The Picard index of a surface with torus action

We consider normal rational projective surfaces with torus action and provide a formula for their Picard index, that means the index of the Picard group inside the divisor class group. As an application, we classify the log del Pezzo surfaces with torus action of Picard number one up to Picard index 10,000.


Introduction
Consider a normal variety X defined over an algebraically closed field K of characteristic zero.If X is normal, then the Picard group Pic(X) embeds into the divisor class group Cl(X) as the subgroup consisting of the Cartier divisor classes, and the Picard index [Cl(X) : Pic(X)] measures the amount of non-invertible reflexive rank one sheaves on X.For rational normal projective surfaces admitting a (non-trivial) action of the multiplicative group K * , we provide the following formula, involving the torsion part Cl(X) tors and the local class groups Cl(X, x), hosting the Weil divisors modulo those being principal near x ∈ X.
Theorem 1.1.The Picard index of a normal rational projective K * -surface X is given by Note that rationality forces our K * -surface X to be Q-factorial and Cl(X) to be finitely generated, see for instance [2,Thm. 5.4.1.5].Moreover, by normality of X, there are only finitely many singular points and these are the only possible contributors of non-trivial local class groups.Thus, all terms in our formula are indeed finite.
Beyond the K * -surfaces, the formula trivially holds for all smooth projective surfaces with a finitely generated and torsion free divisor class group.As soon as we allow torsion, the r.h.s. is no longer integral in the smooth case and thus the formula fails.Concrete examples are the Enriques surfaces, having divisor class group Z 10 × Z /2 Z.A singular counterexample without K * -action is provided by the D 8 -singular log del Pezzo surface of Picard number one: it is Q-factorial with divisor class group Z × Z /2 Z and doesn't satisfy the formula, see Example 7.5.
Our motivation to consider the Picard index arises from the study of log del Pezzo surfaces.Recall that these are normal projective surfaces with an ample anticanonical divisor and at most finite quotient singularities.The log del Pezzo surfaces form an infinite class, which can be filtered into finite subclasses by further conditions on the singularities.Common conditions are bounding the Gorenstein index or the log terminality; for the state of the art we refer to [1,5,16] in the general case and to [7][8][9] in the case of log del Pezzo surfaces with K * -action.The idea of filtering by the Picard index has appeared in [11], where not-necessarily log terminal Fano varieties with divisor class group Z and a torus action of complexity one have been considered.Here, we use Theorem 1.1 to derive in Picard number one suitable bounds on toric and non-toric log del Pezzo K * -surfaces and then present a classification algorithm.Explicit results are obtained up to Picard index 1 000 000 in the toric case and up to Picard index 10 000 in the non-toric case.We use the approach to rational K * -surfaces developed in [10,12], see also [2,Sections 3.4,5.4].A key feature is that rational projective K * -surfaces are realized in natural way as closed subvarieties of non-complete toric varieties with a very specific defining fan.This reduces the computation of the Picard index of a rational K * -surface to computing the Picard index of its non-complete toric ambient variety.Accordingly, we begin in Section 2 with a study of the Picard group in a purely toric setting.In Section 3, we recall the necessary combinatorial theory of K * -surfaces in terms of the integral generator matrices of their ambient toric varieties.Sections 4, 5 and 6 are the technical heart of the proof of Theorem 1.1: there we perform a detailed combinatorial study of the maximal minors of the integral matrices in question.Section 7 completes the proof of Theorem 1.1 and presents examples.Finally, in Section 8, we present the classification algorithm for log del Pezzo K *surfaces of Picard number one with given Picard index, proving Theorem 1.2.
The author would like to thank Prof. Jürgen Hausen for his valuable feedback and advice.In this section, we develop our approach to the Picard group of toric varieties, which yields in Corollary 2.4 a criterion for torsion-freeness and in Proposition 2.7 a first formula involving the Picard index and local class groups.The reader is assumed to be familiar with the basics of toric geometry [4,6].
Construction 2.1.Let Z = Z Σ be a toric variety coming from a fan Σ in the lattice N := Z r .We assume Σ to be non-degenerate, i.e. its primitive ray generators v 1 , . . ., v n span Q r as a convex cone.We allow Σ to be non-complete.With F := Z n , we have the generator map To any maximal cone σ = cone(v i1 , . . ., v in σ ) ∈ Σ max , we associate the lattices We define the local generator map associated to σ by With the inclusion α σ : N σ → N and the map β σ : F σ → F sending e j to e ij , we obtain a commutative diagram Consider the dual lattices By standard toric geometry, we have isomorphisms K ∼ = Cl(Z) and where U σ is the affine toric chart associated to σ.Moreover, the map π σ corresponds to the restriction of divisor classes [D] → [D| Uσ ].In particular, its kernel consists of those divisor classes that are principal on U σ .
Construction 2.2.In the setting of Construction 2.1, we define lattices N and F and a lattice homomorphism P : N → F by Furthermore, we define lattice homomorphisms Let γ : N → N be a kernel of α and δ : F → F be a kernel of β.We obtain an induced map P : F → N making the following diagram commute.
Now consider the dual lattices M := N * and E := F * as well as the abelian group K := σ∈Σmax K σ .We define the map Setting M := M / im(α * ) and Ê := E / im(β * ) as well as K := K / im(π), we obtain a map P ′ : M → Ê fitting into the following commutative diagram with exact rows.
Proposition 2.3.In Construction 2.2, the map β is surjective and there is an exact sequence Moreover, if α is surjective, M is torsion-free and P ′ = P * .
Proof.Every primitive generator of a ray of Σ is a generator of some maximal cone.This implies that β is surjective, hence β * is injective.As a subgroup of K, the Picard group Pic(Z) consists of the Cartier divisor classes, i.e. those that are principal on all affine toric charts U σ for σ ∈ Σ max .This means Applying the snake lemma to the lower diagram of Construction 2. Definition 2.6.The Picard index of a normal Q-factorial variety X is defined as Proposition 2.7.Let Z = Z Σ be a toric variety with a non-degenerate simplicial fan Σ.In the notation of Construction 2.2, we have Proof.Recall that Pic(Z) = ker(π) and K = K / im(π).Since Σ is simplicial, each K σ is finite, hence so is K.We obtain A K * -surface is an irreducible, normal surface X coming with an effective morphical action K * ×X → X.Let us briefly recall the relevant aspects of the geometry of K * -surfaces, the major part of which has been developed in [17][18][19].
Let X be a projective K * -surface.For each point x ∈ X, the orbit map t → t • x extends to a morphism φ x : P 1 → X.This allows one to define The points x 0 and x ∞ are fixed points for the K * -action and they lie in the closure of the orbit K * •x.There are three types of fixed points: A fixed point is called parabolic (hyperbolic, elliptic), if it lies in the closure of precisely one (precisely two, infinitely many) non-trivial K * -orbits.Hyperbolic and elliptic fixed points are isolated, hence their number is finite.Parabolic fixed points form a closed smooth curve with at most two connected components.Every projective K * -surface has a source and a sink, i.e. two irreducible components F + , F − ⊆ X such that there exist non-empty K * -invariant open subsets U + , U − ⊆ X with The source either consists of a single elliptic fixed point or it is a smooth curve of parabolic fixed points.The same holds for the sink.We now describe the combinatorial approach to K * -surfaces that will be our working environment for the rest of this article.It is the approach to varieties with a torus action of complexity one via their Cox Ring developed in [10,12,14]; see also [2,Section 5.4] for the surface case.In a first step, we produce generator matrices of toric varieties that will serve as ambient spaces for our K * -surfaces.Construction 3.1.Fix positive integers r, n 0 , . . ., n r .We start with integral vectors l i = (l i1 , . . ., l ini ) ∈ Z ni ≥1 and The building blocks for our defining matrices are According to the possible constellations of source and sink, we introduce four types of integral matrices: With the canonical basis vectors e 1 , . . ., e r+1 of Z r+1 and e 0 := −(e 1 + • • • + e r ), the columns of P are We call P a defining matrix, if its columns generate Q r+1 as a convex cone.
Next, we will construct the fan for the ambient toric varieties for our K * -surfaces.
Note that Σ is non-degenerate simplicial lattice fan in Z r+1 .However, it is in general not complete.Construction 3.3.Let P be a defining matrix.Consider the toric variety Z = Z Σ , where Σ is as in Construction 3.2.Let U 1 , . . ., U r+1 be the coordinate functions on the acting torus T r+1 of Z. Fix pairwise different 1 = λ 2 , . . ., λ r ∈ K * and set Passing to the closure of the common set of zeroes of h 2 , . . ., h r , we obtain an irreducible rational normal projective surface Since the h i do not depend on the last coordinate U r+1 of T r+1 , we get an effective K * -action on X(P ) as a subtorus of T r+1 by t • x := (1, . . ., 1, t) • x.
Remark 3.4.Consider a K * -surface X = X(P ).Let Z = Z Σ be the ambient toric variety with acting torus T = T r+1 .The fixed points of X are given as follows.For every τ ij ∈ Σ max , the associated toric orbit T • z τij intersects X in a fixed point If 1 ≤ j ≤ n i − 1, the fixed point x ij is hyperbolic and all hyperbolic fixed points arise this way.For j ∈ {0, n i }, the fixed point x ij is parabolic.According to the type of P , we have the following.
(ee) There are two elliptic fixed points x + = z σ + and x − = z σ − and no parabolic fixed points.(pe) There is one elliptic fixed point x − = z σ − .There are parabolic fixed points x i0 ∈ F + and all parabolic fixed points in F + \{x 00 , . . ., x r0 } are smooth.(ep) There is one elliptic fixed point x + = z σ + .There are parabolic fixed points x ini ∈ F − and all parabolic fixed points in F − \{x 0n0 , . . ., x rnr } are smooth.(pp) There are no elliptic fixed points.There are parabolic fixed points x i0 ∈ F + and x ini ∈ F − and all parabolic fixed points in F + \{x 00 , . . ., x r0 } and F − \{x 0n0 , . . ., x rnr } are smooth.
Proposition 3.6.Let X = X(P ) ⊆ Z arise from Construction 3.3.Then we have Proof.The embedding X ⊆ Z is the canonical toric embedding in the sense of

Maximal minors of P
In this section, we consider defining matrices P as in Construction 3.1.We show that the greatest common divisor of its maximal minors is equal to the greatest common divisor of a certain subset of maximal minors.A maximal minor of a defining matrix is given by a square submatrix of maximal size.First, we introduce a special lattice basis that reflects the block structure of P .
For each i = 0, . . ., r, we define Setting we obtain an isomorphism Then we can view P as a lattice map given by We call | det(P A )| ∈ Z ≥0 the maximal minor of P associated to A. The set of maximal minors of P is The following Lemma gives a vanishing criterion for maximal minors of P .In particular, it will allow us to describe the nonzero maximal minors of P explicitly.

The construction of P
In this section, we apply Construction 2.2 to the ambient toric variety of a rational projective K * -surface.In particular, we will give an explicit description of the map P .Construction 5.1.Let Σ be the fan of an ambient toric variety of a K * -surface, as defined in Construction 3.2.Consider the lattices F σ := Z nσ and N σ := lin N Q (σ)∩N from Construction 2.1.We have n σ + = n σ − = r + 1 and n τij = 2.Moreover, we have N σ + = N σ − = N = Z r+1 and N τij = Z e i + Z u ⊆ N .We will work with the identifications Then according to the type of P , a lattice basis of N is given by {e + 1 , . . ., e + r , u + } ∪ S ∪ {e ini , u ini ; i = 0, . . ., r}, (pp) {e i0 , u i0 ; i = 0, . . ., r} ∪ S ∪ {e ini , u ini ; i = 0, . . ., r}, where S := {e ij , u ij ; i = 0, . . ., r, j = 1, . . ., n i − 1}.A lattice basis of F is given by In particular, we have rank(F) = rank(N) = 2n + m(r + 1).With respect to these bases, the maps α and β from Construction 2.2 are , the maps P σ + , P σ − and P τij are then given as Clearly, the map α in Construction 5.1 is surjective in all cases.Hence Corollary 2.4 implies that the Picard group of a projective rational K *surface is torsion-free.Construction 5.3.We continue in the setting of Construction 5.1.We will give explicit descriptions of the maps γ, δ and P from Construction 2.2.According to the type of P , let us set for all i = 0, . . ., r Now define the sets , Let N and F be the free lattices over N and F respectively.In particular, we have rank( N ) = 2n + (m − 1)(r + 1) and rank( F ) = n + mr.According to the type of P , we define a map γ : N → N as follows: Then γ is a kernel of α : N → N .Next, we define a map δ : F → F as follows: (ee Then δ is a kernel of β : F → F .If P is of type (ee), set ẽ0 := −(ẽ Then the induced map P : F → N is given as follows. (ee Proof.In all cases, we can verify that α • γ = 0 and β • δ = 0.Moreover, γ and δ are both injective and we have This shows that γ and δ are kernels of α and β respectively.Furthermore, direct computations verify that γ • P = P • δ holds in all cases.This shows that P is the induced map between the kernels.□ Remark 5.4.We discuss the matrix representation of P from Construction 5.3 for the case (ee).Let Fi and Ni be the free lattices over Fi and Ni respectively.Let Ñ be the free lattice over Ñ .We define the lattice maps .

Maximal minors of P
In this section, we work in the notation of Construction 5.3.We will determine the maximal minors of the lattice map P .
Proof.For (i), we have PA ( fij ) = 0, hence det( PA ) = 0. We show (ii).Consider first the case (ee).Then we have n ′ i = n i − 1. Set NA,i := A ∩ Ni and NA,i := NA ∩ Ni .By (i), we may assume that êij ∈ NA,i or ûij ∈ NA,i holds for all 1 ≤ j ≤ n i − 1.Since (i, j 0 ), (i, j 1 ) ∈ L(A), we thus have | NA,i | > n i .Consider the map PA,i : NA,i → Fi .In the matrix representation from Remark 5.4, we have where the outlined box is a square | NA,i | × | NA,i |-matrix.Since the determinant of the outlined box vanishes, also det( PA ) = 0. Now let P be of type (pe), (ep) or (pp).Then n ′ i = n i .We define the set NA := {û ij ; (i, j) ∈ L(A)} ⊆ N red A .
Writing NA for the free lattice over NA and F for the free lattice over F, we obtain an induced map PA : F → NA as in the commutative diagram That means, P red A contains a row with a single entry l ij and zeroes elsewhere.Doing cofactor expansion, we arrive at det( P red A ) = det( PA ) But since (i, j 0 ), (i, j 1 ) ∈ L(A), we have P * A (û * ij0 ) = P * A (û * ij1 ), i.e.PA contains two equal rows.Hence we have det( PA ) = det( P red A ) = det( PA ) = 0. □ Proposition 6.6.Let P be of (pe), (ep) or (pp).Let A ⊆ N with |A| = | F| such that det( PA ) ̸ = 0. Then we have |L(A)| = r.Furthermore, there exists an i 1 = 0, . . ., r and j i = 1, . . ., n i for all i ̸ = i 1 such that In particular, we have gcd(M red ( P )) = gcd(M ′ (P )).
Proof.By Lemma 6.5 (i), we have êij ∈ A or ûij ∈ A for all i and j.By Lemma 6.5 (ii), for each i there is at most one j with êij ∈ A and ûij ∈ A.
Writing π 1 : Z × Z → Z for the projection onto the first coordinate, this implies that |π 1 (L(A))| = |L(A)|.Together, we obtain This implies that we have L(A) = {(i, j i ) ; i ̸ = i 1 } for some i 1 = 0, . . ., r and j i = 1, . . ., n i .Following the proof of Lemma 6.5 (ii), we proceed to do cofactor expansion and see that | det( PA )| = 1.This proves the claim.The supplement follows directly from the Definition of M ′ (P ).□ The preceding Proposition settles the discussion of maximal minors of P for the cases (pe), (pe) and (pp).The remainder of this section is devoted to the case (ee).In particular, 0 ≤ |L(A)| ≤ r + 1.
For part (ii), assume that there exist 1 ≤ i 0 < i 1 ≤ r such that ẽi0 , ẽi1 / ∈ A and i 0 , i 1 / ∈ π 1 (L(A)).We obtain P red A ( fi0ni 0 ) = d i0ni 0 ũ and P red A ( fi1ni 1 ) = d i1ni 1 ũ.Hence det( P red A ) = 0, a contradiction.The formulas for det( P red A ) again follow from cofactor expansion.□ Definition 6.11.Let P be of type (ee).Let π 1 : Z × Z → Z be the projection onto the first coordinate.We call a subset we define . Proposition 6.12.Let P be of type (ee).We have Proof.We show (i).Proposition 6.10 implies that every element of M red k ( P ) is a Z-linear combination of elements of M ′ k ( P ).This shows that gcd(M ′ k ( P )) divides gcd(M red k ( P )).For the converse, it suffices to show that gcd(M red k ( P )) | x holds for all x ∈ M ′ k ( P ).So, let be arbitraty, where L ⊆ L is a valid subset with |L| = k and 0 Proposition 6.10 implies that det( P red A ) = l i1ni 1 x and det( P red Again, we have |A∩ Ñ | = (r +1)−|L|, hence we can pick A such that det( P red A ) ̸ = 0. Proceeding in the same way as in Case 1, we arrive at gcd(M red k ( P )) | x.Case 3: i 1 = 0.As in Case 2, we can pick some i 0 ∈ π 1 (L) as well as a subset A ⊆ N with |A| = n such that L(A) = L and For all 1 ≤ i ≤ r with i / ∈ π 1 (L), set A i := (A\{ẽ i }) ∪ {ũ}.Then Proposition 6.10 implies that det(P red A ) = l 0n0 x and det(P This implies gcd(M red k ( P )) | gcd(l 0n0 x, d 0n0 x) = x.Part (ii) follows from the fact that every element of M red r+1 ( P ) is an integer multiple of an element of M ′ r ( P ).Part (iii) is a consequence of (i) and (ii).□ Definition 6.13.Let P be of type (ee).For k = 1, . . ., r, we define the set . Lemma 6.14.Let i = 0, . . ., r and j i = 1, . . ., n i − 1.Then we have Proof.By definition we have ν(i, j) = l ini d ij − l ij d ini .Since l ini and d ini are coprime, we find x, y ∈ Z such that xl ini + yd ini = 1.Then we have This implies the claim.□ Proposition 6.15.Let P be of type (ee).We have Proof.We show (i).Since M ′ k ( P ) ⊆ M ′′ k ( P ) and elements of M ′′ k+1 ( P ) are Z-linear combinations of elements of M ′′ k ( P ), we have gcd(M ′′ k ( P )) | gcd(M ′ k ( P )∪M ′′ k+1 ( P )).For the converse, let where We define numbers as well as Then Lemma 6.14 implies gcd(x m−1 , y m ) | x m for all m = 1, . . ., r−k.In particular, we obtain gcd(x 0 , y 1 , . . ., y r−k ) | x r−k .Now set c := (i,j)∈L ν(i, j).Then we have cx r−k = x as well as x 0 c ∈ M ′ k ( P ) and y m c ∈ M ′′ k+1 ( P ) for all m = 1, . . .r − k.Together, we have gcd(M ′ k ( P ) ∪ M ′′ k+1 ( P )) | gcd(x 0 c, y 1 c, . . ., y r−k c) | x r−k c = x.Part (ii) follows from repeated application of (i), together with the fact that M ′ r ( P ) = M ′′ r ( P ).□

Proof of Theorem 1.1 and examples
In this section, we prove the formula for the Picard index of a K * -surface given in Theorem 1.1.We then give two examples where the formula fails: The first one is a toric threefold, the second one is the D 8 -singular log del Pezzo surface of Picard number one.Proposition 7.1.Let P be a defining matrix and P be as in Construction 5.3.Write M (P ) and M ( P ) for the set of maximal minors of P and P respectively.Then we have gcd(M ( P )) = gcd(M (P )).
Remark 7.3.Let Z = P(w 0 , w 1 , w 2 ) be a weighted projective plane.We can assume the weights w i to be pairwise coprime.The divisor class group of a weighted projective space is torsion-free and the orders of the local class groups at the toric fixed points are equal to the weights w i .By Corollary 7.2, we obtain ι Pic (Z) = w 0 w 1 w 2 .
In Proposition 8.1, we will generalize this formula to fake weighted projective planes.For weighted projective planes, there is also a direct way to compute the Picard index: The subgroup of divisor classes that are principle on the i-th standard affine chart Since the weights are pairwise coprime, we have lcm(w 0 , w 1 , w 2 ) = w 0 w 1 w 2 .
The following example shows that a Corollary 7.2 does not hold for higher dimensional toric varieties.
Example 7.4.Consider the three-dimensional weighted projective space Z = P(2, 2, 3, 5).Note that the weights are well-formed, i.e. any three weights have no common factor.The Picard group is given by Hence we have ι Pic (Z) = 30.On the other hand, the product of the orders of the local class groups is 60.
We now consider the D 8 -singular log del Pezzo surface of Picard number one, which does not admit a K * -action.Using the description of its Cox Ring [13, Theorem 4.1], we construct the surface via its canonical ambient toric variety, see also [2, Sections 3.2 and 3.3].
Example 7.5.Consider the integral matrix Let Z = Z Σ be the toric variety whose fan Σ has the following maximal cones: Let p : Ẑ → Z be Cox's quotient presentation of Z, where Ẑ ⊆ Z := K 4 .Consider the polynomial We obtain a commutative diagram ι Here, ι : X → Z is the canonical toric embedding in the sense of [2,Sec. 3.2.5].This implies that X has non-empty intersection with the toric orbits T 3 •z σ for σ ∈ Σ max .We obtain a decomposition into pairwise disjoint pieces By [2, Proposition 3.3.1.5],we have Cl(X, x) ∼ = Cl(Z, z σ ) for x ∈ X(σ).Note that σ 12 , σ 23 and σ 24 are regular and | Cl(Z, We turn to the Picard index.In the notation of Construction 2.2, we have Under the lattice bases {v i , v j } of N σij , we can view P σij as the identity matrix and P σ134 = v 1 v 3 v 4 .Computing matrix representations of the maps involved in Construction 2.2, we obtain 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 We see that P * , is surjective, hence its cokernel K is trivial.Using Proposition 2.7, we conclude ι Pic (X) = ι Pic (Z) = 2. On the other hand, we have This shows that the formula from Theorem 1.1 does not hold for X.

Log del Pezzo K * -surfaces of Picard number one
We contribute to the classification of log del Pezzo K * -surfaces of Picard number one, which in the toric case are fake weighted projective planes.As a special case of fake weighted projective spaces, these have been studied by several authors [8,15].Our classification relies on Theorem 1.1 to produce bounds for the number of log del Pezzo K * -surfaces with fixed Picard index.For a related classification of fake weighted projective spaces by their Gorenstein index, we refer to [3].
We recall some facts about fake weighted projective planes, see also [8,Section 2].Consider K := Z × Z /n Z, where n ∈ Z ≥1 .Let ω 0 , ω 1 , ω 2 ∈ K be given, where ω i = (w i , η i ) and the w i are positive and pairwise coprime.Write C(n) ⊂ K * for the set of n-th roots of unity.The quasitorus H := K * ×C(n) acts on K 3 by The fake weighted projective plane associated to ω 0 , ω 1 , ω 2 is the surface Moreover, the Picard index is where the columns are primitive vectors generating Q 3 as a convex cone.Let K := Z 3 /P * (Z 2 ) and consider the map Q : Z 3 → K. Then we have K ∼ = Z × Z /n Z for some n ∈ Z ≥1 .Setting ω i := Q(e i ), we obtain a fake weighted projective plane Z(P ) := P(ω 0 , ω 1 , ω 2 ).
Remark 8.3.Let Z(P ) be as in Construction 8.2.Then P is the generator matrix of the defining fan of Z(P ).Moreover, we have Z(P ) ∼ = Z(P ′ ) if and only if P ′ = A • P • S holds with a unimodular matrix A and a permutation matrix S.
• For each quadruple (n, w 0 , w 1 , w 2 ) of positive integers with n 2 w 0 w 1 w 2 = ι such that (w 0 , w 1 , w 2 ) are pairwise coprime, do: • For each 0 ≤ x < nw 2 with gcd(x, nw 2 ) = 1, do: • If for all permutation matrices S, the Hermite normal form of P • S differs from the Hermite normal form of all P ′ ∈ L, add P to L. • end do.
Output: The set L. Then every fake weighted projective plane Z with ι Pic (Z) = ι is isomorphic to precisely one Z(P ) with P ∈ L.
Using the formula for the Picard index in Proposition 8.8, we can go through all possible quadruples of positive integers (λ, w 01 , w 02 , M ) whose product equals a given Picard index.Going through all the possible tuples (l 01 , l 02 , l 1 , . . ., l r ) in Proposition 8.10, we can use the equations for the orders of the local class groups as well as the fact that QP * = 0 to derive bounds for all remaining entries of P .This yields an efficient classification of all non-toric rational projective log terminal K * -surfaces of Picard number one and type (ee) by their Picard index.The type (ep) is handled analogously.We arrive at the following classification.Theorem 8.11.There are 1 347 433 families of non-toric, log terminal, rational, projective K * -surfaces of Picard number one and Picard index at most 10 000.The numbers of families for given Picard index develop as follows:

Theorem 1 . 2 .
There are 1 415 486 families of log del Pezzo K * -surfaces of Picard number one and Picard index at most 10 000.Of those, 68 053 are toric and 1 347 433 are non-toric.The number of families for given Picard index develops as follows:

5 . 2 .
Picard group of a toric variety 3 3. Background on K * -surfaces 5 4. Maximal minors of P 7 The construction of P 10 6.Maximal minors of P 14 7. Proof of Theorem 1.1 and examples 20 8. Log del Pezzo K * -surfaces of Picard number one 23 References 27 The Picard group of a toric variety

Definition 6 . 1 ..Construction 6 . 2 .AProposition 6 . 3 .
Let a subset A ⊆ N with |A| = | F| be given.Then we have a sublattice NA := x∈A Z •x ⊆ N and an induced map PA : F → NA as in the commutativeWe call | det( PA )| ∈ Z the maximal minor of P associated to A. The set of all maximal minors of P is defined asM ( P ) := {| det( PA )| ; A ⊆ N , |A| = | F|}.Let A ⊆ N with |A| = | F|.We define N sing A := {ê ij ; êij ∈ A and ûij / ∈ A} ∪ {û ij ; êij / ∈ A and ûij ∈ A} ⊆ A, Fsing A := { fij ; êij ∈ A and ûij / ∈ A} ∪ { fij ; êij / ∈ A and ûij ∈ A} ⊆ F, N red A := A \ N sing A , Fred A := F \ Fsing A .Note that we have | N red A | = | Fred A |. Let N redA and F red A be the free lattices over N red A and Fred A respectively.We obtain an induced map P red A as in the commutative diagram We call | det( P red A )| the reduced minor of P associated to A. We define the set of reduced minors of P as M red ( P ) := {| det( P red A )| ; A ⊆ N , |A| = | F|}.Let A ⊆ N with |A| = | F|.Then we have For A ⊆ N with |A| = | F|, we define L(A) := {(i, j) ; êij ∈ A and ûij ∈ A} ⊆ L. Lemma 6.5.Let A ⊆ N with |A| = | F|.
2, gives the exact sequence of the Proposition.The last statement is clear.□ .1.4.1] expresses the µ i via minors of the generator matrix.The formula for the Picard Index follows from Corollary 7.2.