Classification of singularities of cluster algebras of finite type: the case of trivial coefficients

Abstract We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields of arbitrary characteristic. Furthermore, from the same perspective, we study a family of cluster algebras which are not of finite type and which arise from a star shaped quiver.


INTRODUCTION
Cluster algebras were originally introduced by Fomin and Zelevinsky to study total positivity phenomena and Lusztig's canonical bases in Lie theory, see e.g.[Fom10,Lec10,Lus90].They quickly developed to a vibrant research area going far beyond its initial motivations, and with connections to many other areas, such as algebraic geometry [BFMMNC20, BMRS15, GHK15, Nag13, MS16, Sco06], commutative algebra [GLS13,Mul13,Mul14], combinatorics [RS16,FR05], representation theory of finite dimensional algebras and quivers [MRZ03, BMR + 06, GLS06, DWZ10], higher Teichm üller spaces [FG07,GSV05], or mirror symmetry [GHKK18, GS15,KS14].For further connections and applications see e.g. the surveys [Kel12,Wil14].A cluster algebra is a subring of the field of rational functions in n variables over a base field K 1 .It is a commutative ring that is constructed differently than most rings that usually are considered in commutative algebra: instead of generators and relations, one starts with a set of distinguished generators (the cluster variables) and then iteratively constructs (via the process of mutation) all other generators of the ring.In this article, we will mostly consider cluster algebras A(Q) that are constructed from a quiver Q.We will also assume that Q is totally mutable, that is, we assume trivial coefficients.For the precise definitions and an outline of the more general construction via skew-symmetrizable matrices we refer to Section 2. Our main theme here is to investigate cluster algebras from the perspective of singularity theory, in particular, resolution of singularities.Our studies were motivated by an interesting coincidence in classifications: on the one hand, cluster algebras A(Q) of finite type are classified by ADE-Dynkin diagrams [FZ03a], whereas on the other hand the dual resolution graphs of the Kleinian surface singularities are classified by the same diagrams, see [Art66,Bri66,Lip69], as well as simple hypersurface singularities in the sense of Arnold [Arn72].For an overview, see e.g., [Slo80].Thus we were guided by the following Question 1.1.Let A be a cluster algebra.Which types of singularities can Spec(A) have?Can one classify these singularities for certain types of cluster algebras?Question 1.2.How can one describe resolutions of singularities of cluster algebras and do these resolutions take into account the combinatorial structure of the cluster algebras?So far, there are only few results in this direction.In [BMRS15], Benito, Muller, Rajchgot, and Smith proved that locally acyclic cluster algebras are strongly F-regular (when defined over a field of prime characteristic) and that they have at worst canonical singularities (over a field of characteristic 0).Further, Muller, Rajchgot, and Zykoski [MRZ18] showed that the lower bound cluster algebra (which is an approximation of a given cluster algebra obtained by a suitable truncation of the construction process) is Cohen-Macaulay and normal.
In this paper we study cluster algebras of finite type, which can be classified in terms of Dynkin diagrams (as mentioned above, finite type cluster algebras from quivers are of type ADE, and more generally, all cluster algebras of finite type are classified by the crystallographic Coxeter groups [FZ03a]).We provide a complete classification of their singularities and describe their embedded desingularization in the case of trivial coefficients.Due to the combinatorial nature of cluster algebras, the characteristic of the base field K does not play an essential role.
Notation.For a Dynkin diagram X n ∈ {A n 1 , B n 2 , C n 3 , D n 4 , E 6 , E 7 , E 8 , F 4 , G 2 | n i ≥ i}, we denote by A(X n ) the corresponding cluster algebra with trivial coefficients.Note that the corresponding variety Spec(A(X n )) is a different object to what is called a cluster variety.The latter will not play a role in the present work.
Let us briefly introduce notions in the context of simple singularities, which we need to state our classification theorem.For the entire list of simple singularities in arbitrary characteristics, we refer to [GK90, Definition 1.2].Let K be a field of arbitrary characteristic.A formal power series f ∈ K [[x, y, z]] is of type A m , for some m ∈ Z ≥1 , if K [[x, y, z]]/ f is isomorphic to K [[x, y, z]]/ z m+1 + xy .Note that if K is algebraically closed and char(K) = 2, then we may perform a change in the variables such that z m+1 + xy = z m+1 + x 2 + y 2 .Let n ∈ Z with n ≥ 3. A formal power series f ∈ K[[z, x 1 , . . ., Let N > n ≥ 2. We say that an n-dimensional variety X ⊂ A N  K with an isolated singularity at a closed point x is locally isomorphic to an isolated hypersurface singularity of type A 1 (resp. of type A 2 if n = 2 and N = 3), if the completion of the local ring of X at x is isomorphic to K[[z 0 , x 1 , . . ., x n ]]/ f , where f is a power series of type A 1 (resp. of type A 2 ).If dim(Sing(X)) ≥ 1, we say that X is locally at some U ⊆ Sing(X) isomorphic to a cylinder over an isolated hypersurface singularity of type A 1 in A m+1 Using the introduced notions, we can state our main result on the classification of singularities of cluster algebras of finite type with trivial coefficients.Theorem A. Let K be a field of characteristic p ≥ 0.
(1) Spec(A(A n )), n ≥ 2, is singular if and only if p = 2 and n ≡ 3 mod 4, or if p = 2 and n ≡ 1 mod 2. In the singular case, Spec(A(A n )) is locally isomorphic to an isolated hypersurface singularity of type A 1 .(2) Spec(A(B n )), n ≥ 2, is singular if and only if p = 2 and n ≡ 3 mod 4, or if p = 2.In the singular case, Spec(A(B n )) is locally isomorphic to an isolated hypersurface singularity of type A 1 .(3) Spec(A(C n )), n ≥ 3, is singular if and only if p = 2.In the singular case, we have Sing(Spec(A(C n ))) ∼ = Spec(A(A n−2 )).
(a) If n ≡ 0 mod 2, then Sing(Spec(A(C n ))) is regular and Spec(A(C n )) is locally isomorphic to a cylinder over an isolated hypersurface singularity of type A 1 in A 3 K .(b) If n ≡ 1 mod 2 and n > 3, then Sing(Spec(A(C n ))) has an isolated singularity of type A 1 at the origin and, locally at the origin, Spec(A(C n )) is isomorphic to a hypersurface of the form (where n = 2m + 1) Spec(k[x 1 , . . . ,x 2m , y, z]/ yz while at a singular point different from the origin, Spec(A(C n )) is locally isomorphic to a cylinder over an isolated hypersurface singularity of type A 1 in A 3 K .(c) If n = 3, then Sing(A(C n )) is isomorphic to two lines intersecting transversally at the origin.All other statements of (2) remain true for m = 1.(4) (a) Spec(A(D 4 )) is isomorphic to a subvariety of A 6 K and Sing(Spec(A(D 4 ))) consists of the 6 coordinate axes.At the origin, Spec(A(D 4 )) is locally isomorphic to the intersection of two hypersurface singularities of type A 1 , while at a singular point different from the origin, Spec(A(D 4 )) is locally isomorphic to a cylinder over an isolated hypersurface singularity of type A 1 in A 4  K intersected with a regular hypersurface which is transversal to the cylinder.(b) If p = 2 and n ≡ 0 mod 4 or if p = 2 and n ≡ 1 mod 2, then the singular locus of Spec(A(D n )) has a single irreducible component Y 0 , which is regular and of dimension n − 3.Moreover, Spec(A(D n )) is locally at the singular locus isomorphic to a cylinder over a hypersurface singularity of type A 1 in A 4 K .(c) Let n > 4. If p = 2 and n ≡ 0 mod 4 or if p = 2 and n ≡ 0 mod 2, then , where Y i are isomorphic to coordinate axes, for i ≥ 1, and Y 0 is irreducible, singular at the origin, and of dimension n − 3.At the origin, Spec(A(D n )) is locally isomorphic to the intersection of two hypersurface singularity of type A 1 , while Y 0 is locally isomorphic to a hypersurface singularity of type A 1 .Away from the origin, the situation is analogous to the two D n -cases before.
(5) Spec(A(E 7 )) is singular if and only if p = 2.In the singular case, Sing(Spec(A(E 7 ))) is a regular surface and locally at the singular locus, Spec(A(E 7 )) is isomorphic to a cylinder over an isolated hypersurface singularity of type A 1 in A 6 K intersected with a regular hypersurface which is transversal to the cylinder.(6) Spec(A(G 2 )) is singular if and only if p = 3.In the singular case, Spec(A(G 2 )) is locally isomorphic to an isolated hypersurface singularity of type A 2 in A 3 K .(7) The varieties corresponding to the cluster algebras A(E 6 ), A(E 8 ), and A(F 4 ) are regular.
Cluster algebras of finite type arise in applications very often with non-trivial coefficients.The presence of frozen variables (i.e., directions in which one cannot mutate) can affect the existence and type of singularities.Therefore an interesting question would be to extend the above classification in the case of non-trivial coefficients.This is the subject of further future studies.
Part (1) of Theorem A has previously been proven for p = 2 in [MRZ18, Proposition A.1].Note that the statement in loc.cit. is characteristic free, but the special case char(K) = 2 has been overseen.We note that from our classification follows that there is no obvious direct link between the singularities of cluster algebras of finite types and the rational double point singularities.For example, for cluster algebras of type ADE, only hypersurface singularities of type A 1 (cluster algebras of type A) or more complicated configurations (cluster algebras of type D) appear.
As a consequence of Theorem A, we can construct an embedded resolution of singularities for cluster algebras of finite type.

Corollary B.
Let K be any field and let X := Spec(A), where A is any cluster algebra of finite type.There exists a finite sequence π of blowups in regular centers such that the strict transform of X along π is regular and it has simple normal crossings with the exceptional divisors.
In order to prove Theorem A, we choose first a suitable presentation of the cluster algebra A(X n ), which arises from an acyclic seed.The latter has the benefit that the cluster algebra can be described as a quotient of a polynomial ring in 2n variables over K by an ideal generated by n relations determined by the initial seed, the exchange relations.We determine the singular locus by applying Zariski's criterion for regularity [Zar47, Theorem 11, p. 39].The latter is a variant of the Jacobian criterion for smoothness [Cut04, §2.2],where derivatives with respect to a fixed p-basis of the base field K have to be taken into account in the Jacobian matrix.Since the coefficients appearing in the exchange relations are contained in Z, we do not have to consider a p-basis of K.In particular, K can be any field and is not necessarily perfect.In general, it is not very pleasant to handle the maximal minors of a matrix of size n × 2n.Via subtle eliminations of variables, we deduce from the mentioned presentation a new one, which is better suited for our task.In particular, the number of generators in the resulting set diminishes to at most three and often only one.From this, we can then detect and classify the singularities of the corresponding variety and thus of Spec(A(X n )).
A key ingredients in our studies are continuant polynomials, as they naturally appear in the elimination process.Therefore, as a preparation for Theorem A, we examine them from a perspective of singularity theory in Section 3. Furthermore, we also take a look beyond cluster algebras of finite type.More precisely, we investigate the singularities of a class of cluster algebras which arise from a star shaped quiver S n , where n ≥ 2: Observe that the case n ≤ 4 has already been treated in Theorem A since the corresponding quivers are coming from the Dynkin diagrams A 2 , A 3 , and D 4 , respectively.
Theorem C. Let K be any field and n ≥ 4. Let A(S n ) be the cluster algebra over K arising form the star shaped quiver S n .The singular locus Sing(Spec(A(S n ))) has (n − 1)(n − 2)2 n−4 irreducible components, where each of them is regular and of dimension n − 3. Locally at a generic point of such a component, Spec(A(S n )) is isomorphic to an A 1 -hypersurface singularity.On the other hand, locally at the closed point determined by the intersection of all these components, Spec(A(S n )) is isomorphic to a toric variety, defined by the binomial ideal The singularities of Spec(A(S n )) are resolved by first separating the irreducible components of its singular locus and then blowing up their strict transforms.
The appearing integer sequence ((n − 1)(n − 2)2 n−4 ) n≥4 can be found in the The On-Line Encyclopedia of Integer Sequences, [Slo21, Sequence A001788].In Remark 6.5, we explain the connection to one of the descriptions given in loc.cit.
Let us give a brief summary of the contents: In Section 2, we recall basic notions and results on cluster algebras.In particular, we address the classification of finite type via Dynkin diagrams.After that we study the singularities of continuant polynomials in Section 3, as they play an essential role in our investigations.Then, we show Theorem A and Corollary B by studying case-by-case the cluster algebras A(X n ) of different Dynkin types in Sections 4 (quiver case) and 5 (non-quiver case).We end with the proof of Theorem C in Section 6.

Acknowledgments:
The authors want to thank Bernhard Keller for comments on an earlier version of this manuscript.We also thank the anonymous referee for helpful comments to improve the exposition.This project was initiated during a three weeks stay at the Mathematisches Forschungsinstitut Oberwolfach in the context of an Oberwolfach Research Fellowship in 2020.All authors are grateful to the Institute for their support and hospitality during their visit.

CLUSTER ALGEBRAS BASICS
Since we do not require that the reader is familiar with the theory of cluster algebras, we first briefly recall the basics on cluster algebras associated to quivers and the necessary notions to deal with all cluster algebras of finite type.That is, we also outline the more general theory using skew-symmetrizable matrices.However, for most of the paper, we will be dealing with cluster algebras associated to quivers, so we provide a more detailed exposition for this case.For more details on the general theory, we refer the reader to literature, [FZ02,FZ03a,FWZ16,FWZ17,FWZ20].
A quiver Q is a finite directed graph.So, Q = (Q 0 , Q 1 ) is a pair of two finite sets, where Q 0 = {1, . . ., n} is the set of vertices and Q 1 is the set of arrows between the vertices.An element of Q 1 can be identified with a pair (i, j) with i, j ∈ Q 0 , where the corresponding arrow goes from i to j; we also write i → j.Note that multiple arrows are allowed between two vertices.Additionally, we always assume: For example, the pictures of the quivers (In the set of arrows for Q, we wrote (1, 2) α , for α ∈ {1, 2}, in order to indicate that there are two different arrows from 1 → 2 appearing in Q.) Definition 2.1.Let Q = (Q 0 , Q 1 ) be a quiver and k ∈ Q 0 be a vertex.The quiver mutation µ k (in direction k) transforms Q into a new quiver Q ′ = µ k (Q), which is obtained in the following way: (1) for every directed path i → k → j in Q, we add a new arrow i → j; (2) we reverse the arrows incident to the vertex k; (3) we remove oriented 2-cycles until there is none left.
Two quivers Q (1) and Q (2) are called mutation-equivalent, if there exists a sequence of mutations transforming Q (1) into a quiver Q ′ , which is isomorphic to Q (2) (i.e., there exists a bijection f : 0 between the set of vertices such that (i, Let us illustrate the mutation procedure for an example.Here, we mutate at the vertex k = 1. (3) Remark 2.2.In general, one subdivides the set of vertices into two disjoint sets: the mutable vertices, for which we are allowed to perform a mutation, and frozen vertices, which cannot be mutated, see [FWZ16, Section 2.1].In this paper, we only deal with quivers where all vertices are mutable, so we will not go into details of frozen variables.
From now on, we fix a field K and a field F , which is isomorphic to the field of rational functions over K in n variables.
Definition 2.3.A labeled seed of geometric type in F is a pair (x, Q), where ) is a n-tuple of algebraically independent elements and such that F ∼ = K(x 1 , . . . ,x n ); • Q is a quiver with n vertices, which neither contains loops nor 2-cycles.
The n-tuple x is called the cluster of the seed and x 1 , . . ., x n are the cluster variables.The number n of vertices is called the rank of the seed.Since all seeds appearing in this article are labeled seeds of geometric type, we simply speak of seeds in F .The mutation of a quiver extends in the following way to a seed.
, which is obtained in the following way: Two seeds (x (1) , Q (1) ) and (x (2) , Q (2) ) are called mutation-equivalent, if there exists a sequence of mutations transforming one seed into the other (up to permutation of the cluster variables, which also induces an isomorphism of quivers).If this is the case, we write (x (1) , Note that µ k is an involution, i.e., µ k (µ k (x, Q)) = (x, Q).On the other hand, there exist examples for which , where ℓ = k.For example, one can verify that µ 3 (µ 1 (Q)) = µ 1 (µ 3 (Q)) in the example given above.
Definition 2.5.Let (x, Q) be a seed in F .We set The cluster algebra A := A(x, Q) (of geometric type, over K) determined by the seed (x, Q) is defined as the sub-K-algebra of F generated by all cluster variables, Remark 2.6 (cf.[FWZ16, Section 3.1]).The data of Q can be encoded in an n × n integer matrix B = B(Q) with entries b i,j , which are equal to the number of arrows i → j in Q and where an arrow j → i is counted with negative sign for b i,j , i.e., Then B is called the exchange matrix.Note that B(Q) is skew-symmetric and that B determines Q, so that sometimes the notion (x, B) for the seed (x, Q) is used.Moreover, mutation µ k (Q) can also be defined on the matrix B, where the mutation More generally, the notion of a seed (x, B) and its corresponding cluster algebra A(x, B) can be extended to the following setting: • B := (b i,j ) i,j∈{1,...,n} a skew-symmetrizable integer matrix, i.e., there exists a diagonal matrix D with integer entries such that DB is skew-symmetric, • where the mutation rule µ k (B) is given by (2), and • the exchange relations (1) become Let us point out that the sign pattern of a skew-symmetrizable matrix is skew-symmetric.
The sign pattern of a such a matrix can be encoded in terms of a quiver.More precisely, to a skew-symmetrizable matrix B, one associates the quiver Γ(B) in the following way: if b i,j > 0, then we have an arrow i → j.Note that if B is skew-symmetric, then B = B(Q) and Γ(B) is the quiver Q with multiple arrows collapsed into 1.
Example 2.7.Let us discuss the first non-trivial example.Consider the seed (x, Q), where x = (x 1 , x 2 ) and Q is the quiver with two vertices and one arrow between them, 1 2 .
The following table describes the behavior of (x, Q) along repeated mutation: quiver cluster expression in terms of initial cluster x = (x 1 , x 2 ) 2 ) x (1) 2 ) x (2) 2 ) 1 .Therefore, we have where the second equality holds since Observe that the singular locus of Spec(A) is empty.
In the example above, all cluster variables can be expressed as Laurent polynomials in the initial cluster variables x 1 , x 2 .Indeed, this is always true by [FZ02, Theorem 3.1].In our context this can be stated as follows: Theorem 2.8 (Laurent phenomenon).Let (x, Q) be a seed in F .Every cluster variable can be expressed as a Laurent polynomial with integer coefficients in x.
It can be quite tedious to determine all seeds mutation-equivalent to a given initial seed (x, Q).A useful tool for determining mutation equivalent quivers and related invariants is the Java applet [Kel].Sometimes these calculations can be avoided by considering the lower cluster algebra, which can be easily determined and which coincides with the cluster algebra in many interesting cases, see Theorem 2.12 below.Note that the following results (Definition 2.9, Lemma 2.10, Theorem 2.12) also hold in the skew-symmetrizable case, i.e., for A(x, B) where B is skew-symmetrizable.
Definition 2.9.Let (x, Q) be a seed in F .The lower bound cluster algebra L(x, where x ′ 1 , . . ., x ′ n are the elements that we obtain by the exchange relation (1) after mutating Q once in direction 1, . . ., n, respectively.
We immediately see that the inclusion holds and whenever we have equality, then it is easy to provide a set of generators for A(x, Q).
Let J be the ideal of relations among the generators of L(x, Q).Clearly, the exchange relations provide the elements In general, it may happen that these are not all relations between the generators, see [MRZ18, subsection 1.2].Nonetheless, the following useful result holds for acyclic quivers.Recall that a quiver is called acyclic, if it does not contain an oriented cycle.In the case (x, B) we say that A(x, B) is acyclic if Γ(B) is an acyclic quiver.
Lemma 2.10 (cf.[BFZ05, Corollary 1.17]).If Q is acyclic, then the exchange relations (1) generate the ideal J of relations among the generators of L(x, Q).Moreover, the polynomials . ., n}, form a Gröbner basis for J with respect to any term order for which x ′ 1 , . . ., x ′ n are much larger than x 1 , . . ., x n .
In particular, the dimension of the corresponding variety Spec Remark 2.11.In the skew-symmetrizable case, when Γ(B) is acyclic, the exchange relations (3) generate the ideal of relations among the generators of L(x, B).
A seed (x, Q) is called totally mutable if it admits unlimited mutations in all directions.Since we assume in this article that all vertices of a given quiver are mutable, the seeds (x, Q), which we consider, are always totally mutable.
Theorem 2.12 (cf.[BFZ05, Theorem 1.20]).The cluster algebra A(x, Q) associated with a totally mutable seed (x, Q) is equal to the lower bound L(x, Q) if and only if Q is acyclic.
By the previous result, the cluster algebra Recall that a vertex i of a quiver Q is called a sink (resp.source) if i is the target (resp.source) of every arrow in Q incident to i.In Example 2.13, the vertex 1 is a source, while 3 is a sink.As a consequence of [FZ03a, Proposition 9.2], one has the following lemma.
Lemma 2.14.All orientations on a tree are mutation-equivalent via sequences of mutations at sinks and sources.
Example 2.15.Let us continue Example 2.13.The exchange relations imply: Using the Jacobian criterion, one determines that the singular locus of Spec(A) is the origin V(x 1 , y 1 , y 2 , y 3 ).Locally at the origin, 1 + y 2 y 3 is invertible.In particular, we may introduce the local variable z 1 := y 1 (1 + y 2 y 3 ) + y 3 and we obtain Therefore, Spec(A) has an singularity of type A 1 at the origin.In particular, the blowup of the origin resolves the singularities.
The determinants arising above are examples of continuants.They will play a central role in our considerations, which is why we study some of their properties in the next section.
2.1.Finite type classification.The central object of the present paper are cluster algebras of finite type.We end the section by recalling this notion as well as a classification theorem connecting cluster algebras of finite type with Dynkin diagrams.Precise references for more details are [FZ03a], [FWZ17], or [Mar13, 5.1], for example.
Definition 2.16.Recall that we fixed a field F , which is isomorphic to the field of rational functions in n variables over a field K.
(1) Let (x, B) be a seed in F .The cluster algebra A(x, B) is said to be of finite type if there are only finitely many distinct seeds mutation-equivalent to (x, B).
(2) For any n × n square integer matrix B, its Cartan counterpart A(B) = (a i,j ), is defined to be the integer matrix a i,i := 2 and a i,j Recall that a Cartan matrix A = (a i,j ) is called of finite type if all its principal minors are positive.For the 2 × 2 principal minors, this implies the condition a i,j a j,i ≤ 3 for i = j.
Definition 2.17 ([FWZ17, Definition 5.2.4]).Let A = (a i,j ) be an n × n Cartan matrix of finite type.The Dynkin diagram of A is a graph with vertices {1, . . ., n}, for which the edges are determined as follows: Let i, j ∈ {1, . . ., n} with i = j.If a i,j a j,i ≤ 1, then the vertices i and j are joined by an edge if a i,j = 0. Whenever a i,j a j,i > 1, the following rule is applied for the edge between i and j: Here are two examples.The graph on the right hand side is the Dynkin diagram of the corresponding matrix on the left hand side: There is the following classification of cluster algebras of finite type, cf.[FZ03a, Theorem 1.4].

Theorem 2.18. Let (x, B) be a seed. The cluster algebra A(x, B) is of finite type if and only if the Cartan counterpart of one of its seeds is a Cartan matrix of finite type.
Recall, that the Cartan matrices A(B) of finite type are classified by the Dynkin diagrams Remark 2.19.In the proof for the classification of finite type cluster algebras (see [FZ03b] or [FWZ17, Chapter 5]), the non-quiver cases B n , C n , F 4 , G 2 are connected to the quiver cases via the process of folding.The latter corresponds to taking a quotient with respect to a suitable group action on the quiver, see [FWZ17,§4.4].More precisely, one has: • The seed pattern of type G 2 can be obtained from D 4 via folding ([FWZ17, § 5.7]); • the seed pattern of type F 4 arises from E 6 through folding ([FWZ17, Exercise 4.4.12 and § 5.7]); • the seed pattern of type C n comes from A 2n−1 via folding ([FWZ17, Proof of Theorem 5.5.2]);• we get the seed pattern of type B n from D n+1 by folding ([FWZ17, Proof of Theo- rem 5.5.1]).

CONTINUANT POLYNOMIALS
Continuants are classic in the study of determinants and were already considered by Euler.
Definition 3.1.A continuant of order n is the determinant of a tri-diagonal matrix of the form We will consider the special case where b i = c i = −1 for all 1 ≤ i ≤ n − 1, and denote this continuant by P n (y 1 , . . ., y n ).We set P 0 := 1.
These special continuants also appear in the work of Dupont [Dup12] under the name generalized Chebyshev polynomials.
Moreover, for 1 ≤ r ≤ n − 1 we have Lemma 3.5 (Derivative).For 1 ≤ k ≤ n, one has: From the description of the continuant of Example 3.2 it is straightforward to verify that the terms of order ≤ 2 of P n (y 1 , . . ., y n ), written P n (y 1 , . . ., y n ) ≤2 , depend on n mod 4 and are of the following form: Lemma 3.6.We have As a preparation for the remainder of the article, we study the singularities of the varieties determined by the continuants and deformations of them.
Proof.(1) A straightforward induction using Lemma 3.4 shows that P n , ∂P n ∂y 1 = 1 .Thus, the Jacobian criterion implies the first claim.
By Lemma 3.6, it is clear that P 4k+2 + 1 and P 4k − 1 are singular at the origin.Hence, the analog of Lemma 3.7(2) is not true for n = 2m and singularities appear.
We have In particular, X 2m,λ has at most an isolated singularity at the origin.If Sing(X 2m,λ ) = ∅, then X 2m,λ has a singularity of type A 1 at the origin.Therefore, blowing up the origin resolves the singularities of X 2m,λ .
Proof.Set n := 2m and Q n := P n + λ.We compute the singular locus of Q n using the Jacobian criterion.As in the proof of Lemma 3.7(2), we prove by induction on k that we have In particular, we get y 1 = • • • = y n = 0 by (a m ) and (c m ).In conclusion, we have which implies our claim on the singular locus, (4).
Observe that the statement and the proof of Proposition 3.8 are independent of the characteristic p = char(K) ≥ 0 of the field.Nonetheless, the characteristic plays a role when it comes to the condition λ = (−1) m+1 .For example, if λ = 1 and m = 2k, for some k ∈ Z + , then P 4k + 1 is regular if p = 2 since (−1) 2k+1 = −1 = 1, while it is singular if p = 2.

SINGULARITIES OF FINITE TYPE CLUSTER ALGEBRAS COMING FROM QUIVERS
4.1.A n cluster algebras.Assume that Q is a simply laced Dynkin diagram of type A n with any orientation.Since all trees with the same underlying undirected graph are mutation equivalent (Lemma 2.14), we may choose the following orientation: Recall that we denote by A(A n ) the corresponding cluster algebra.

Lemma 4.1. The cluster algebra
Here, P n+1 is the continuant polynomial defined in Section 3. In particular, the variety This result can also be found in [Dup12, Corollary 4.2].We provide a simpler and shorter proof.
Remark 4.2.Observe that the technique of the proof of Lemma 4.1 to reduce the number of generators using continuant polynomials can be applied for any quiver Q = (Q 0 , Q 1 ), which contains a string of n vertices such that one them is a sink or a source.More generally x k , for a unique vertex k ∈ Q 0 , and hence x k can be eliminated.Lemma 4.1, Lemma 3.7, and Proposition 3.8 immediately imply Theorem A in the A ncase, where it states: Corollary 4.3.Let A(A n ) be the cluster algebra of type A n over a field K.If char(K) = 2, then we have: In particular, the resolution of singularities of Spec(A(A 4m−1 )) is given by the blowup of the singular point.
On the other hand, if char(K) = 2, then we have: ) is isomorphic to an isolated hypersurface singularity of type A 1 .In particular, the resolution of singularities of Spec(A(A 2m−1 )) is given by the blowup of the singular point.
4.2.D n cluster algebras.Next, we consider the quiver Q, whose underlying graph is a simply laced Dynkin diagram of type D n , for some n ≥ 4. Since all orientations on a tree are mutation equivalent (Lemma 2.14), we choose the following orientation and numbering of the vertices for Q: The corresponding cluster algebra, which we denote by A(D n ), coincides with the lower cluster algebra of Q (Theorem 2.12) and the latter is completely described by its exchange relations by Lemma 2.10.
Lemma 4.4.The cluster algebra A(D n ) is isomorphic to where In particular, the variety Spec(A(D n )) is isomorphic to a subvariety of A n+2 K of codimension 2. (As before, P n−2 an P n−3 are the continuant polynomials discussed in Section 3.) Proof.As mentioned before, we have A(D n ) ∼ = K[x 1 , . . . ,x n , y 1 , . . . ,y n ]/I, where I is the ideal generated by As in the proof of Lemma 4.1, we obtain from (5) x k = P k (x 1 , y 1 , . . ., y k−1 ), for all k ∈ {2, . . ., n − 2}.The last generator in (6) can be replaced by If we substitute x 2 , . . ., x n−2 in the remaining two generators, we obtain On the other hand, g 3 + y n−2 g 2 ∈ I yields that we may eliminate By the previous result, Spec ( Using this presentation, we determine the singular locus of Spec(A(D n )).
Lemma 4.5.Let A(D n ) be the cluster algebra of type D n over a field K.If char(K) = 2, then we have where, for i ∈ {1, 2, 3, 4}, the component Y i is the u i -axis and so dim(Y i ) = 1, while and dim(Y 0 ) = n − 3. The only possible singular point of Y 0 is the origin, Observe that Y 0 has two irreducible components if n = 4 since P 2 (z 1 , z 2 ) On the other hand, if char(K) = 2, the same statement holds true if we replace every condition n ≡ 0 mod 4 by n ≡ 0 mod 2.
Proof of Lemma 4.5.By Lemma 4.4, Spec(A(D n )) is isomorphic to the subvariety of A n+2 K determined by Observe that there is an ordering on the variables such that u 1 u 2 is the leading monomial of h 1 and u 3 u 4 is the one of h 2 .Hence, the dimension of Spec(A(D n )) is equal to n and by applying the Jacobian criterion for smoothness, the singular locus of Spec(A(D n )) is determined by the vanishing of the 2 × 2 minors of the Jacobian matrix of (h 1 , h 2 ).We abbreviate Jac(D n ) := Jac(h 1 , h 2 ; u 1 , u 2 , u 3 , u 4 , z 1 , . . ., z n−2 ).The first four columns of Jac(D n ) are while the remaining columns are (Here, we use the obvious abbreviations P n−2 and P n−3 .)Clearly, the maximal minors of the first matrix are of the form u 3 (. ..) and u 4 (. ..).Suppose u 3 = u 4 = 0.The vanishing of h 1 and h 2 provides that for a singular point, we have to have u 1 u 2 = 0 and P n−2 + 1 = 0.
Note that this covers all cases, where the minors of Jac(D n ) vanish.Hence, we determined all components of the singular locus.Furthermore, observe that ( 9) First, assume char(K) > 2. By Lemma 3.7 and Proposition 3.8, we have This implies where Sing(Y 0 ) = V(u 1 , . . ., u 4 , z 1 , . . ., z n−2 ) = 4 i=1 Y i is the origin if n ≡ 0 mod 4, while Y 0 is regular in the second case.Observe that Y i is the u i -axis and so dim(Y i ) = 1, for i ∈ {1, . . ., 4}, and dim(Y 0 ) = n − 3.
Let us turn to the case char(K) = 2.The same arguments apply and the only difference appears, when we apply Proposition 3.8, which leads to the condition n ≡ 0 mod 2 instead of n ≡ 0 mod 4.
As before in the A n -case, we can classify the singularities and construct a desingularization from this.Proposition 4.6.Let A(D n ) be the cluster algebra of type D n over a field K.We use the notation of Lemma 4.5.
(1) If Sing(Spec(A(D n ))) ∼ = Y 0 , then the variety Spec(A(D n )) is locally at Y 0 isomorphic to a cylinder over a hypersurface singularity of type A 1 .In particular, the blowup with center Y 0 resolves the singularities.Proof.By Lemma 4.4, Spec(A(D n )) is isomorphic to the subvariety of A n+2 K given by First, suppose Sing(Spec(A(D n ))) ∼ = Y 0 , where Y 0 = V(u 1 , . . ., u 4 , P n−2 (z 1 , . . ., z n−2 ) + 1) , is regular and of dimension n − 3.Moreover, Lemma 3.7 and Proposition 3.8 provide that H := Spec(K[u 1 , . . ., u 4 , z 1 , . . ., z n−2 ]/ h 2 ) is regular.Locally at Y 0 , the element 1 − u 3 u 4 is invertible and thus we may introduce the local variable w Using the latter, we get h 1 = w 1 u 2 − u 3 u 4 locally.Therefore, locally at Y 0 , the variety Spec(A(D n )) is isomorphic to an intersection of a cylinder over an A 1 -hypersurface singularity and a regular variety H, which is transversal to the cylinder.In particular, the blowup of Y 0 is a desingularization of Spec(A(D n )).This ends the proof of part (1).

Let us come to the case Sing(Spec(A(D
. By Lemma 4.5, this can only happen if n ≡ 0 mod 2. (Note that n ≡ 0 mod 4 implies n ≡ 0 mod 2.) Here, the singular locus of Spec(A(D n )) has five components, Y 0 above and the u i -axes Y i for i ∈ {1, 2, 3, 4}.The only singular point of Sing(Spec(A(D n ))) is the origin 0, which is also the singular locus of Y 0 , as well as the intersection of the four other components Y 1 , . . ., Y 4 .The same argument as above shows that, locally at a singular point, which is contained in Y 0 \ {0}, the variety Spec(A(D n )) is isomorphic to a cylinder over an A 1 -singularity intersected with a regular hypersurface, which is transversal to the cylinder.
It remains to study the situation at the origin, which is the singular locus of Y 0 and also equal to 4 i=1 Y i .By (10), Y 0 is isomorphic to a hypersurface singularity in A n−2 K of type A 1 .In particular, we get (2)(b) and blowing up the origin resolves the singularities of Y 0 .Finally, for (2)(c), the same argument as above (for Y 0 ) provides that h 1 = w 1 u 2 − u 3 u 4 locally at the origin.In particular, h 1 and h 2 are both homogeneous of degree 2. This implies, if we blow up the origin, then the singular locus of the strict transform of Spec( where Y ′ i denotes the strict transform of Y i .Furthermore, for every i = j, we have i is regular and after blowing up with center Z all singularities are resolved by (2)(a).

4.
3. E 6 , E 7 , E 8 cluster algebras.Let us now turn our attention to the missing skew-symmetric cluster algebras of finite type, which are those arising from orientations on E 6 , E 7 , E 8 Dynkin diagrams.As before, we fix a field K. Proposition 4.7.Let A(E 6 ) (resp.A(E 8 )) be the cluster algebra of type E 6 (resp.E 8 ) over K.
(1) There exist presentations of A(E 6 ) and A(E 8 ) of codimension three.
The statement for E 6 follows by applying the analogous arguments for the quiver: We leave the details as an easy exercise for the reader.Proposition 4.8.Consider the variety Spec(A(E 7 )) over any field K corresponding to the cluster algebra of type E 7 .
(2) If char(K) = 2, then Spec(A(E 7 )) is isomorphic to 7-dimensional subvariety of A 10 K , whose singular locus is a regular surface.Locally at the singular locus, Spec(A(E 7 )) is isomorphic to a cylinder over an isolated hypersurface singularity of type A 1 in A 6 K intersected with a regular hypersurface, which is transversal to the cylinder.In particular, the resolution of singularities of Spec(A(E 7 )) is given by the blowup of the singular locus.

SINGULARITIES OF FINITE TYPE CLUSTER ALGEBRAS NOT COMING FROM QUIVERS
Next, let us discuss the singularities of cluster algebras of finite type for which it is necessary to work with skew-symmetrizable matrices.Recall the exchange relations (3) in the matrix setting (Remark 2.6) as well as the definitions of subsection 2.1.5.1.B n cluster algebras.A possible exchange matrix B for type B n , n ≥ 2, is given as [FWZ17, §5.5, (5.31)]) and the corresponding Dynkin diagram is of type B n (where n is the number of vertices);

• • • .
Lemma 5.1.Let K be any field and n ≥ 2. The cluster algebra A(B n ) is isomorphic to Proof.Since the underlying diagram Γ(B) is acyclic of type B n , we get the presentation As in the proof of Lemma 4.1 (type A n ), one can express x k in terms of x 1 , y 1 , . . ., y k−1 : If we plug this into the remaining generators x n , y 1 , . . ., y n ]/ g n , h n , which yields the assertion after renaming the variables.Proposition 5.2.Let K be any field and n ≥ 2. For char(K) = 2 the following holds: (1) Spec(A(B n )) is singular if and only if n ≡ 3 mod 4.
(2) If n = 4m − 1, for some m ∈ Z >0 , then Spec(A(B 4m−1 )) has an isolated singularity at the origin and locally at the singular point, the variety is isomorphic to an A 1 -hypersurface singularity.In particular, its resolution of singularities is given by the blowup of the singular point.
5.2.C n cluster algebras.We choose the exchange matrix B for type C n , n ≥ 3, as [FWZ17, §5.5, (5.32)]).The corresponding Dynkin diagram is of type C n (where n is the number of vertices);

• • • .
For this cluster algebra, the characteristic of the field makes a significant difference.First, we provide a suitable presentation of A(C n ).
Lemma 5.3.Let K be any field and n ≥ 3. The cluster algebra A(C n ) is isomorphic to where P n is the continuant polynomial defined in Section 3. In particular, the variety Proof.Since the underlying diagram Γ(B) is acyclic, the cluster algebra A(C n ) has a presentation as K[x 1 , . . . ,x n , y 1 , . . . ,y n ]/ h 1 , . . ., h n where we define h , for k ∈ {2, . . ., n − 1}.Similar as for type A n (see Lemma 4.1) one can for 2 ≤ k ≤ n stepwise express x k in terms of x 1 , y 1 , . . ., y k−1 : The only difference is in the last generator h n , which becomes After renaming the variables, we obtain the assertion.
Proposition 5.4.Let n ≥ 3 and K be a field with char(K) = 2.The variety Proof.The proof follows the steps of the proof of Proposition 3.8.We apply the Jacobian criterion to the presentation is not a square in K, the last equality cannot hold and we have shown that the claim.Assume −1 ∈ K 2 and let λ ∈ K \ {0} be such that λ 2 = −1 and P n−1 (z 1 , . . ., z n−1 ) = λ.Using Lemma 3.5, we get and thus z n+1 = 0. We prove by induction on k that h n = ∂h n ∂z n+1 = . . .
Let us discuss the case char(K) = 2, which turns out to be more complicated.
Proposition 5.5.Let K be a field of characteristic two.The singular locus of Spec(A(C n )) is isomorphic to Spec(A(A n−2 )) and is of dimension n − 2.Moreover, we have: (1) If n ≡ 0 mod 2, then Sing(A(C n )) is regular and Spec(A(C n )) is locally at the singular locus isomorphic to a cylinder over a A 1 -hypersurface singularity in A 3 K .In particular, the blowup with center Sing(A(C n )) resolves the singularities of Spec(A(C n )).
(2) If n ≡ 1 mod 2 and n > 3, then Sing(A(C n )) has an isolated singularity of type A 1 at the origin.Locally at the origin, Spec(A(C n )) is isomorphic to a hypersurface singularity of the form (where m is defined by n = 2m + 1) , where n = 2m + 1, while locally at a singular point different from the origin, Spec(A(C n )) is again isomorphic to a cylinder over the A 1 -hypersurface singularity given by V(x 2 + yz) ⊂ A 3 K .The singularity Spec(A(C n )) is resolved by three blowups; the first center is the origin, the second is the strict transform of the original singular locus, and the third center is the strict transform of an exceptional component created after the first blowup.
(3) If n = 3, then Sing(A(C n )) is isomorphic to two regular lines intersecting transversally at the origin.All other statements of (2) remain true for m = 1.
Remark 5.6.Observe that case (2) is not among the simple singularities.
The partial derivatives are This implies that we can replace the condition h n (z 1 , . . ., z n+1 ) = 0 by f n−2 (z 1 , . . ., z n−1 ) = P n−1 (z 1 , . . ., z n−1 ) + 1 = 0 when determining the singular locus, where f n−2 (z 1 , . . ., z n−1 ) is the polynomial providing a hypersurface presentation for the variety Spec(A(A n−2 )) (see Lemma 4.1 and use char(K) = 2).Since the second factor of ∂h n ∂z n = z n+1 P n−1 (z 1 , . . ., z n ) cannot vanish if f n−2 = 0 and since P n (z 1 , . . ., z n ) = z n P n−1 (z 1 , . . ., z n−1 ) + P n−2 (z 1 , . . ., z n−2 ), we obtain where we introduce u n := z n + P n−2 (z 1 , . . ., z n−2 ).Observe that Corollary 4.3(1') implies that Sing(A(C n )) is singular if and only if n ≡ 1 mod 2. First assume n ≡ 0 mod 2. Then V( f n−1 ) is regular and we may take u n−1 := f n−1 as a local variable locally at the singular locus.Furthermore, 1 + f n−1 is a unit and we may introduce the local variable w n := u n (1 + f n−1 ) + f n−1 P n−2 .Hence, h n = u 2 n−1 + z n+1 w n and claim (1) follows.Suppose that n ≡ 1 mod 2. By Corollary 4.3(2'), Sing(A(C n )) has an isolated singularity of type A 1 at the origin V(z 1 , . . ., z n−1 , u n , z n+1 ).The statement about the the type of singularity away from origin follows with the same argument as in the case n ≡ 0 mod 2. Let us study the situation at the origin.Write n − 2 = 2m − 1, for m ∈ Z + .As we have seen in the proof of Proposition 3.8, there is a coordinate transformation (z 1 , . . ., z 2m ) → (t 1 , . . ., t 2m ) such that f n−2 (t 1 , . . ., t 2m ) = ∑ 2m i=1 t 2i−1 t 2i , locally around the origin.Furthermore, we can again introduce w n above such that, locally at the origin, we obtain h n = (∑ m i=1 t 2i−1 t 2i ) 2 + w n z n+1 , as desired.Let us discuss the desingularization of the variety Spec(A(C n )).First, we blow up with center the origin.In order to simplify the presentation of the charts, we abuse notation and write z n := u n .
We fix the notation when considering explicit charts of a blowup: (We only discuss this for the blowup of the origin, but it can be adapted for any blowup with a smooth center.)Recall that the blow-up in the origin of A n+1 K = Spec(R), where R := K[z 1 , . . ., z n+1 ], is given by Bl 0 (A n+1 K ) := Proj(R[Z 1 , . . ., Z n+1 ]/ z i Z j − z j Z i | i, j ∈ {1, . . ., n + 1} ), where (Z 1 , . . ., Z n+1 ) are projective variables.In particular, Bl 0 (A n+1 K ) is covered by the open subsets D i given by Z i = 0, for i ∈ {1, . . ., n + 1}.We also say that D i is the Z i -chart.
Fix i ∈ {1, . . ., n + 1}.Since z i Z j − z j Z i = 0, for j = i, we obtain that z j = z i Z j Z i in the Z i -chart.This provides that the Z i -chart is isomorphic to Spec(K[z ′ 1 , . . ., z ′ n+1 ]), where we set z ′ i := z i and z ′ j := Z j Z i for j = i.In order to keep the notation light, we abuse it by using the same letter for the variables after the blowup as before, i.e., the transformation of the variables will be written as z j = z i z j for j = i.
Z n -chart.We have z i = z n z i for every i = n.The strict transform of h n is where we denote by f ′ n−1 (resp.P ′ n−2 ) the strict transform of f n−2 (resp.P n−2 ).The strict transform D ′ of D is empty in this chart.Hence, the only singularities which may appear have to be contained in the exceptional divisor V(z n ).On the other hand, we have ∂h The analogous argument applies for the Z n+1 -chart.
Z 1 -chart. (All other charts remaining are analogous.)We get z i = z 1 z i for every i = 1 and ) is regular and thus the same is true for ) and we may introduce the variable u 2 := f ′ n−1 locally at D ′ .Since any newly created singularities have to be contained in the exceptional divisor V(z 1 ), the element 1 + z 2 1 f ′ n−1 is invertible locally at the singular locus of V(h ′ n ).We introduce the local variable Obviously, we have Sing(V(h ).Both components are regular and V(h ′ n ) is an A 1 -singularity at every point expect their intersection.Let us define E := V(z 1 , w n , z n+1 ).Next, we blow up with center D ′ .Observe that this is a well-defined global center, which is seen in any Z i -chart with i ∈ {1, . . ., n − 1} as it is the strict transform of Sing(Spec(A(C n ))).There are no new singularities contained in the exceptional divisor of the second blowup and hence the singular locus of V(h ′′ n ) has to be the strict transform E ′′ of E (where h ′′ n is the strict transform of h ′ n after the second blowup).Locally at E ′′ , the hypersurface is given by an equation of the form x 2 − yz = 0. Therefore, after blowing up E ′′ all singularities are resolved.Again observe that E ′′ is a well-defined global center at this step of the resolution process since it is the singular locus of the strict transform after the second blowup.Case (3) is seen by explicit computation.5.3.F 4 and G 2 cluster algebras.Let us discuss the remaining cases of A(F 4 ) and A(G 2 ).
For the cluster algebra A(F 4 ), we pick the exchange matrix B = (cf. [FWZ17, Exercise 4.4.12]),whose corresponding Dynkin diagram is of type F 4 ; . Lemma 5.7.For any field K, the variety Spec(A(F 4 )) is isomorphic to a regular hypersurface in A 5 K .
Proof.The underlying graph Γ(B) is acyclic and thus A(F 4 ) is isomorphic to its lower bound cluster algebra.Similar as above we obtain the presentation Hence, Spec(A(F 4 )) is isomorphic to a hypersurface in A 5 K .Moreover, the Jacobian criterion shows that the latter is regular.
Next, let us come to A(G 2 A possible exchange matrix is B = 0 1 −3 0 (cf.[FWZ17, § 5.7]) and the corresponding Dynkin diagram is of type G 2 ; . Lemma 5.8.Let K be any field.The variety Spec(A(G 2 )) is isomorphic to a hypersurface in A 3 K .(1) If char(K) = 3, then Spec(A(G 2 )) is regular.
(2) If char(K) = 3, then Spec(A(G 2 )) has an isolated singularity of type A 2 at a closed point.
In particular, the singularities of the variety are resolved by two point blowups.
Proof.Since Γ(B) is acyclic, the cluster algebra A(G 2 ) is isomorphic to its lower bound cluster algebra by Theorem 2.12.We get the presentation Since x 1 can be expressed in term of the other variables, we find By the Jacobian criterion, this algebra is regular for char(K) = 3, and for characteristic 3 the Jacobian criterion yields Sing(A(G 2 )) ∼ = V(x + 1, y, z + 1), a closed point.Assume char(K) = 3 and set x ′ := x + 1, and z where the singular point is the origin in the new coordinates.Applying the local coordinate change which is a singularity of type A 2 , see [GK90, Definition 1.2].

STAR CLUSTER ALGEBRAS
The final section is devoted to the question how singularities of cluster algebras may look for more general quivers.We examine the cluster algebra of a star shaped quiver.Consider the star quiver S n with n vertices and the following orientation: By Lemma 2.14, all orientations on this tree are equivalent.Since the quiver is acyclic (and all vertices are mutable), the corresponding cluster algebra, denoted by A(S n ), is isomorphic to its lower bound cluster algebra by Theorem 2.12.Remark 6.1.Observe that S 2 = A 2 , S 3 = A 3 , and S 4 = D 4 .By Corollary 4.3, Spec(A(S 2 )) is regular, while Spec(A(S 3 )) is an A 1 -hypersurface singularity which is resolved by blowing up the singular locus.Moreover, by Lemma 4.5 and Proposition 4.6, the singular locus of Spec(A(S 4 )) has six irreducible components of dimension one and we obtain a desingularization of Spec(A(S 4 )) by first blowing up the intersection of these irreducible components, followed by the blowup of their strict transforms.
In particular, if suffices to restrict to the case n ≥ 5 in the following, if needed.Lemma 6.2.The cluster algebra A(S n ) has a presentation of the form In particular, the n-dimensional variety Spec(A(S n )) can be embedded into A 2n−2 K .Proof.Since A(S n ) is isomorphic to its lower bound cluster algebra, we have A(S n ) ∼ = K[x 1 , . . . ,x n , y 1 , . . . ,y n ]/I , where I is the ideal generated by the exchange relations The first generator allows us to substitute x n = x 1 y 1 − 1 and we get We can eliminate another generator via , and z 2k−1 := x k , z 2k := y k , for k ∈ {1, . . ., n − 2}, provides the assertion.
Theorem 6.3.Let n ≥ 4. Let A(S n ) be the cluster algebra arising form the star shaped quiver S n over a field K. Using the notation of Lemma 6.2, we have In particular, the singular locus consists of ( n−1 2 )2 n−3 = (n − 1)(n − 2)2 n−4 irreducible components, where each of them is regular and of dimension n − 3. Furthermore, locally at a generic point of such a component, Spec(A(S n )) is isomorphic to an A 1 -hypersurface singularity.On the other hand, locally at the closed point determined by the intersection of all irreducible components, Spec(A(S n )) is isomorphic to a toric variety, defined by the binomial ideal We have that the intersection of all irreducible components is the origin of A 2n−2 K and after blowing up the latter, we obtain in each chart a singularity which is of the same kind as the one of Spec(A(S n−1 )) ⊂ A 2n−4 K .In other words, the singularities of Spec(A(S n )) are resolved by first separating the irreducible components of its singular locus and then blowing up their strict transforms.
Proof.Notice that the inclusion V(x 1 , x 2 , y 1 , y 2 ) ⊆ Sing(V(h)) is obvious.It remains to show the equality.We have If x 1 x 2 = 0, then the vanishing of the derivatives with respect to y 1 and y 2 implies y 1 = y 2 = 0. Since ρ ∈ y 1 ,we obtain x 1 = x 2 = 0 from the other two derivatives.Thus, all components of the singular locus with x 1 x 2 = 0 are contained in V(x 1 , x 2 , y 1 , y 2 ).
Finally, locally at a point of V(x 1 , x 2 , y 1 , y 2 ), the element 1 + ρ is a unit since ρ ∈ y 1 and hence V(h) is locally isomorphic to the hypersurface singularity V(x 1 x 2 + y 1 y 2 ) ⊂ A 4+a K .This implies the remaining statements.
Proof of Theorem 6.3.By Lemma 6.2, we have A(S n ) ∼ = K[z 1 , . . . ,z 2n−2 ]/I, where I := h 1 , h 2 , . . ., h n−2 and Observe that each generator is of the form as h in Lemma 6.4.Furthermore, for every ℓ ∈ {2, . . ., n − 2} fixed, we may interchange the role of z 1 z 2 and z 2ℓ−1 z 2ℓ using the relation h ℓ = 0. Hence, Lemma 6.4 implies that It remains to prove that this is an equality.Suppose there exists C ⊂ Sing(Spec(A(S n ))) with C D. We deduce a contradiction via an induction on the number of generators h 1 , . . . ,h n−2 .If n − 2 = 2, then n = 4, i.e., S 4 = D 4 (by Remark 6.1).By Lemma 4.5, the singular locus of Spec(A(D 4 )) consists of the six lines determined by D k,ℓ .Suppose n − 2 > 2. The induction hypothesis implies that (12) is an equality for any K[z 1 , . . . ,z 2m−2 , u 1 , . . . ,u a ]/ g 1 , . . ., g m−2 with m − 2 < n − 2 and where ρ ∈ z 2m−3 ⊂ K[z 1 , . . ., z 2m−2 , u 1 , . . ., u a ] and where we have z 1 z 2 = 0 or ρ is of the form as in (ii) of Lemma 6.4.We blow up the origin, which is the intersection of all irreducible components in D. Since C D, the strict transform C ′ of C must appear in one of the charts.Since (h 1 , . . ., h n−2 ) is a Gr öbner basis of the ideal I, the strict transform of I is generated by their strict transforms h ′ 1 , . . ., h ′ n−2 .We go through the different charts of the blowup.Z 2k−1 -chart, k ∈ {2, . . ., n − 2}.Without loss of generality, we assume k = 2.We have z i = z 3 z i for every i = 3. (By abuse of notation, we denote the variables after the blowup by the same letter.)Hence, we get , for k ∈ {3, . . ., n − 2} .Since z 4 appears only in h ′ 2 , we can eliminate it and forget the generator h ′ 2 without changing the other h ′ k , k ≥ 3. Notice that h ′ 1 is of the form as h in Lemma 6.4 and thus, we can apply the induction hypothesis for K[z 1 , z 2 , z 5 , z 6 , . . . ,z 2n−2 , u 1 ]/ h ′ 1 , h ′ 3 , . . ., h ′ n−2 , where u 1 := z 3 .Therefore, the corresponding singular locus is equal to the strict transform of D. In particular, C ′ has to be empty in this chart.Z 2k -chart, k ∈ {2, . . ., n − 2}.Without loss of generality, we choose k = 2.We get z i = z 4 z i for every i = 3.The strict transforms of h 1 , . . ., h n−2 are , for k ∈ {3, . . ., n − 2} .We eliminate h ′ 2 by replacing z 3 = z 1 z 2 .Observe that this changes h ′ 1 as z 3 appears in the product.Since we have already treated the Z 3 -chart, it is sufficient to consider only those points of the Z 4 -chart, which are not contained in the Z 3 -chart.Therefore, the singular points, which we have to determine here, fulfill the extra condition z 3 = 0. (Using the precise distinction of the variables before and after the blowup as discussed in the proof of Proposition 5.5, we have z 3 = z 4 z ′ 3 , where z ′ 3 := Z 3 Z 4 , and hence, we avoid the chart Z 3 = 0 by setting Z 3 = 0, which leads to z ′ 3 = 0).The relation z 3 = z 1 z 2 implies that we must have z 1 z 2 = 0, which is Lemma 6.4(1).Therefore, we can apply the induction hypothesis and obtain C ′ = ∅ in the Z 4 -chart.
Z 1 -chart.Here, z i = z 1 z i for every i = 1 and we obtain Combining the arguments of the Z 1 -and the Z 4 -charts shows that C ′ = ∅ in the Z 2 -chart.Z 2n−3 -chart.We have z i = z 2n−3 z i for every i = 2n − 3. The strict transforms of h 1 , . . ., h n−2 are , for k ∈ {2, . . ., n − 2} .Since we already handled the Z i -charts for i ∈ {1, . . ., 2n − 4}, we only have to take those points into account which are not contained in these charts.Therefore, analogous to the Z 4 -chart, it suffices if we determine only those singular points, for which we have additionally z 1 = . . .= z 2n−4 = 0.
We may rewrite h ′ 1 as and, by the previous, 1 − z 1 z 2 z 2 2n−3 is a unit.This implies that we may eliminate z 2n−2 and forget h ′ 1 .The resulting ideal is binomial.In particular, we can apply the induction hypothesis with u 1 := z 2n−3 (as ρ = 0 is a possible choice) and we get C ′ = ∅ in the present chart.
The analogous arguments can be applied for the remaining Z 2n−2 -chart.This concludes the proof that ( 12) is an equality.
The results on the desingularization, on the type of the singularity at a generic point of an irreducible component of Sing(Spec(A(S n ))), and on the local description of Spec(A(S n )) at the origin follow: In each of the charts above, we blow up the intersection of the irreducible components of strict transform D and continue this process until we eventually reach the case, where there is only one irreducible component left.Since we eliminate after every blowup one generator, the strict transform of the variety is isomorphic to a hypersurface as in Lemma 6.4 in every chart.In particular, we get a hypersurface singularity of type A 1 and all singularities are resolved after the next blowup.Finally, after localizing at z 1 , . . ., z 2n−2 , the factor in parentheses of h 1 becomes a unit, which we abbreviate as ǫ.Therefore, we may introduce x 2n−2 := ǫ −1 z 2n−2 and the ideal generated by h k , for k ∈ {1, . . ., n − 2}, is binomial.
• The partition in 2-blocks is determined by the monomials appearing in h 1 , . . ., h n−1 , namely, the blocks are X i := {z 2i−1 , z 2i }, for i ∈ {1, . . ., n}. • A subset with n + 2 elements determines a regular (n − 2)-dimensional subvariety C ⊂ A 2n K .The condition that every X i has to be intersected ensures that we have C ⊂ Spec(A(S n+1 )).Furthermore, the number of elements n + 2 provides that there is a generator which is singular at C and that C ⊂ Sing(Spec(A(S n+1 )) is an irreducible component.
then we have: (a) For every i ∈ {0, . . ., 4}, Spec(A(D n )) is locally at a singular point different from the origin isomorphic to a cylinder over a hypersurface singularity of type A 1 .(b) Y 0 is isomorphic to an (n − 3)-dimensional A 1 -hypersurface singularity; (c) The singularities of Spec(A(D n )) are resolved by first blowing up the origin and then choosing the strict transform of 4 i=0 Y i as the next center.