and Y are given by correspondences with R-coefficients modulo rational equivalence. Here, X is assumed to be of pure dimension d (otherwise one works component-wise) and the composition law is given by respectively. This R-linear category is far from being abelian, so that one formally adds to this R-linear category the images of idempotents. This is called taking the pseudo-abelian, or Karoubi, envelope. This new category is called the category of effective Chow motives, and objects are pairs (X, p), where X is a smooth projective variety of dimension d and p ∈ CH d (X × X) ⊗ Z R is an idempotent. When p is the class of the diagonal ∆ X in CH d (X × X), we write h(X) for (X, ∆ X ). In general, the object (X, p) should be thought of as the image of p acting on the motive h(X) of X. For example, in the category of effective Chow motives, the motive h(P 1 ) of the projective line becomes isomorphic to (P 1 , p) ⊕ (P 1 , q), where p := {0} × P 1 and q := P 1 × {0} are idempotents in Hom(h(P 1 ), h(P 1 )) := CH 1 (P 1 × P 1 ). The object (P 1 , p) is isomorphic to 1 := h(Spec k), and the motive (P 1 , p) is called the Lefschetz motive and is written 1(−1). The fiber product of two smooth projective varieties induces a tensor product in the category of effective Chow motives, for which 1 is a unit. The category of Chow motives with R-coefficients is then obtained by formally inverting the Lefschetz motive.
Concretely, a Chow motive with R-coefficients is a triple (X, p, n) consisting of a smooth projective variety X of pure dimension d over k, of a correspondence p ∈ CH d (X × X) ⊗ Z R such that p • p = p, and of an integer n. A morphism γ ∈ Hom((X, p, n), (Y, q, m)) is an element The simplest motives are the motive 1 := h(Spec k) of a point Spec k and its Tate twists, that is, the motives 1(n) = (Spec k, id, n) for n ∈ Z. These are called the Lefschetz motives.
As in the case of triangulated categories, it is natural, given a motive M , to split off copies of Lefschetz motives. Given a morphism γ ∈ Hom(1(−n), h(X)), there is an obvious obstruction to the existence of a splitting to that morphism : if γ ∈ CH n (X) is a non-zero numerically trivial cycle, then γ does not admit a left-inverse. Even if γ is not numerically trivial, the existence of a left-inverse is in general a problem of existence of algebraic cycles (in this case, the existence of a cycle γ ∈ CH d−n (X) such that deg(γ · γ ) = 1). In Section 2, we prove : Theorem 2 (Theorem 2.7). Let S be a smooth projective surface over a field k. Assume that S has a full exceptional collection. Then the integral Chow motive of S is isomorphic to a direct sum of Lefschetz motives.
One may naturally ask if the converse to Theorem 2.7 holds. In Remark 2.9, we give evidence that the converse should fail to be true.
Numerical constraints. Although the problem of classifying smooth projective complex surfaces that admit a full exceptional collection seems out of reach at present (it is conjectured that only the surfaces that are rational have a full exceptional collection), a fair amount of work [1,7,8,16] has been carried out in order to construct exceptional collections of maximal length on complex surfaces with p g = q = 0. (As usual, for a smooth projective surface S, the geometric genus is p g := h 0 (Ω 2 S ) = h 2 (O S ) and the irregularity is q := h 1 (O S ).) A first step in constructing exceptional collections consists in constructing numerically exceptional collections.
Recall that, given a k-linear triangulated category T and two objects E and F in T , the Euler pairing χ is the integer The Euler pairing defines a bilinear pairing on the Grothendieck groups K 0 (T ) that we still denote χ.

Definition 2.
An object E is said to be numerically exceptional if A numerically exceptional collection (E 1 , . . . , E r ) on a smooth projective variety X over k is said to be of maximal length if E 1 , . . . , E r span the numerical Grothendieck group, or equivalently if r is equal to the rank of K num 0 (X). (Here, K num 0 (X) := K 0 (X)/(ker χ) ; note that the left and right kernels of χ are the same so that the notation ker χ is unambiguous.) Obviously, an exceptional object is numerically exceptional, an exceptional collection is a numerically exceptional collection, and a full exceptional collection is a numerically exceptional collection of maximal length.
In this work, we give a complete classification of smooth projective complex surfaces, with p g = q = 0, that admit numerically exceptional collections of maximal length : Theorem 3 (Theorem 3.10). Let S be a smooth projective complex surface, with p g = q = 0. As usual, κ(S) denotes the Kodaira dimension of S.
• If S is not minimal, then S has a numerically exceptional collection of maximal length. Assume now that S is minimal.
• If κ(S) = −∞, then S has a numerically exceptional collection of maximal length.
In particular, for surfaces of general type with p g = q = 0, there is no numerical obstruction to the existence of exceptional collections of maximal length. In the case of Enriques surfaces, a general Enriques surface admits ten different elliptic pencils |2F 1 |, . . . |2F 10 | and the line-bundles O(F 1 ), . . . , O(F 10 ) provide an exceptional collection of length 10 ; see Zube [47]. (Any reordering of this length-10 exceptional collection is still an exceptional collection.) Our Theorem 3.10 says in particular that it is not possible to find an exceptional collection consisting of 12 exceptional objects. On the other hand, Theorem 3.10 says that an Enriques surface blown up at a point admits a numerically exceptional collection of maximal length, that is, there is no numerical obstruction to the existence of an exceptional collection of maximal length on an Enriques surface blown up at a point ; cf. Remark 3.12.
So far, exceptional collections have almost only be considered for varieties defined over algebraically closed fields. Our analysis of numerically exceptional collections of maximal length leads, intuitively, to the conclusion that exceptional collections of maximal length, consisting of rank one objects, do not exist for surfaces that are not "split". First, we have a general result : Theorem 4 (Theorem 3.3). Let S be a smooth projective surface over a field k, with χ(O S ) = 1 and with H 1 et (Sk, Q ) = 0. Here,k is a separable closure of k, Sk = S × Spec k Speck, and is a prime = char k. Assume that S admits a numerically exceptional collection (E 0 , E 1 , . . . , E n+1 ) of maximal length, consisting of rank one objects. Then the cycle class map (1)) is surjective modulo torsion, that is, it induces a surjective map (1))/torsion. In particular, for all field extensions K/k, the collection ((E 0 ) K , (E 1 ) K , . . . , (E n+1 ) K ) for S K = S × Spec k Spec K is numerically exceptional of maximal length, and the base-change is an isometry.
Second, we find arithmetic obstructions for geometrically rational surfaces to admit a numerically exceptional collection of maximal length : Theorem 5 (Theorem 3.7). Let S be a minimal smooth projective surface defined over a perfect field k, such that Sk is rational. If S admits a numerically exceptional collection of maximal length, then S is rational.
In particular, the folklore conjecture of Orlov stating that a surface over an algebraically closed field admits a full exceptional collection only if it is rational can be extended to surfaces defined over arbitrary fields. Note however that a surface may be rational but not admit a numerically exceptional collection of maximal length ; see Remark 3.9.
Surprisingly, Theorems 2, 3, 4 and 5 are obtained essentially by exploiting the linear algebra constraints on the Néron-Severi lattice of S imposed by the existence of a numerically exceptional collection of maximal length. The rich linear algebra stemming from the Riemann-Roch formula is treated independently in the appendix. In this work, the Néron-Severi lattice of a smooth projective surface S over a field k, denoted N 1 (S), refers to the group of codimension-1 cycles of S modulo numerical equivalence. Note that, by definition of numerical equivalence, N 1 (S) is torsion-free.
Our main theorem, from which all theorems stated above arise from, is : Theorem 6 (Main Theorem 3.1). Let S be a smooth projective surface over a field k, with χ(O S ) = 1. The following statements are equivalent : (i) S admits a numerically exceptional collection (L 0 , L 1 , . . . , L n+1 ) of line-bundles which is of maximal length, that is, n = rk N 1 (S). (ii) S admits a numerically exceptional collection (E 0 , E 1 , . . . , E n+1 ) of rank one objects in D b (S) which is of maximal length, that is, n = rk N 1 (S). (iii) We have (K S ) 2 = 10 − rk N 1 (S), and the lattice N 1 (S) and the canonical divisor K S , when seen as an element of N 1 (S), satisfy one of the following properties : • N 1 (S) is unimodular of rank 1, and K S = 3D for some primitive divisor D ; • N 1 (S) is the hyperbolic plane, and K S = 2D for some primitive divisor D ; • N 1 (S) is unimodular and odd of rank > 1, and K S is a primitive divisor.
Note that a complex smooth projective surface with p g = q = 0 always satisfies the equation K 2 S = 10 − rk N 1 (S). It is thus surprising that, over any field, the existence of a numerically exceptional collection of line-bundles of maximal length imposes that equation.
A recent result of Perling [40] (see Theorem 2.2) implies that, for complex surfaces with p g = q = 0, the conditions of Theorem 3.1 are further equivalent to the existence of a numerically exceptional collection of maximal length (without any assumptions on the ranks of the objects of the collection). Theorem 3.1 suggests then that if an exceptional collection of maximal length exists on a surface over an algebraically closed field, then an exceptional collection of maximal length that consists of line-bundles should exist. (As far as I am aware, all examples of complex surfaces that admit an exceptional collections of maximal length admit such a collection consisting of line-bundles.) Finally, given a smooth projective variety X, note that if X has an exceptional object of nonzero rank or a numerically exceptional line-bundle, then the structure sheaf O X is numerically exceptional, that is, χ(O X ) = 1. (Note also that, since the classes of rank 0 objects sit in a codimension 1 subspace of K 0 (X), if X has an exceptional collection of maximal length, then X has an exceptional object of non-zero rank and hence χ(O X ) = 1.) Therefore, the running assumption that χ(O S ) = 1 is innocuous.
Notations. Given a smooth projective surface S defined over a field k, N 1 (S) denotes the Néron-Severi lattice of S. The following numerical invariants of S will be used : , the Betti numbers of S, where is a prime = char k. Acknowledgments. Thanks to Magdalene College, Cambridge, the Simons Center for Geometry and Physics, and the Institute for Advanced Study for excellent working conditions. Thanks to Marcello Bernardara, Fabrizio Catanese, Dmitri Orlov, Markus Perling, and Burt Totaro for useful discussions. Thanks to Pierre Deligne for his interest and for fruitful discussions [12] related to the formula (6). Many thanks to Alexander Kuznetsov and to the referees for their insightful comments.

A characterization of projective space
In this section, upon which the rest of this paper does not depend, the base field k is assumed to be of characteristic zero. Galkin, Katzarkov, Mellit, and Shinder [15] have recently considered so-called minifolds. These are smooth projective varieties of dimension n whose bounded derived category D b (X) admits a full exceptional collection of objects in D b (X) of length n + 1. (1) The only two-dimensional minifold is P 2 .
(2) The minifolds of dimension 3 are : the projective space P 3 , the quadric three-fold, the del Pezzo quintic three-fold, and a six-dimensional family of three-folds V 22 ; see [15] for more details. For other examples of minifolds, we refer to [42,26]. Here, although we allow the base-field k to be non-algebraically closed, we consider a more restrictive class of varieties, namely smooth projective varieties of dimension n whose bounded derived category D b (X) admits a full exceptional collection of line-bundles of length n + 1. We show in Theorem 1.2 below that such a property characterizes completely projective space. As Theorem 1.1 shows, it is important to consider full exceptional collections of line-bundles rather than full exceptional collections consisting of objects in the derived category D b (X). In fact, it is important to consider full exceptional collections of line-bundles, rather than merely full exceptional collections of pure sheaves or even vector-bundles. For example, Kapranov [22] showed that quadrics of odd dimension, say d, over an algebraically closed field have a full exceptional collection consisting of d + 1 vector-bundles. On a slightly different perspective, Bernardara [6] showed that a Severi-Brauer variety X of dimension r has a full semi-orthogonal collection of objects E i , 0 ≤ i ≤ r, such that Hom(E i , E i [l]) = 0 for all l = 0 and Hom(E i , E i ) = A ⊗i for the central division algebra A over k that has same class as X in the Brauer group Br(k). Theorem 1.2. Let X be a smooth projective variety of dimension n over a field k of characteristic zero. Assume that L 0 , . . . , L n is a full exceptional collection of D b (X) for some line-bundles L 0 , . . . , L n . Then X is isomorphic to the projective space P n .
Proof. First, note that the assumption implies that Pic X is torsion-free (see Lemma 2.6 below) and of rank 1, so that Pic X = ZH for some divisor H. By Bondal-Polishchuk [9, Theorem 3.4], a d-dimensional smooth projective variety with a full exceptional collection consisting of d + 1 pure sheaves is necessarily Fano, that is, −K X is ample. The theorem then follows from Proposition 1.3 below. Proposition 1.3. Let X be a smooth projective variety of dimension n with Pic X = ZH. Assume either that the dimension n of X is odd, or that X is Fano. If X admits a numerically exceptional collection (L 0 , . . . , L n ) of line-bundles, then X ∼ = P n .
Proof. Since Pic X = ZH, we may write L i = O X (a i H) for some integers a i . By Riemann-Roch, χ(O X (aH)) is a polynomial P with rational coefficients of degree n in the variable a. Since χ(O X ) = 1, this polynomial vanishes at most n times. We know, by semi-orthogonality, that χ(L j , L i ) = χ(L i ⊗ (L j ) −1 ) = P (a i − a j ) = 0 for all i < j. Therefore the set {a i − a j : 0 ≤ i < j ≤ n} has order at most n. Moreover, a i = a j for all i = j, because otherwise χ(O X (a i H − a j H)) = χ(O X ) = 1 would not be zero. This easily implies that there exist an integer m and a non-zero integer k such that a i = m + ki for all 0 ≤ i ≤ n. Up to replacing H with −H if necessary, we may assume that k is a positive integer. Now the polynomial P vanishes exactly at −lk for 1 ≤ l ≤ n. By Riemann-Roch, we have Having in mind that k is positive, it follows that k = 1 and then that deg(H n ) = 1.
We can also compute deg(H n−1 · c 1 (X)) : by looking at the coefficient of a n−1 in (1), we find n l=1 l = n 2 deg(H n−1 · c 1 (X)), which gives deg(H n−1 · c 1 (X)) = n + 1. Therefore c 1 (X) = (n + 1)H. We now distinguish whether X is odd-dimensional, or Fano. In the latter case, by definition, c 1 (X) is ample. In the odd-dimensional case, since either H or −H is ample and since deg(H n ) = 1, we find that H is ample. In both cases, c 1 (X) is n + 1 times an ample divisor. By [23], it follows that the base-change of X to the complex numbers is isomorphic to P n C . Thus X is a Severi-Brauer variety. It has a zero-cycle of degree 1, namely H n . Therefore it is split, i.e., X is isomorphic to P n . Remark 1.4. When n is odd, Proposition 1.3 shows that, provided Pic(X) is torsion-free of rank 1, the assumptions of Theorem 1.2 can be relaxed, essentially by dropping the condition that L 0 , . . . , L n is full. This is possibly related to the fact that there are no fake projective spaces of odd dimension ; cf. [41]. As a corollary to Proposition 1.3, one sees that a quadric hypersurface of odd dimension n ≥ 3 or a non-split Severi-Brauer variety of odd dimension does not admit a numerically exceptional collection of line-bundles of length n + 1.

Full exceptional collections on surfaces and Lefschetz motives
Let X be a smooth projective variety defined over a field k. Assume that X has a full exceptional collection. Then, by flat base-change [27], for a universal domain Ω containing k (this means that Ω is algebraically closed of infinite transcendence degree over its prime subfield), the pull-back X Ω of X along Spec Ω → Spec k also has a full exceptional collection. It follows that K 0 (X Ω ) is of finite rank (see e.g. Lemma 2.6), and hence by applying the Chern character that the Chow ring CH * (X Ω ) ⊗ Z Q is a finite-dimensional vector space over Q. By Kimura [25], it follows that the Chow motive with rational coefficients of X Ω is isomorphic to a direct sum of Lefschetz motives. (In fact, one may choose such an isomorphism to be defined over the base field k, so that the Chow motive with rational coefficients of X is isomorphic to a sum of Lefschetz motives ; see [45,Corollary 3.5]). Such a result was also obtained in [34], via the theory of non-commutative motives. The main result of this section is Theorem 2.7 : we show that when S is a surface with a full exceptional collection, then the result above can be improved by showing that the integral Chow motive of S is isomorphic to a sum of Lefschetz motives. The question of understanding the links between the derived category of coherent sheaves on X and the Chow motive of X is of course not new and we refer to Orlov's [39].
2.1. The Riemann-Roch formula for surfaces. In this section, S is a smooth projective surface defined over a field k. Let K S be the canonical divisor on S and let D be a divisor on S. The Riemann-Roch formula is More generally, there is a Riemann-Roch formula for any object E in D b (S). The rank of an object E in D b (S) is defined as follows : if E • is a complex representing E, then rk E := i (−1) i rk E i . If E and F are two objects in D b (S) of respective ranks e and f , then the Riemann-Roch formula is Assume from now on that χ(O S ) = 1. If E is an object in D b (S) of rank 1 that is numerically exceptional, that is χ(E, E) = 1, then the above Riemann-Roch formula gives c 2 (E) = 0. This justifies referring to a numerically exceptional object of rank one as a numerical line-bundle. Finally, note that if E and F are two numerical line-bundles in D b (S), then the Riemann-Roch formula takes the simple form :

2.2.
Numerically exceptional collections of line-bundles and the Riemann-Roch formula. If (E 0 , E 1 , . . . , E r+1 ) is a numerically exceptional collection, then by definition the matrix χ(E i , E j ) 0≤i,j≤r+1 is upper-triangular with 1's as diagonal entries. If moreover one assumes that the objects E i have rank 1 for all i, then the Riemann-Roch formula relates the upper-triangularity of the matrix χ(E i , E j ) 0≤i,j≤r+1 to the trigonality of the matrix Proposition 2.1. Let S be a smooth projective surface with χ(O S ) = 1 and let r be a nonnegative integer. The following statements are equivalent.
where all blank entries consist of zeroes.

2.3.
Numerically exceptional collections of maximal length and the Néron-Severi lattice. In order to prove Theorem 2.7, we will have to prove that the existence of a (numerically) exceptional collection of maximal length consisting of objects in D b (S) implies that the Néron-Severi lattice N 1 (S) is unimodular. For that matter, M. Perling [ [40]). Let S be a smooth projective surface with χ(O S ) = 1. Then any numerically exceptional collection of maximal length on S can be transformed by mutations into a numerically exceptional collection of maximal length Moreover, if p g = q = 0 and ρ = b 2 (or more generally if ρ = b 2 − 2b 1 ), then any numerically exceptional collection of maximal length on S can be transformed by mutations into a numerically exceptional collection of maximal length that consists of rank one objects.
From the arguments developed by Perling, one can extract the following : Proposition 2.3. Let S be a smooth projective surface with χ(O S ) = 1. Assume that S has a numerically exceptional collection of maximal length. Then the Néron-Severi lattice N 1 (S) is unimodular, and ρ = b 2 − 2b 1 modulo 8.
Proof. By Theorem 2.2, it suffices to prove that if (Z 0 , Z 1 , . . . , Z t , F 0 , . . . , F s ) is a numerically exceptional collection of maximal length, with rk Z i = 0 for 0 ≤ i ≤ t and rk F j = 1 for 0 ≤ j ≤ s, then the Néron-Severi lattice is unimodular.
Suppose Z is a numerically exceptional object of rank zero. From the Riemann-Roch formula, one immediately sees that c 1 (Z) 2 = −1. Suppose now that (Z 1 , Z 2 ) is a numerically exceptional collection consisting of objects of rank zero. From the Riemann-Roch formula again, one finds that Consider now a numerically exceptional collection (Z 0 , Z 1 , . . . , Z t , F 0 , . . . , F s ), where rk Z i = 0 for 0 ≤ i ≤ t. The Riemann-Roch formula gives for all i and all j Thus we see that , we see that the matrix D i · D j 1≤i,j≤s , which is trigonal by the proof of Proposition 2.1, is degenerate. By Proposition A.3, D i · D j 1≤i,j≤s−1 is unimodular. We conclude that the lattice is unimodular, and therefore that N 1 (S) is unimodular.
On the other hand, since the lattice N 1 (S) is unimodular, van der Blij's lemma (see [20,Lemma II.(5.2)], or Remark A.11), together with the Hodge index theorem, gives K 2 S = 2 − ρ modulo 8. This concludes the proof. Remark 2.4. At least in the case when the base field k is algebraically closed, Sasha Kuznetsov has mentioned to me the following geometric argument, which is directly inspired from [5,Theorem 4.7], showing that the existence of a numerically exceptional collection of maximal length implies the unimodularity of the Néron-Severi lattice. Choose a basis of N 1 (S) consisting of classes of smooth curves C i which intersect pairwise transversally. Let F i be a thetacharacteristic on C i , considered as a torsion sheaf on S supported on C i . Then it is clear that On the other hand, if there is a numerically exceptional collection then the determinant is 1, so one can conclude that the determinant of the intersection form is A straightforward corollary to Proposition 2.3 is : Assume that the surface S admits a numerically exceptional collection of maximal length. Then S has a zero-cycle of degree 1.
For example, a smooth quadric surface in P 3 with no rational point does not admit a numerically exceptional collection of maximal length, and hence does not admit a full exceptional collection. (See Theorem 3.6 and Remark 3.9 for stronger statements.) Note that the intersection pairing on N 1 (S) for a smooth projective complex surface S with p g = 0 is always unimodular by Poincaré duality and by the Lefschetz (1, 1)-theorem. Thus Proposition 2.3 provides an arithmetic obstruction for a surface defined over a non-algebraically closed field to admit a numerically exceptional collection of maximal length.

Exceptional collections of maximal length and Chow motives.
Lemma 2.6. Let X be a smooth projective variety. Assume that K 0 (X) is torsion-free. Then CH 1 (X) is torsion-free. If additionally X is a smooth projective surface, then the Chow ring CH * (X) is torsion-free.
Proof. First, we note that CH 0 (X) = Z[X] is torsion-free for all varieties X.
We show that if a smooth variety X has torsion-free K 0 (X), then CH 1 (X) is torsion-free. This was already proved in [15,Lemma 2.2], and we reproduce their proof for the sake of completeness. Recall that the first Chern class provides a group isomorphism Pic(X) ∼ = CH 1 (X), so that it is equivalent to show that Pic(X) is torsion-free. Assume that L is a line-bundle with L ⊗r = 0 for some integer r ≥ 2. The element [L] − 1 ∈ K 0 (X) has rank zero and thus belongs to F 1 K 0 (X), where F • denotes the topological filtration on K-groups. By multiplicativity of the topological filtration, we find that ( It remains to see that if S is a smooth projective surface such that K 0 (S) is torsion-free, then CH 2 (S) is torsion-free. This follows immediately from the fact [14,Ex. 15.3.6] that the second Chern class induces an isomorphism c 2 : spanned by coherent sheaves supported in codimension 2). Since K 0 (S) is assumed to be torsion-free, we obtain that CH 2 (S) is torsion-free.
Theorem 2.7. Let S be a smooth projective surface over a field k, with χ(O S ) = 1. Assume that S has a numerically exceptional collection of maximal length. Then 1 ⊕ 1(−1) ⊕ρ ⊕ 1(−2) is a direct summand of the integral Chow motive of S. Moreover, if S has a full exceptional collection, then the integral Chow motive of S is isomorphic to a sum of Lefschetz motives.
Proof. First assume that S has a numerically exceptional collection (E 0 , . . . , E n+1 ) of maximal length. By Proposition 2.3, there exists a n-tuple (D 1 , . . . , D n ) of elements of CH 1 (S), such that the matrix M = (D i · D j ) 1≤i,j≤n is unimodular. (In particular, ZD 1 ⊕ · · · ⊕ ZD n is a sub-group of CH 1 (S).) Let then (D ∨ 1 , . . . , D ∨ n ) be the basis of ZD 1 ⊕ · · · ⊕ ZD n that is dual to the basis (D 1 , . . . , D n ) with respect to the intersection pairing and let, by Corollary 2.5, a ∈ CH 0 (S) be a zero-cycle of degree 1 on S. We define the correspondences π 0 := a × S, π 4 := S × a, and Assume now that the collection (E 0 , . . . , E n+1 ) is full exceptional. It is enough to show that the idempotent correspondence is equal to 0. The key is to show that Γ acts as zero on the integral Chow groups CH i (S) for i = 0, 1 and 2. For this, it is enough to show that CH 1 (S) = ZD 1 ⊕ · · · ⊕ ZD n and that CH 2 (S) = Za. First recall that, in general, if X is a smooth projective variety with a full exceptional collection that consists of N exceptional objects, then K 0 (X) is free of rank N . The Chern character induces an isomorphism K 0 (S) ⊗ Z Q −→ CH * (S) ⊗ Z Q. Since rk CH 0 (S) = 1 and since rk CH 2 (S) ≥ 1, we get that rk CH 1 (S) ≤ n. Since the intersection pairing restricted to the subspace of CH 1 (S) spanned by D 1 , . . . , D n is unimodular, and since CH 1 (S) is torsion-free by Lemma 2.6, we find that (D 1 , . . . , D n ) forms a Z-basis of CH 1 (S). It follows that CH 2 (S) has rank 1. Since CH 2 (S) is also torsion-free by Lemma 2.6, we find that CH 2 (S) is spanned by any zero-cycle of minimal positive degree and thus that CH 2 (S) = Za.
Finally, by flat base-change [27], the sequence (E 0 ) K , (E 1 ) K , . . . , (E n+1 ) K is a full exceptional collection for S K = S × Spec k Spec K for all field extensions K/k. Here (E i ) K is the pull-back of the object E i along the projection S K → S. Hence, the arguments above show that (Γ K ) * CH * (S K ) = 0 for all field extensions K/k. Therefore the correspondence Γ is nilpotent ; see e.g. [43,Proposition 3.2]. Because Γ is an idempotent, we conclude that Γ = 0.
Remark 2.8. One may in fact prove the following generalization of Theorem 2.7 : If S is a surface such that the base-change K 0 (S) → K 0 (S K ) is surjective for all field extensions K/k, then the integral Chow motive of S is isomorphic to a direct sum of Lefschetz motives. (This is indeed a generalization of Theorem 2.7 because of the base-change theorem [27] for full exceptional collections.) For that matter, one uses a recent result of Totaro [44,Theorem 4.1], combined with the integral version of the Riemann-Roch formula as used in the proof of Lemma 2.6. Remark 2.9. It seems that the converse to Theorem 2.7 is not true, that is, for a surface to have a full exceptional collection seems more restrictive than its integral Chow motive being isomorphic to a direct sum of Lefschetz motive. Consider for instance a complex Barlow surface S. On the one hand, it is proved in [2, Proposition 1.9] and [46, Corollary 2.2] that the Chow group of zero-cycles of S is universally trivial, in the sense that CH 2 (S K ) = Z for all field extensions K/C. By [44,Theorem 4.1], it follows that the integral Chow motive of S is a direct sum of Lefschetz motives. On the other hand, Böhning, Graf von Bothmer, Katzarkov, and Sosna [8] have exhibited a complex Barlow surface S (a determinantal Barlow surface) with an exceptional collection whose orthogonal complement is a phantom category, that is, a nontrivial strictly full triangulated category with vanishing K 0 . Of course this does not say that the Barlow surface S does not admit a full exceptional collection, but it looks like a possibility that it won't. Finally, as yet another reason why having a full exceptional collection is stronger than having an integral motive isomorphic to a direct sum of Lefschetz motives, it is believed and conjectured that a surface that admits a full exceptional collection must be rational.

Numerically exceptional collections of maximal length on surfaces
In the previous section, we saw that the existence of a full exceptional collection for a surface S gives serious constraints on the integral motive of S. In this section, we show that, for a surface S, the weaker condition of having a numerically exceptional collection of maximal length, consisting of rank one objects, is still very restrictive. The main result, which builds up on Proposition 3.2, is Theorem 3.1 : we give a necessary and sufficient condition for a smooth projective surface S defined over a field k, with χ(O S ) = 1, to admit a numerically exceptional collection of maximal length, consisting of line-bundles. Although its proof consists mostly of elementary linear algebra and lattice theory, Theorem 3.1 has surprising consequences. On an arithmetic perspective, Theorem 3.3 roughly says that a "non-split" surface over a field k (e.g. a surface that is not rational over k but that becomes rational after some field extension ; see Theorem 3.7) does not admit a numerically exceptional collection of maximal length, consisting of line-bundles. On a geometric perspective, Theorem 3.10 determines exactly which complex surfaces with p g = q = 0 admit a numerically exceptional collection of maximal length.
3.1. Main theorem. Before we proceed to the statement of Theorem 3.1, let us recall some facts about lattices for which we refer to [20]. A lattice Λ is a free Z-module of finite rank equipped with a symmetric bilinear form b : Λ × Λ → Z. A lattice is said to be even if the norm of every vector is even ; it is said to be odd otherwise. A lattice is said to be unimodular if the determinant of its bilinear form (expressed in any Z-basis) is equal to ±1. An odd unimodular lattice of signature (1, N ) is always isomorphic to the lattice −1 ⊕N ⊕ 1 . Here, ±1 denotes the lattice of rank 1 with generator of norm equal to ±1, and the direct sum is understood as being orthogonal. An even unimodular lattice of signature (1, N ) exists only when N − 1 is divisible by 8, in which case it is isomorphic to U ⊕ E 8 (−1) ⊕ N −1 8 . Here, U is the hyperbolic plane and E 8 (−1) is the opposite of the E 8 -lattice.
Theorem 3.1. Let S be a smooth projective surface over a field k, with χ(O S ) = 1. The following statements are equivalent : (i) S admits a numerically exceptional collection (L 0 , L 1 , . . . , L n+1 ) of line-bundles which is of maximal length, that is, n = rk N 1 (S). (ii) S admits a numerically exceptional collection (E 0 , E 1 , . . . , E n+1 ) of numerical line-bundles in D b (S) which is of maximal length, that is, n = rk N 1 (S). (iii) We have (K S ) 2 = 10 − rk N 1 (S), and the lattice N 1 (S) and the canonical divisor K S , when seen as an element of N 1 (S), satisfy one of the following properties : • N 1 (S) ∼ = 1 and K S = 3D for some primitive divisor D ; • N 1 (S) ∼ = U and K S = 2D for some primitive divisor D ; • N 1 (S) ∼ = 1 ⊕ −1 ⊕n with n > 0 and K S is a primitive divisor.
A first step towards proving the theorem consists in characterizing the Néron-Severi lattice, together with the way the canonical divisor sits in it, of surfaces that admit a numerically exceptional collection of maximal length : (ii) ⇒ (iii) : By Proposition 3.2, the Néron-Severi lattice N 1 (S) is trigonal and unimodular, and K S is a special characteristic element in the sense of Definition A.2, i.e., K S · c 1 (E i ) = −c 1 (E i ) 2 − 2 for all 0 ≤ i ≤ n + 1. We can conclude by invoking Theorem A.4 (and specifically the implication (i) ⇒ (ii) therein).
(iii) ⇒ (i) : This is the conjunction of Proposition 3.2, and of the implication (ii) ⇒ (i) of Theorem A.4. Actually, thanks to item (iii) of Theorem A.4, we can be more precise : we can find an orthogonal basis of N 1 (S) in which K S has a nice expression, and we can express explicitly a numerically exceptional collection of maximal length in terms of that orthogonal basis.
Assume that N 1 (S) is an odd unimodular lattice of rank n. Since it has signature (1, n − 1) and since (K S ) 2 = 10 − n by assumption, Theorem A.4 implies that there exists a Z-basis (D 1 , . . . , D n ) such that Again it is straightforward to check that the collection (O S , O S (D 1 ), O S (D 2 ), O S (D 1 + D 2 )) is numerically exceptional. (Note the analogy : suppose S is the Hirzebruch surface Σ n , n ≥ 0, that is the projective bundle P(O ⊕ O(−n)) over P 1 , and let F be a fiber and C the zero section. In particular we have F 2 = 0, F · C = 0, C 2 = −n and K S = −(2 + n)F − 2C. If n is even, say n = 2m, then D 1 := F and D 2 := C + mF is a hyperbolic basis of N 1 (Σ 2m ) such that K S = −2D 1 − 2D 2 ; moreover the collection (O S , O S (F ), O S (C + mF ), O S (C + (m + 1)F )) is numerically exceptional -in fact, full exceptional. If n is odd, say n = 2m − 1, then the Néron-Severi lattice is odd and we are in the previous situation. The divisors D 1 := C + (m − 1)F and D 2 := C + mF provide an orthogonal basis such that (D 1 ) 2 = −1 and (D 2 ) 2 = 1 and such that is numerically exceptional -in fact, full exceptional.)

3.2.
Consequence for the cycle class map. Let k be a field and denotek a separable closure. Given a field extension K/k and a scheme X over k, we write X K := X × Spec k Spec K.  (1))/torsion. In particular, the collection ((E 0 ) K , (E 1 ) K , . . . , (E n+1 ) K ) for S K is numerically exceptional of maximal length, and the base-change is an isometry for all field extensions K/k.
In other words, in the same way a full exceptional collection remains full exceptional after extension of the base field [27], a numerically exceptional collection of numerical line-bundles of maximal length on a surface S with p g = q = 0 remains of maximal length after any field extension. Note that, by Theorem 2.7, if E 0 , E 1 , . . . , E n+1 is full exceptional, then the cycle class map CH 1 (S) ⊗ Z → H 2 et (Sk, Z (1)) is surjective. Note that surfaces S with χ(O S ) = 1 and b 1 = 0 include surfaces with p g = q = 0 ; see e.g. [32, §3.4]. These conditions are equivalent in characteristic zero, or if the surface lifts to characteristic zero. However, in positive characteristic, there are examples of surfaces with b 1 = 0 and p g = q > 0, e.g., non-classical Godeaux surfaces [31].
Proof of Theorem 3.3. By Noether's formula, (K S ) 2 = 10 − b 2 . By Theorem 3.1, we also have (K S ) 2 = 10 − ρ. This implies that ρ = b 2 . On the other hand, Theorem 3.1 also says that the intersection pairing on N 1 (S) is unimodular. This finishes the proof of the theorem. [19]. The aim of this paragraph is to extend the main result of Hille-Perling [19] to surfaces that are defined over non-algebraically closed fields and that admit numerically exceptional collections of maximal length (rather than full exceptional) consisting of line-bundles. Hille and Perling proved the following (we refer to [40,Theorem 11.3] for a precise statement) :

On a result of Hille and Perling
Theorem 3.4 (Hille-Perling [19]). Let S be a smooth projective surface defined over an algebraically closed field k. Assume that S admits a full exceptional collection (E 0 , . . . , E n+1 ) consisting of line-bundles. Set E n+2 := E 0 (−K S ). Then to this sequence there is associated in a canonical way a smooth complete toric surface with torus invariant prime divisors ∆ 0 , . . . , ∆ n+1 such that ∆ 2 i + 2 = χ(E i+1 ⊗ E −1 i ) for all 0 ≤ i ≤ n + 1.
In light of Theorem 3.1, the assumption that the base field k is algebraically closed in Theorem 3.4 can be lifted : Theorem 3.5. Let S be a smooth projective surface defined over a field k. Assume that χ(O S ) = 1 and that S admits a numerically exceptional collection (E 0 , . . . , E n+1 ) of maximal length, consisting of line-bundles. Then the conclusion of Theorem 3.4 holds.
Proof. Let us define as before, for 1 ≤ i ≤ n + 1, D i := c 1 (E i ) − c 1 (E i−1 ). Let us also define, following Hille and Perling [19, p. 1242 holds. Conditions (i), (ii) and (iii) yield (K S ) 2 = 2(n + 2) + n+2 i=1 (D i ) 2 . But then, by our main Theorem 3.1, we have (K S ) 2 = 10 − n. Therefore (iv) does indeed hold, so that {D i , 1 ≤ i ≤ n} does define an abstract toric system. The theorem then follows because the proof of [19, Theorem 3.5] depends only on the combinatorial data of an abstract toric system.
Theorem 3.6. Let S be a smooth projective minimal surface defined over a perfect field k. Assume that Sk is rational. Then S is one of the following : • P 2 ; • S ⊂ P 3 a smooth quadric with Pic(S) = Z ; • a del Pezzo surface with Pic(S) = ZK S ; • a conic bundle f : S → C over a conic, with Pic(S) ∼ = Z ⊕ Z.
The following theorem is a consequence of Theorem 3.1 ; it shows that geometrically rational, but non-rational, minimal surfaces defined over a perfect field do not admit an exceptional collection of maximal length. Theorem 3.7. Let S be a geometrically rational, smooth projective surface defined over a perfect field k that admits a numerically exceptional collection (E 0 , . . . , E n+1 ) of maximal length. Assume either that S is minimal, or that the objects E i are numerical line-bundles. Then S is rational.
Lemma 3.8. Let S be a smooth projective surface over a perfect field k, and denote Σ a minimal model of S. Assume that the Néron-Severi lattice N 1 (S) is unimodular. Then S is obtained from Σ by successively blowing up rational k-points on Σ. Moreover, N 1 (Σ) is unimodular.
Proof. First note that S is obtained from Σ by successively blowing up Galois-invariant closed points ; see for instance [18]. LetT be the blow-up of a smooth projective surface T over k along a Galois-invariant closed point of degree d ≥ 1. Such a blow-up produces a Galois-invariant collection of pairwise disjoint (−1)-curves, say E 1 , . . . , E d , and the Néron-Severi lattice ofT splits orthogonally as E ⊕ N , where E = E 1 + · · · + E d and E 2 = −d, for some lattice N . This establishes the lemma.
Proof of Theorem 3.7. Let S be a geometrically rational, smooth projective surface defined over a perfect field k that admits a numerically exceptional collection of maximal length. By Proposition 2.3, its Néron-Severi lattice is unimodular. It follows from Lemma 3.8 that S is obtained from one of the minimal surfaces listed in Theorem 3.6 by successively blowing up rational points. Denote Σ a minimal model for S ; N 1 (Σ) is unimodular.
If Σ = P 2 , then S is obviously rational.
If Σ is a smooth quadric in P 3 with Pic(Σ) = Z, then ρ(Σ) = b 2 (Σ) = 2 modulo 8. It follows that ρ(S) = b 2 (S) modulo 8. Therefore, by Proposition 2.3, S does not admit a numerically exceptional collection of maximal length and we get a contradiction.
If Σ is a conic bundle f : Σ → C over a conic, with Pic(Σ) ∼ = Z⊕Z, then by the unimodularity Σ has a degree-one zero-cycle and hence C = P 1 . Note that N 1 (Σ) is spanned by a fiber F and by a multi-section D. Indeed N 1 (Σ) cannot be spanned by two vertical components, i.e., by irreducible components of some fibers, since otherwise the intersection pairing on N 1 (Σ) would be negative, contradicting the Hodge index theorem. Suppose now that N 1 (Σ) is spanned by two multi-sections D and D . Then there exist co-prime integers u and v such that uD + vD is torsion in CH 1 (Σ η ), where Σ η is the conic that is the generic fiber of f . Thus, by localization for Chow groups, we see that N 1 (Σ) is spanned by D and by a vertical component. But since N 1 (Σ) has rank two we see [18, §3.2] that in fact N 1 (Σ) is spanned by D and a fiber F , as claimed. Since F 2 = 0, the unimodularity yields D · F = 1, and hence that D is in fact a section. We conclude by showing that this implies that f is a smooth P 1 -bundle : let F be a fiber of f and denote k 1 /k the field of definition of F . Since D · F = 1, we see that F has a k 1 -rational point and that F is smooth (since otherwise the two geometric components of F would be defined over k 1 and thus would not be in the same Galois orbit and hence N 1 (Σ) would have rank ≥ 3). This implies that F = P 1 k1 . This proves that Σ is a P 1 -bundle over P 1 , and hence that S is rational.
If Σ is a del Pezzo surface with Pic(Σ) = ZK Σ , then we distinguish between two cases. Assume S admits a numerically exceptional collection of maximal length, consisting of numerical line-bundles. Note that if a surface S is obtained from a surface Σ by successively blowing up k-rational points, then, since each such blowup increases rk(N 1 ) by 1 and decreases K 2 by 1, (K S ) 2 = 10 − rk N 1 (S) if and only if (K Σ ) 2 = 10 − rk N 1 (Σ). Therefore, by Theorem 3.1, we have on the one hand that N 1 (Σ) is unimodular and hence K 2 Σ = 1, and on the other hand that K 2 Σ = 10 − rk N 1 (Σ) = 9, thus yielding a contradiction. Assume now that S = Σ (i.e., S is minimal). Then Theorem 2.2 gives a numerically exceptional collection of maximal length, consisting of 3 numerical line-bundles, for S. We conclude to a contradiction as before.
Remark 3.9. It should be noted that the converse to Theorem 3.7 does not hold : a rational surface over a field k need not admit a numerically exceptional collection of maximal length. Consider for example the rational surface X defined over a non-algebraically closed field obtained as the blow-up of the projective plane P 2 along a non-rational Galois-invariant closed point. Then the Néron-Severi lattice of X is not unimodular and, by virtue of Theorem 3.1, X does not admit a numerically exceptional collection of maximal length.
In fact, a minimal rational surface over a field k need not admit a numerically exceptional collection of maximal length. Indeed, a smooth quadric Q with a rational point is rational, but, if Pic(Q) = Z, then, by Proposition 2.3, Q does not admit a numerically exceptional collection of maximal length since ρ = b 2 − 1.

Numerically exceptional collections of maximal length for complex surfaces.
Let us now turn to geometric consequences. Until the end of this paragraph, the base field is the field of complex numbers. In Theorem 3.10, we determine exactly which smooth projective complex surfaces with p g = q = 0 admit a numerically exceptional collection of maximal length.
Let S be a smooth minimal projective complex surface with q = p g = 0. Then, by the Enriques-Kodaira classification of compact complex surfaces according to their Kodaira dimension κ, S is one of the following (see e.g. [24]) : • κ = −∞, a minimal rational surface. The minimal rational surfaces are P 2 and the Hirzebruch surfaces Σ n , n = 0, 2, 3, 4, . . ., where Σ n is the P 1 -bundle P(O ⊕ O(−n)) over P 1 . For instance, Σ 0 = P 1 × P 1 . Note that Σ 1 is not minimal, it is P 2 blown up once. Note also that K 2 = 9 for P 2 and K 2 = 8 for Σ n .
• κ = 0 or 1, a minimal properly elliptic surface. Let X 9 be the rational elliptic surface obtained from P 2 by blowing up the nine base points of a generic cubic pencil. Then Dolgachev [13] proved that the minimal complex surfaces with p g = q = 0 of Kodaira dimension 0 or 1 are obtained from X 9 by performing logarithmic transformations on at least two different smooth fibers. We denote the surfaces obtained in this way by X 9 (p 1 , ..., p n ) (p 1 ≤ p 2 ≤ . . . ≤ p n ), where the p i are the multiplicities of the logarithmic transformations, and call them Dolgachev surfaces. (Some authors reserve this name for the case when there are only two multiple fibers and their multiplicities are relatively prime.) Now X 9 (2, 2) is the Enriques surface ; it is the only Dolgachev surface with Kodaira dimension 0. Note that K 2 S = 0 for all of these surfaces. • κ = 2, a minimal surface of general type. For a minimal surface of general type, we have K 2 S > 0, as well as Castelnuovo's inequality c 2 > 0. If in addition q = p g = 0, then K 2 S + c 2 = 12 by Noether's formula, and K 2 S ≤ 9. In fact, unless S has Kodaira dimension = 0 or 1, there are no obstructions to the existence of a numerically exceptional collection of maximal length : Theorem 3.10. Let S be a smooth projective complex surface with p g = q = 0.
• If S is not minimal, then S has a numerically exceptional collection of maximal length. Assume now that S is minimal.
• If κ(S) = −∞, then S has a numerically exceptional collection of maximal length.
Proof. First note that under the condition p g = q = 0, we have χ(O S ) = 1, b 2 = ρ, and the intersection pairing on N 1 (S) is unimodular. Indeed, the first Chern class induces an isomorphism Pic(S) ∼ = H 2 (S, Z), so that N 1 (S) ∼ = H 2 (S, Z)/torsion ; furthermore, by Poincaré duality, it follows that the intersection pairing on N 1 (S) is unimodular.
If S is not minimal, that is if S is the blow-up of a smooth projective surface, then N 1 (S) is odd of rank ≥ 2 and K S is clearly primitive. Hence, by Theorem 3.1, if S is not minimal, then S admits a numerically exceptional collection of line-bundles of maximal length.
From now on, we assume that S is a minimal surface. By Theorem 2.2 and Theorem 3.1, note that, since b 2 = ρ, S has a numerically exceptional collection of maximal length if and only if it has one consisting of line-bundles.
• If κ = −∞, then in fact S has a full exceptional collection of line-bundles : if S = P 2 , then the Beilinson collection O S , O S (1), O S (2) is full exceptional ; if S = Σ n , n = 0, 2, 3, 4, . . ., is a Hirzebruch surface, then denoting respectively F and C a fiber and a section of the corresponding P 1 -bundle (so that F 2 = 0, F · C = 1 and C 2 = −n), we have that O S , O S (F ), O S (C + sF ), O S (C + (s + 1)F ) is a full exceptional collection for all integers s (this is essentially contained in [38]).
• If κ = 2, then we know that K 2 S ∈ {1, 2, . . . , 9}. Thus if K S = rD for some positive integer r and some primitive divisor D, then r ∈ {1, 2, 3}. Also, by Noether's formula, the Néron-Severi lattice N 1 (S) has rank ∈ {1, 2, . . . , 9}, and, by the classification of unimodular lattices of signature (1, n − 1), we see that N 1 (S) is even if and only if it is isomorphic to the hyperbolic plane U . Now, we have Lemma 3.11. Let S be a complex surface with p g = q = 0. Then, the Néron-Severi lattice N 1 (S) is even if and only if K S = 2D for some divisor D ∈ N 1 (S).
Proof of the lemma. Recall that K S is a characteristic element in N 1 (S), that is, E · (E − K S ) is even for all E ∈ N 1 (S) (this goes by the name of Wu's formula ; it follows from the Riemann-Roch formula (2) whereby E · (E − K S ) = 2(χ(O S (E)) − χ(O S ))). Therefore E 2 is even for all E ∈ N 1 (S) if and only if K S · E is even for all E ∈ N 1 (S). Thus, if K S = 2D, then N 1 (S) is even. Conversely, the pairing on N 1 (S) is unimodular by Poincaré duality, and hence induces an isomorphism from N 1 (S) to its dual. Since K S is characteristic and N 1 (S) is assumed to be even, the element K S ∈ N 1 (S) is mapped to 2w, for some w ∈ N 1 (S) ∨ , under this isomorphism. It is then apparent that K S = 2D for some divisor D ∈ N 1 (S). By Lemma 3.11, if the intersection pairing on N 1 (S) is odd, then K S is either primitive or equal to 3D for some primitive divisor D. In order to conclude, we need to show that K S = 3D if and only if n = 1, which is further equivalent by Noether's formula to K 2 S = 9. Clearly if K S = 3D, then K 2 S = 9D 2 so that K 2 S must be equal to 9. If now n = 1, then N 1 (S) = ZH for some divisor H, which by Poincaré duality satisfies H 2 = 1. The canonical divisor K S is then equal to aH for some integer a. Since K 2 S = 9, we find that a = ±3 and we are done. By Theorem 3.1, we deduce that every minimal smooth projective complex surface of general type with p g = q = 0 admits a numerically exceptional collection of maximal length.
• If κ = 0 or 1, then S is a Dolgachev surface. The Néron-Severi lattice of a Dolgachev surface S has rank 10, so that by Theorem 3.1 (combined with Lemma 3.11) S admits a numerically exceptional collection of maximal length if and only if K S is primitive.
Let us first consider the case κ = 0, that is, the case where S is a classical Enriques surface. It is known that the canonical sheaf ω S is 2-torsion, and hence that K S = 0 in N 1 (S). Therefore, S does not admit a numerically exceptional collection of maximal length. (Recall also that the Néron-Severi lattice of an Enriques surface is U ⊕ E 8 (−1) ; it is even of rank 10.) Consider now a Dolgachev surface S = X 9 (p 1 , . . . , p n ) of Kodaira dimension 1 (i.e., which is not X 9 (2, 2)). Its canonical divisor K S is not torsion and is given by [13, p. 129] where F is the class of a general fiber and F i is the class of the multiple fiber corresponding to p i (in particular, p i F i = F ∈ Pic(S)). The canonical divisor K S may or may not be primitive. First we show by elementary arithmetic that if S is not one of X 9 (2, 3), X 9 (2, 4), X 9 (3, 3), X 9 (2, 2, 2), then K S is not primitive. Let us assume that S is X 9 (p 1 , . . . , p n ) with 2 ≤ p 1 ≤ p 2 ≤ . . . ≤ p n and with distinct multiple fibers F i such that p i F i = F . Let us write c := gcd(p 1 , p 2 ), d := lcm(p 1 , p 2 ), and let u and v be integers such that up 1 + vp 2 = c. Consider G := vF 1 + uF 2 ∈ Pic(S) ; then note that Note also that, in the Néron-Severi lattice N 1 (S), the fibers F , F 1 , F 2 are integral multiples of G (namely, F = dG, F 1 = q 2 G and F 2 = q 1 G, where p 1 = cq 1 and p 2 = cq 2 ) and the fibers F 3 , . . . , F n are all rational multiples of G.
If n > 2, then K S is primitive only if n = 3 and (p 1 , p 2 , p 3 ) = (2, 2, 2). Indeed, we have On the one hand, we have (note that q 1 ≤ q 2 ) with equality if and only if p 1 = p 2 = 2. On the other hand, each divisor F − F i is a positive rational multiple of G. Together with the inequality (2d − q 1 − q 2 ) ≥ 2q 2 ≥ 2, it follows that and equality holds only if n = 3 and p 1 = p 2 = 2. If we can write K S = λG in N 1 (S) for some rational number λ > 1, then K S cannot be primitive. We deduce that K S can only be primitive when n = 3, p 1 = p 2 = 2 and F 3 = G, that is, when S = X 9 (2, 2, 2).
Finally we check that K S is primitive for the Dolgachev surfaces X 9 (2, 3), X 9 (2, 4), X 9 (3, 3) and X 9 (2, 2, 2). In the first three cases, by (3), we have K S = G. More generally, when n = 2, we claim that the divisor G is primitive. We proceed as in [3, p. 384], by contradiction. If, for some rational number 0 < λ < 1, the class λG is represented by a divisor, then by the Riemann-Roch formula, either λG or K S − λG is effective. Note then that an effective divisor D such that deg(D · F ) = 0 is linearly equivalent to a 1 F 1 + a 2 F 2 for some non-negative integers a 1 and a 2 . Since clearly λG is not effective, K S − λG must be effective, that is, we can write K S − λG = a 1 F 1 + a 2 F 2 for some non-negative integers a 1 and a 2 . But then we obtain and hence we find that λ is an integer, which gives a contradiction.
In the last case (S = X 9 (2, 2, 2)), we have Again, if, for some rational number 0 < λ < 1, the class λK S is represented by a divisor, then by the Riemann-Roch formula, either λK S or (1 − λ)K S is effective. Assume that µK S is effective for some 0 < µ < 1. Then there exist non-negative integers a 1 , a 2 , a 3 such that µK S = a 1 F 1 + a 2 F 2 + a 3 F 3 . Since F 1 , F 2 and F 3 are numerically equivalent, we find that µ = a 1 + a 2 + a 3 , in particular we find that µ is an integer. Theorem 3.10 is now proved.
Remark 3.12. Although an Enriques surface does not admit an exceptional collection of maximal length, it would be very interesting to decide whether or not an Enriques surface blown up at a point admits an exceptional collection of maximal length. This would give an example of a triangulated category with an exceptional collection of maximal length that admits an exceptional object whose orthogonal complement does not admit an exceptional collection of maximal length. Indeed, denoting p :S → S the blow-up of S along a point P and denoting E the exceptional divisor, we have by Orlov's blow-up formula a semi-orthogonal decomposition D b (S) ∼ = O E (−1), p * D b (S) . Then the right-orthogonal complement of the exceptional object O E (−1) in D b (S) does not admit an exceptional collection of maximal length by Theorem 3.1. This is related to the failure of the Jordan-Hölder property for semi-orthogonal decompositions ; cf. [28].
N.B. Cho and Lee [10] have recently constructed exceptional collections on some Dolgachev surfaces of type X 9 (2, 3) of maximal length whose orthogonal complements provide examples of phantom categories.

Appendix A. On trigonal unimodular lattices
The main result is Theorem A.4. The equivalence (i) ⇔ (ii) therein reduces the equivalence (i) ⇔ (iii) of Theorem 3.1 to a purely linear algebraic statement.
We refer to [20] for the basics of lattice theory. A lattice (Λ, b) is a free Z-module Λ of finite rank equipped with a symmetric bilinear form b : Λ × Λ → Z. The norm of a vector x ∈ Λ is b(x, x) ∈ Z. We denote a the rank-one lattice Λ = Zλ such that b(λ, λ) = a. The signature of a lattice (Λ, b) is (n + , n − , n 0 ) if Λ ⊗ Z R splits as the orthogonal sum 1 ⊕n + ⊕ −1 ⊕n − ⊕ 0 ⊕n 0 . If n 0 = 0, that is, if Λ is non-degenerate, we will omit the term n 0 from the signature of Λ.
A lattice is said to be even if the norm of every vector is even ; it is said to be odd otherwise. A lattice is said to be unimodular if the determinant of its bilinear form (expressed in any Zbasis) is equal to ±1. Let us denote U the hyperbolic lattice, that is the lattice Ze 1 ⊕ Ze 2 with b(e 1 , e 2 ) = 1 and b(e i , e i ) = 0 for i = 1, 2 ; it is up to isomorphism the only even unimodular lattice of rank 2.
Definition A.1 (Trigonal lattices). We will say that a lattice (Λ, b) of rank n is trigonal if there exists a Z-basis (e 1 , . . . , e n ) of Λ such that the matrix of b expressed in that basis has the trigonal form (4) M := Mat (e1,...,en where the entries outside the 3 diagonals consist solely of zeroes. For simplicity, we will write M = [a 1 , a 2 , . . . , a n ] and sometimes M = trig(a 1 , a 2 , . . . , a n ), for clarity. A Z-basis in which b takes a trigonal form will be called a trigonal basis for b. Such a trigonal reduction is fairly special for unimodular matrices ; see Remark A.11, but also [35] where it is shown that for any unimodular bilinear form over Z there is a basis in which its matrix takes the form (4) with the (n − 1, n) and (n, n − 1) entries replaced by some positive integer d . (Note that such a special characteristic element always exists if the trigonal lattice Λ is unimodular.) The motivation for introducing special characteristic elements comes from Proposition 2.1.
First we characterize trigonal unimodular lattices (endowed with a special characteristic element) : Proposition A.3. Let (Λ, b) be a lattice of rank n and signature (n + , n − , n 0 ). The following statements are equivalent.
Conversely, pick a trigonal basis (e 1 , . . . , e n ) of the unimodular lattice (Λ, b) such that b(ω, e i ) = −b(e i , e i )−2 for all 1 ≤ i ≤ n. We note that ω +e 1 +· · ·+e n identifies with the dual of −(e 1 +e n ) with respect to the unimodular symmetric bilinear form b. Therefore, for e 0 and e n as defined in the proof of (i) ⇒ (iii), ω = −(e 0 + · · · + e n ). This concludes the proof of the proposition.
We now state our main result ; it gives a characterization of pairs consisting of a trigonal unimodular lattice of signature (1, n − 1) endowed with a special characteristic element.
(iii) There exists a Z-basis (e 1 , . . . , e n ) of Λ such that : and such that b(ω, e i ) = −b(e i , e i ) − 2 for all 1 ≤ i ≤ n.
Before we prove the theorem, we state and prove a few useful lemmas that give constraints on the coefficients of trigonal unimodular matrices.
Lemma A.5. Let n ≥ 2 and let a 1 , a 2 , . . . , a n be real numbers. Consider a trigonal matrix M = [a 1 , a 2 , . . . , a n ] as in (4). Assume that det M = ±1. Then there exists an index i such that |a i | < 2.
Proof. Assume for contradiction that for all i we have |a i | ≥ 2. Let V be an n-dimensional R-vector space with basis (e 1 , . . . , e n ) ; we view M as the matrix of a symmetric bilinear form b on V expressed in the basis (e 1 , . . . , e n ). Since |a i | ≥ 2 for all i, it is possible to define inductively b 1 := a 1 and b i := a i − 1 bi−1 for i ≥ 2 ; in fact one sees by induction that |b i | > 1 for all i. Let us consider the following volume-preserving change of basis for V : e 1 := e 1 , and e i := e i − 1 bi−1 e i−1 . Expressed in the basis (e 1 , . . . , e n ), the matrix M of the bilinear form b has determinant det M = det M = ±1. Moreover, M is diagonal with diagonal terms given by the real numbers b i . Thus | det M | = i |b i | > 1, which is a contradiction.
Observations A.6. Let (Λ, b) be a trigonal unimodular lattice of rank n equipped with a special characteristic element ω. Consider a basis (e 1 , . . . , e n ) of Λ such that Mat (ei) (b) = [a 1 , . . . , a n ] for some integers a i , and such that b(ω, e i ) = −b(e i , e i ) − 2 for all 1 ≤ i ≤ n. We make the following observations : (i) Suppose there exists j < n such that a j = 0 and, given an integer x, consider the new basis (e i ) 1≤i≤n := (e 1 , . . . , e j , e j+1 + xe j , e j+2 , . . . , e n ).
Then that new basis is trigonal for b ; in fact the matrix of b in that basis is . . , a j−1 , 0, a j+1 + 2x, a j+2 , . . . , a n ].
Moreover, b| ei ⊥ is unimodular and negative definite. Therefore, by (iii), we have −1 ∈ {a 1 , . . . , a i−2 , a i−1 − 1, a i+1 − 1, a i+2 , . . . , a n }, and this shows that there is an index j such that a j = −1 or 0, which contradicts our assumption. Proof. Let (e i ) 1≤i≤n be a trigonal basis for b and denote a i := b(e i , e i ). By Lemma A.5, for the bilinear form b to be unimodular, one of the a j has to be equal to −1, 0 or 1. The pairing is assumed to be even, so that one of the a j is equal to 0. The integer a j+1 is even. Consider, as in Observation A.6(i), the change of Z-basis e i → e i for i = j + 1 and e j+1 → e j+1 − aj+1 2 e j . In that new basis, the matrix of the form b is [a 1 , . . . , a j−1 , 0, 0, a j+2 , . . . , a 2n ]. Performing several similar changes of bases shows that there is a basis (e 1 , . . . , e 2n ) of Λ such that the matrix of the form b expressed in that basis is [0, 0, . . . , 0]. A straightforward calculation shows that det b = 0 if n is odd, and that det b = (−1) m if n = 2m for some integer m. Thus n is even. Consider then the basis (e 1 , e 2 , e 3 − e 1 , e 4 , e 5 − e 3 , e 6 , . . . , e 2m−1 − e 2m−3 , e 2m ) of Λ. It becomes apparent that Λ ∼ = U ⊕m .
The following proposition and its corollary prove (i) ⇒ (ii) of Theorem A.4, and is the heart of the proof of (ii) ⇒ (iii) of our Main Theorem 3.1 (and hence of the arithmetic application thereof given by Theorem 3.6).
Proposition A.9. Let (Λ, b) be a trigonal unimodular lattice of signature (1, n − 1) and let ω be a special characteristic element in Λ. Then there exists a Z-basis (e 1 , . . . , e n ) of Λ such that Proof. The case n = 1 is trivial. Assume that n = 2 ; in that case any trigonal basis (e 1 , e 2 ) for b is such that Mat (ei) (b) = [a 1 , a 2 ] with a 1 a 2 = 0 (since det b = −1). Thus, by Observation A.6(i), there is a basis (e 1 , e 2 ) for b is such that Mat (ei) (b) = [0, a] with a = 0 or −1, and such that ω is special with respect to that basis. The former case is the case where Λ is even, while in the latter case Observation A.6(ii) gives us a new basis, namely (e 1 + e 2 , e 2 ), in which the matrix of b is diag(1, −1) and with respect to which ω is special.
For the sake of the induction argument to come, let us consider a negative definite trigonal unimodular lattice (Λ, b) of rank 2. Let (e 1 , e 2 ) be a trigonal basis ; the unimodularity of b shows that up to reordering e 1 and e 2 , we have Mat Assume now that n ≥ 3 ; we are going to proceed by induction. We suppose that for all m < n, if (Λ, b) is a trigonal unimodular lattice of signature (1, m − 1) or (0, m) endowed with a special characteristic element ω, then there exists a basis (e 1 , . . . , e m ) of Λ such that b(ω, e i ) = −b(e i , e i ) − 2 for all 1 ≤ i ≤ m, and such that Mat (ei) (b) is either equal to [0, 0] or to diag(±1, −1, . . . , −1). We now fix a trigonal unimodular lattice (Λ, b) of signature (1, n − 1) or (0, n) endowed with a special characteristic element ω. By Lemma A.8, the trigonal lattice (Λ, b), which has signature (1, n − 1) or (0, n), has to be odd. Let (e 1 , . . . , e n ) be a trigonal basis of Λ with respect to which ω is special. By Lemma A.7(i), there exists 1 ≤ j ≤ n, such that b(e j , e j ) = −1 or 0. Suppose that b(e j , e j ) = 0. Then, by repeated use of Observation A.6(i), we find a trigonal basis (e 1 , . . . , e n ) of Λ with respect to which ω is special, and such that b(e k , e k ) = −1 for some k. Therefore, we may assume that b(e j , e j ) = −1 in the first place. By Observation A.6(ii), we find a basis (f 1 , . . . , f n ) of Λ such that By the induction hypothesis, we obtain a basis (f 1 , . . . , f n ) for which b(ω, f i ) = −b(f i , f i ) − 2 for all 1 ≤ i ≤ m, and for which Λ is either −1 ⊕n , 1 ⊕ −1 ⊕n−1 or −1 ⊕n−2 ⊕ U . In order to finish off the induction, we note that if there is a basis (e 1 , e 2 , e 3 ) of a rank-3 unimodular lattice Λ such that Mat Corollary A. 10. Let (Λ, b) be a trigonal unimodular lattice of signature (1, n − 1). If ω is a special characteristic element in Λ, then Proof. This follows immediately from Proposition A.9 : • If Λ = [0, 0], then ω = −2e 1 − 2e 2 and hence b(ω, ω) = 8.
Remark A.11. In fact, it is possible to generalize Proposition A.9 to unimodular lattices of any signature : it can be shown that if Λ is unimodular and trigonal, then there exists a basis (e 1 , . . . , e n ) of Λ such that b(ω, e i ) = −b(e i , e i ) − 2 for all 1 ≤ i ≤ n, and such that with respect to that basis Λ splits as a direct orthogonal sum of lattices isomorphic to [0, 0], [1],[−1], [1,2] and [1,3,1]. (Note that in particular a unimodular lattice is trigonal if and only if it is isomorphic to a direct sum of copies of U , 1 , and −1 .) As a consequence, if (Λ, b) is a trigonal unimodular lattice of signature (n + , n − ) and if ω is a special characteristic element in Λ, then (6) b(ω, ω) = 8 n + + 1 2 + n + − n − .
The formula (6) should be compared to van der Blij's lemma [20,Lemma II.(5.2)]. Let (Λ, b) be a unimodular lattice ; it is easy to see that a characteristic element ω always exists since the function Λ → Z/2Z, λ → b(λ, λ)[mod 2] is Z/2Z-linear. It is also easy to check that the integer b(ω, ω) is an invariant modulo 8. Van der Blij's lemma states that in fact b(ω, ω) = n + − n − [mod 8]. Thus, for a trigonal unimodular lattice, a special characteristic element ω (that is, the element ω ∈ Λ such that b(ω, e i ) = −b(e i , e i ) − 2 for a basis (e 1 , . . . , e n ) of Λ in which the matrix of b is trigonal) can be thought of as an integral characteristic element in the lattice Λ, and (6) gives an integral version of van der Blij's lemma for trigonal unimodular lattices.
The following Witt-type proposition proves (ii) ⇒ (iii) of Theorem A.4 for odd unimodular lattices, and is the heart of the proof of (iii) ⇒ (i) of our Main Theorem 3.1 (and hence of Theorem 3.10).
Proof. In the case n > 10, that is, in the case b(ω, ω) < 0, the proposition was already proved in greater generality by Nikulin [36]. Indeed, assume n > 10 and let ω be a primitive characteristic element of norm b(ω, ω) = 10−n in the lattice Λ of signature (1, n−1). In particular, b(ω, ω) < 0 and the restriction of b to the orthogonal complement ω ⊥ of ω is even and indefinite. Therefore, we may invoke [36, Prop. 3.5.1] (which applies since our lattice has rank ≥ 4), which says that there is only one orbit under O(q) of primitive characteristic elements of given negative norm.
(When n = 1 or 2, the latter inequality should be ignored ; and when n = 3, it should be understood to read x 3 + x 2 ≤ x 1 ). In other words, the orbit of any element of non-negative norm under the action of O(q) contains an element whose coordinates satisfy (7).
Proof of Claim 1. Given a vector v ∈ Λ of norm b(v, v) that divides 2, we define the reflection R v ∈ O(q) across the hyperplane orthogonal to v by the formula Let ξ := x 1 e 1 + · · · + x n e n be an element of Λ. Up to applying the reflections R ei , we see that all vectors ±x 1 e 1 ± · · · ± x n e n belong to the orbit of ξ under the action of O(q). Consider the action of the symmetric groups S n−1 on the set {2, . . . , n} ; this induces an action on the lattice Λ, given by f σ (ξ) := x 1 e 1 + x σ −1 (2) e 2 + · · · + x σ −1 (n) e n for all σ ∈ S n−1 . Clearly σ → f σ defines a homomorphism S n−1 → O(q). Therefore, f σ (ξ) belongs to the orbit of ξ for all σ ∈ S n−1 .
Consider now a non-zero element ξ of non-negative norm. By the above, the vector ξ has in its orbit a vector x 1 e 1 + · · · + x n e n with 0 ≤ x 1 and 0 ≤ x n ≤ x n−1 ≤ · · · ≤ x 2 . Since b(ξ, ξ) ≥ 0, we actually have 0 ≤ x n ≤ · · · ≤ x 1 . Choose now such a vector in the orbit of ξ with minimal non-negative x 1 . Assume that n ≥ 4 (we indicate how to treat the case n ≤ 3 at the end of the proof). We claim that x 4 + x 3 + x 2 ≤ x 1 . If that is not case, consider the reflection R v with v := e 1 + e 2 + e 3 + e 4 . Then we have + e 5 + · · · + e n .
Given that 0 ≤ x 4 ≤ x 3 ≤ x 2 ≤ x 1 and x 4 + x 3 + x 2 > x 1 (note also that x 4 < x 1 because b(ξ, ξ) ≥ 0), we have Therefore, making all the coordinates of R v (ξ) non-negative and reordering them in decreasing order, we obtain a vector ξ := x 1 e 1 + · · · + x n e n in the orbit of ξ with 0 ≤ x n ≤ · · · ≤ x 1 < x 1 , thus yielding a contradiction. (The cases n = 1 and n = 2 are obvious, while in the case n = 3 one proceeds similarly by considering the reflection R v with v := e 1 + e 2 + e 3 .) Claim 2. Assume n < 10. Then there is only one orbit of characteristic elements of Λ of norm 10 − n ; it is the orbit of the element 3e 1 + e 2 + · · · + e n .
The proof of Proposition A.12 is now complete. (ii) ⇒ (iii) : Given Proposition A.12, it only remains to treat the case where Λ is isomorphic to the hyperbolic plane and ω is twice a primitive vector. Let (e 1 , e 2 ) be a basis of Λ such that the matrix of b is [0, 0]. Let us then write ω = ae 1 + be 2 . By assumption we have b(ω, ω) = 2ab = 8.
(iii) ⇒ (i) : The even case is obvious, and so is the case where Λ = 1 . Let us thus consider an odd unimodular lattice (Λ, b) of signature (1, n − 1) with n > 1, and let (e 1 , . . . , e n ) be a basis of Λ such that Mat