This is the accepted version of the journal article: Heusener, Michael; Porti, Joan. «THE SCHEME OF CHARACTERS IN SL2». Transactions of the American Mathematical Society, Vol. 376, Núm. 9 (May 2023), p. 6283-6313. DOI 10.1090/tran/8910 This version is available at https://ddd.uab.cat/record/287846 under the terms of the license THE SCHEME OF CHARACTERS IN SL2 MICHAEL HEUSENER AND JOAN PORTI Abstract. The aim of this article is to study the SL2 (C)–character scheme of a finitely generated group. Given a presentation of a finitely generated group Γ, we give equations defining the coordinate ring of the scheme of SL2 (C)–characters of Γ (finitely many equations when Γ is finitely presented). We also study the scheme of abelian and non- simple representations and characters. Finally we apply our results to study the SL2 (C)– character scheme of the Borromean rings. 1. Introduction For a finitely generated group Γ, we fix a presentation (1) Γ = ⟨γ1 , . . . , γn ∣ rl , l ∈ L⟩ with L possible infinite. Let π ∶ Fn ↠ Γ denote the natural surjection from the free group Fn on n generators. Hence the kernel of π is the subgroup of Fn normally generated by the relations rl , l ∈ L. Our first goal is to describe the scheme of characters of Γ in SL2 (C) from the scheme of Fn and the presentation (1). The scheme of representations and characters in SL2 (C) are denoted respectively by R(Γ, SL2 (C)) and X(Γ, SL2 (C)). The corresponding algebras of functions are the uni- versal algebra of SL2 (C)-representations A(Γ) = C[R(Γ, SL2 (C))] and the universal algebra of SL2 (C)-characters B(Γ) = C[X(Γ, SL2 (C))] ≅ A(Γ)SL2 (C) . We give the definitions in Section 2. The algebras A(Γ) and B(Γ) may be non-reduced, i.e. they may contain non-zero nilpotent elements (see the examples in Section 8, for instance, or the more general result of [18] that provides the existence of arbitrary sin- gularities). If instead of the scheme we consider the underlying variety, then the algebra of functions of the variety is the quotient B(Γ)red ∶= B(Γ)/N (Γ), where N (Γ) is the nilradical of B(Γ). For γ ∈ Fn , let tγ ∈ B(Fn ) denote the the evaluation function of characters at γ. Theorem 1. Let Γ = ⟨γ1 , . . . , γn ∣ rl , l ∈ L⟩ be a finitely generated group. Then B(Γ) ≅ B(Fn )/(trl − 2, tγi rl − tγi , tγj γk rl − tγj γk ∣ l ∈ L, 1 ≤ i ≤ n, 1 ≤ j < k ≤ n) Date: January 20, 2023. 1991 Mathematics Subject Classification. Primary 57K31; Secondary 20C99, 57M50. Key words and phrases. Representation variety, character variety. Both authors partially supported by FEDER-AEI (grant number PID2021-125625NB-100 and Marı́a de Maeztu Program CEX2020-001084-M) and by the Laboratoire de Mathématiques Blaise Pascal of the Université Clermont Auvergne. 1 We compare this result to Theorem 3.2 in [14] by González-Acuña and Montesinos- Amilibia, who give a presentation for the variety instead of the scheme. More precisely, they deal with the reduced algebra and prove that the reduction B(Γ)red of B(Γ) is given by B(Γ)red ≅ B(Fn )/ rad (trl − 2, tγi rl − tγi ∣ l ∈ L, 1 ≤ i ≤ n) . Our motivation for Theorem 1 is that B(Γ) may be non-reduced and that, even when B(Γ) is reduced, the radical in the ideal of González-Acuña and Montesinos-Amilibia is needed, see Example 28). In order to work with shorter words and to simplify computations, we may write each relation as a product of two words (2) rl = u−1 l vl , ∀l ∈ L. Addendum to Theorem 1. Let Γ = ⟨γ1 , . . . , γn ∣ ul = vl , l ∈ L⟩ be a finitely generated group. Then B(Γ) ≅ B(Fn )/(tvl − tul , tγi vl − tγi ul , tγj γk ul − tγj γk vl ∣ l ∈ L, 1 ≤ i ≤ n, 1 ≤ j < k ≤ n). The first step of the proof uses the isomorphism between B(Γ) and the skein algebra of Γ, proved by Przytycki and Sikora in [33], but that follows also from a result of Procesi [31]. The proof uses trace identities of Vogt and Magnus to find the explicit presentation. In this paper we also define the scheme of non-simple representations and characters (some times called reducible representations, but we already use the word reduced with another meaning). In Proposition 37 we prove that the scheme of non-simple characters is isomorphic to the scheme of characters of its abelianization. We also describe the scheme of characters for abelian groups in Theorem 38. In this article we are mainly interested in the field of complex numbers. Neverthe- less, all results (in particular Theorem 1) are valid over any algebraically closed field of characteristic 0. Remark 2. Recently G. Miura and S. Suzuki have given a similar description of B(Γ) when Γ has three generators [25]. We discuss the differences between both results in Remark 27. The paper is organized as follows. Preliminaries on affine schemes are provided in Section 2, which can be skipped by readers familiar with schemes and used as a reference. Then in Section 3 we discuss the scheme of representations, and the local properties of these are given in Section 4. Section 5 is devoted to the proof of Theorem 1. The schemes of non-simple and of abelian characters are studied respectively in Sections 6 and 7. In Section 8 we provide examples of non-reduced character schemes of three-manifolds, and in the final section we compute the scheme of characters of the Borromean rings exterior. 2. Preliminaries on affine algebraic schemes In this section we review the basic tools in affine algebraic schemes required. For more details see for instance [9, 23, 27]. 2.1. Affine algebraic schemes. Let A be a finitely generated, commutative C-algebra. Recall that A is the quotient of a polynomial algebra by an ideal, i.e. there exists n ∈ N and an ideal I ⊂ C[x1 , . . . , xn ] such that A ≅ C[x1 , . . . , xn ]/I. It follows from Zariski’s lemma that there is a one-to-one correspondence between the maximal ideals m ⊂ A and algebra morphisms A → C. In what follows we let Spm(A) denote the maximal spectrum of A i.e. Spm(A) = {m ⊂ A ∣ m is a maximal ideal.} 2 We can think of elements of A as complex-valued functions on Spm(A) in the following way: if f ∈ A and m ∈ Spm(A), then A/m ≅ C and f (m) ∶= f mod m. Therefore, we have f (m) = 0 if and only if f ∈ m ⊂ A. Notice that we might have that f (m) = 0 for all m ∈ Spm(A), but f ≠ 0; if his happens then f ∈ A is nilpotent. Also each m ∈ Spm(A) determines, by evaluation, an algebra morphism φm ∶ A → C given by φm (f ) = f (m) for f ∈ A. The two important properties are the following: ● The elements of A separate points. Indeed, if m, m′ ∈ Spm(A) then m ≠ m′ implies that there exists f ∈ m and f ∈/ m′ , and therefore 0 = f (m) ≠ f (m′ ) ≠ 0. ● Every algebra morphism φ ∶ A → C is the evaluation at a unique point i.e. there exists a unique point m = Ker(φ) ∈ Spm(A) such that for all f ∈ A we have φ(f ) = f (m). For any subset M of A we define V (M ) ∶= {m ∈ Spm(A) ∣ m ⊃ M } = {m ∈ Spm(A) ∣ ∀f ∈ M, f (m) = 0} . It is clear that V (M ) = V (I) if I is the ideal generated by M . We will endow Spm(A) with the Zariski topology, where the closed sets are of the form V (I) for I and ideal of A. Notice that V (0) = Spm(A) and V (1) = ∅. For more details see Appendix A in [23]. We will call a pair X = (Spm(A), A) an affine algebraic scheme. Given an affine alge- braic scheme X, we define the coordinate algebra of X as C[X] ∶= A, and the underlying space ∣X∣ = Spm(A). In what follows it will be convenient to write x ∈ ∣X∣ for an element in ∣X∣ = Spm(A). In this case the corresponding maximal ideal in A will be denoted by mx . Remark 3. It is worth noticing the word “algebraic” in the definition of affine algebraic scheme. Affine schemes in general are constructed from the prime spectrum of a com- mutative ring with unity, though here we work with the maximal spectrum of a finitely generated commutative C-algebra. See [27] or [9]. Example 4. The n-dimensional affine space has the structure on an affine algebraic scheme An ∶= (Cn , C[x1 , . . . , xn ]). A point p = (p1 , . . . , pn ) ∈ Cn corresponds to the max- imal ideal mp = (x1 − p1 , . . . , xn − pn ), and each maximal ideal of C[x1 , . . . , xn ] is of this form. The value of f ∈ C[x1 , . . . , xn ] at mp = (x1 − p1 , . . . , xn − pn ) is simply f (p). Example 5. We consider the dual numbers A2 ∶= C[]/(2 ). We have that Spm(A2 ) = {∗} has only one point, the maximal ideal ∗ ∶= (). The value of f = a + b  ∈ A2 at the point ∗ is f (∗) = a. Hence for  ∈ A2 we have (∗) = 0, but 0 ≠  and 2 = 0 in A2 . The scheme ({∗}, A2 ) is called the double point, and similarly An ∶= C[]/(n ) gives us a point of multiplicity n. A morphism between two affine algebraic schemes X = (Spm(A), A) and Y = (Spm(B), B) is a pair (α, α∗ ) such that α ∶ Spm(A) → Spm(B) is a map, α∗ ∶ B → A is an algebra ho- momorphism satisfying (3) g(α(m)) = α∗ (g)(m) for all m ∈ Spm(A) and for all g ∈ B i.e. for all g ∈ B the two maps g ○ α and α∗ (g) are equal as maps on ∣X∣ = Spm(A). Notice that α is continuous in the Zariski-topology, for if J ⊂ B is an ideal we have α−1 (V (J)) = {m ∈ Spm(A) ∣ α(m) ⊃ J} = {m ∈ Spm(A) ∣ ∀g ∈ J, α∗ (g)(m) = g(α(m)) = 0} = V (α∗ (J)) . 3 In what follows we will write α ∶ X → Y for a morphism between the schemes X and Y . It is understood that α ∶ ∣X∣ → ∣Y ∣, and α∗ ∶ C[Y ] → C[X] satisfy equation (3) i.e. for all g ∈ C[Y ] and for all x ∈ ∣X∣ we have α∗ (g)(x) = g(α(x)). Remark 6. ● An algebra homomorphism α∗ ∶ B → A determines a unique map α ∶ Spm(A) → Spm(B) in the following way: for m ∈ Spm(A) we consider the map φm ∶ B → C given by φm (g) = α∗ (g)(m). This map is an algebra homomorphism and we can put α(m) ∶= Ker(φm ), and we obtain g(α(m)) = α∗ (g)(m) for all m ∈ Spm(A). Notice that α(m) = (α∗ )−1 (m) since Ker(φm ) = (α∗ )−1 (m). ● The map α ∶ Spm(A) → Spm(B) does not determine α∗ uniquely. This only happens if A has no nilpotent elements. A scheme morphism α ∶ X → Y is called a closed immersion if the underlying continuous map α ∶ ∣X∣ → ∣Y ∣ is a homeomorphism between ∣X∣ and a closed subset of ∣Y ∣, and the algebra homomorphism α∗ ∶ C[Y ] → C[X] is surjective. 2.2. Tangent space. Let X be an affine algebraic scheme. For every x ∈ ∣X∣, the quotient mx /m2x is a finite dimensional C-vector space. By definition its dual space is the tangent space of X at x which will be denoted by Tx (X) ∶= (mx /m2x ) . ∗ Lemma 7. Let m ∈ Spm(A). Then there is a one-to-one correspondence between linear forms ` ∈ (m/m2 )∗ and scheme morphisms (α, α∗ ) ∶ (Spm(A2 ), A2 ) → (Spm(A), A) such that α(∗) = m. Proof. Recall that Spm(A2 ) = {∗} has only one point which corresponds to the maximal ideal (). We consider the map α ∶ Spm(A2 ) → Spm(A) given by α(∗) = m. In order to get a morphism of schemes, we are looking for an algebra homomorphism α∗ ∶ A → A2 which satisfies equation (3). That is α∗ ∶ A → A2 given by α∗ (f ) = α0∗ (f ) +  α1∗ (f ) where αi∗ ∶ A → C are C-linear maps, such that α∗ (f )(∗) = α0∗ (f ) = f (α(∗)) = f (m) , and (4) ∀ f1 , f2 ∈ A, α1∗ (f1 f2 ) = f1 (m)α1∗ (f2 ) + α1∗ (f1 )f2 (m) . Notice that equation (4) implies that α1∗ (1) = 0, and hence α1∗ ∣C ≡ 0. Let ` ∈ (m/m2 )∗ be given. We define a linear map α1∗ ∶ A → C satisfying equation (4) by putting α1∗ (f ) ∶= `(f − f (m)). Indeed, α1∗ (f1 f2 ) = `(f1 f2 − f1 (m)f2 (m)) = `((f1 − f1 (m))(f2 − f2 (m))) + f2 (m)`(f1 − f1 (m)) + f1 (m)`(f2 − f2 (m)) , and equation (4) follows since (f1 − f1 (m))(f2 − f2 (m)) ∈ m2 and `∣m2 ≡ 0. Let (α, α∗ ) ∶ (Spm(A2 ), A2 ) → (Spm(A), A) be a scheme morphisms such that α(∗) = m i.e. for all f ∈ A α∗ (f ) = f (m) +  α1∗ (f ) where α1∗ ∶ A → C is linear and satisfies equation (4). The restriction α1∗ ∣m is linear, and by equation (4) α1∗ vanishes on m2 . Hence α1∗ defines a linear form on m/m2 .  4 2.3. Closed subschemes. Let I ⊂ A be an ideal, and we let π ∶ A → A/I denote the projection. We obtain the affine algebraic scheme (Spm(A/I), A/I). Notice that there is a one-to-one correspondence between the ideals in A/I and the ideals in A which contain I. Hence, there is a natural scheme morphism (α, α∗ ) ∶ (Spm(A/I), A/I) → (Spm(A), A) given by α(m) = π −1 (m) and α∗ = π. We have Im(α) = V (I), and α ∶ Spm(A/I) → V (I) a homeomorphism since it is closed; that is for every ideal J¯ ⊂ A/I we have ¯ = V (I) ∩ V (π −1 (J)) α(V (J)) ¯ . We endow the closed set V (I) with the ring A/I, and we call the pair (V (I), A/I) a closed subscheme of (Spm(A), A). In contrast to classical algebraic geometry, the closed subschemes of (Spm(A), A) and the ideals of A are in one-to-one correspondence. It follows that every affine algebraic scheme is isomorphic to a closed subscheme of some affine space An . For two closed subschemes (V (Ik ), A/Ik ), k = 1, 2, the subscheme (V (I1 ∩ I2 ), A/(I1 ∩ I2 )) is called their union and (V (I1 + I2 ), A/(I1 + I2 )) is called their intersection. Since and these are actually set-theoretic unions and intersections of the underlying spaces. See Section I.2.1 in [9]. We call an affine algebraic scheme X reduced if C[X] is reduced. If N ⊂ C[X] denotes the nilradical of C[X], then the algebra C[X]red ∶= C[X]/N is reduced. Moreover, N is the intersection of all maximal ideals of A. Hence V (N ) = V (0) = ∣X∣, and it follows that Xred ∶= (∣X∣, C[X]red ) is a reduced, closed subscheme of X. 2.4. Local to global properties. For a finitely generated algebra A we let N (A) denote its nilradical. For an ideal I ⊂ A we let Ann(I) ∶= {a ∈ A ∣ aI = 0} denote the annihilator of I. The point of Spm(A) represented by a maximal ideal m is called reduced if the local ring Am is reduced. The non-reduced points of Spm(A) form a Zariski closed subset (which might be empty). This follows from the the following lemma. Lemma 8. Let A be a finitely generated, commutative C-algebra A, I ⊂ A an ideal, and m ⊂ A a maximal ideal. Then (1) N (Am ) = N (A)m ; (2) V ( Ann(I)) = {m ∈ Spm(A) ∣ Im ≠ 0}; (3) The set of non-reduced points in Spm(A) is Zariski-closed. n n Proof. (1) If as ∈ N (Am ) then there exists n ∈ N such that ( as ) = asn = 01 . Hence there exists t ∈ A ∖ m such that an t = 0 in A. Hence (ta)n = 0 in A, and as = at st ∈ N (A)m . The other inclusion is obvious. (2) We have m ∈/ V ( Ann(I)) ⇔ m ⊃/ Ann(I) ⇔ ∃ s ∈ A ∖ m, ∀ b ∈ I s b = 0 and Im ≠ 0 ⇔ ∀ b ∈ I, ∃s ∈ A ∖ m s b = 0 Clearly the first statement implies the second. But also the second implies the first since A is noetherian. For if I = (b1 , . . . , bk ) and si ∈ A ∖ m such that si bi = 0 then for the product s = s1 ⋯sk we have sb = 0 for all b ∈ I. (3) We have N (Am ) = N (A)m ≠ 0 if and only m ∈ V ( Ann(N (A))). Hence the non- reduced points m ∈ Spm(A) form the Zariski-closed set V ( Ann(N (A))).  The first local to global property is the following (see [24, Corollary 5.19]): 5 Lemma 9. Let A be a ring. Then A is reduced if and only if for every maximal ideal m ⊂ A the local ring Am is reduced. Let A be a subring of B. An element b ∈ B is called integral over A if it is a root of a monic polynomial with coefficients in A. An integral domain A is called integrally closed or normal if it is integrally closed in its field of fractions F i.e. α ∈ F , and α integral over A implies α ∈ A. There is an other local to global property (see [24, Prop. 6.16]: Lemma 10. Let A be an integral domain. Then A is normal if and only if for every maximal ideal m ⊂ A the localization Am is normal. 2.4.1. Tangent cone. Let B be a noetherian, local ring with maximal ideal m. In this situation we have by Krull’s intersection theorem that (5) ⋂ mn = {0}, n∈N and we will define the associated associated graded ring of B as gr(B) = B/m ⊕ m/m2 ⊕ m2 /m3 ⊕ ⋯ . The multiplication of two homogeneous elements ā ∈ mn /mn+1 and b̄ ∈ mk /mk+1 is defined as follows: ā ⋅ b̄ ∶= a b mod mn+k+1 where a and b are elements of B representing ā and b̄ respectively. For any finitely generated, commutative C-algebra A the tangent cone of the schema X = (Spm(A), A) at x ∈ X is the schema ( Spm(gr(Am )), gr(Am )) where m = mx . A local noetherian ring B with maximal ideal m inherits some properties of its asso- ciated graded algebra gr(B). For example, if gr(B) is reduced, normal or regular, then so is the local ring B. More precisely, it follows from (5) that for a ∈ B, a ≠ 0, there is a minimal n such that a ∈ mn , a ∈/ mn+1 , and we define the initial term in(a) ∈ gr(B) of a as in(a) ∶= a mod mn+1 ∈ mn /mn+1 ⊂ gr(B) . It is easy to see that in(ab) = in(a) ⋅ in(b). This implies Proposition 11. Let B be a local noetherian ring with maximal ideal m. (1) If the associated graded algebra gr(B) is reduced, then B is reduced. (2) If the associated graded algebra gr(B) is normal, then B is normal. See [22, Section 17] and [8, Chapter 5] for more details. 3. The scheme of representations and characters The general reference for this section is Lubotzky’s and Magid’s book [19]. In what follows we let Γ denote a finitely generated group with presentation (1). 3.1. The universal algebra A(Γ). The C-algebra A(Γ) comes with a representation ρu ∶ Γ → SL2 (A(Γ)), and the pair (A(Γ), ρu ) has the following universal property: for every commutative C-algebra A and every representation ρ ∶ Γ → SL2 (A) there exists a unique algebra morphism f ∶ A(Γ) → A such that ρ = f∗ ○ ρu where f∗ ∶ SL2 (A(Γ)) → SL2 (A) is the induced map. 6 The algebra A(Γ) can be constructed from the presentation of Γ in (1), see [19, Prop. 1.2]. In more detail, the algebra A(Γ) is a quotient of the finitely generated polynomial algebra, and the construction goes at follows: put (k) (k) x11 x12 X (k) = ( (k) (k) ) for 1 ≤ k ≤ n. x21 x22 and consider the polynomial ring on 4k variables C[xij ], where 1 ≤ i, j ≤ 2 and 1 ≤ k ≤ n. (k) Then A(Γ) is the quotient of C[xij ] by the ideal J generated by (k) det(X (k) ) − 1, 1 ≤ k ≤ n, and (ml )ij − δij , 1 ≤ i, j ≤ 2, l ∈ L where ml = rl (X (1) , . . . , X (n) ) is a matrix in GL(C[xij ]), (ml )ij are the entries of ml , (k) and δij is the Kronecker delta. 3.2. The representation scheme and the representation variety. For a finitely generated group Γ the SL2 (C)-scheme of representations R(Γ, SL2 (C)) is defined as pair R(Γ, SL2 (C)) ∶= ( Spm(A(Γ)), A(Γ)) where A(Γ) is the universal algebra of Γ (see Section 3.1). A single representation ρ ∶ Γ → SL2 (C) corresponds to C-algebra morphism A(Γ) → C i.e. to a maximal ideal mρ ⊂ A(Γ), and each maximal ideal m ⊂ A(Γ) determines a representation ρm ∶ Γ → SL2 (C). Hence maximal ideals in A(Γ) correspond exactly to representations Γ → SL2 (C), and we will use this identification in what follows. Regular functions on R(Γ, SL2 (C)) are exactly the elements of A(Γ). More precisely, if f ∈ A(Γ) and ρ ∈ R(Γ, SL2 (C)), then f (ρ) = f mod mρ ∈ A(Γ)/mρ ≅ C . If N is the nil-radical of A(Γ), then the reduced algebra A(Γ)red = A(Γ)/N is iso- morphic to the coordinate ring of the representation variety. This justifies the notation R(Γ, SL2 (C))red for the representation variety. The surjection A(Γ) ↠ A(Γ)red induces a closed immersion R(Γ, SL2 (C))red → R(Γ, SL2 (C)), the underlying continuous map is a homeomorphism in the Zariski topology. In particular the scheme R(Γ, SL2 (C)) and the variety R(Γ, SL2 (C))red have the same dimension. The scheme R(Γ, SL2 (C)) might support more regular functions than R(Γ, SL2 (C))red . More precisely, different regular functions on the scheme can have the same values on maximal ideals. If the nil-radical of A(Γ) is trivial then we call the algebra A(Γ) and the representation scheme R(Γ, SL2 (C)) reduced. 3.3. The universal algebra B(Γ), the character scheme, and trace functions. The group SL2 (C) acts on the 2 × 2 matrices by conjugation. This induces an action of SL2 (C) on the algebra A(Γ), and the algebra B(Γ) = A(Γ)SL2 (C) ⊂ A(Γ) of invariants is called the universal algebra of SL2 (C)-characters. The algebra B(Γ) is also a quotient of a finitely generated polynomial algebra, and the affine GIT quotient is naturally defined as the scheme of characters X(Γ, SL2 (C)) ∶= (Spm(B(Γ)), B(Γ)) , see [19]. A single character χ ∶ Γ → C corresponds to C-algebra morphism B(Γ) → C i.e. to a maximal ideal in B(Γ). Also the algebra B(Γ) might be non-reduced. Remark 12. Notice that both R(Γ, SL2 (C)) and X(Γ, SL2 (C)) can be reducible. 7 Given an element γ ∈ Γ there exists a word wγ (γ1 , , . . . , γn ) ∈ Fn which represents γ. The matrix wγ (X (1) , . . . , X (n) ) ∈ GL2 (C[xij ]), and (k) tγ ∶= tr (wγ (X (1) , . . . , X (n) )) represents an element in B(Γ). We call tγ the trace function associated to γ ∈ Γ. The first fundamental theorem of invariant theory asserts that the trace functions tγ , γ ∈ Γ, generate the algebra B(Γ) (see [19] for instance). Besides finding an explicit finite set of generators, we aim to describe the relations satisfied by the tγ . For the moment we just state some basic relations that follow from properties of the trace. Namely for all a, b ∈ Γ and the neutral element e ∈ Γ we have: (6) te = 2, ta = ta−1 , tab = tba , and tab + tab−1 = ta tb . The last equality follows from the Cayley–Hamilton theorem. 3.4. Free groups. The general reference for this section is Goldman’s Handbook article [12] and the references therein. Let Fn be a free group of rank n. It is a classical result that the algebra B(Fn ) is reduced. By works of Cartier [6], the coordinate ring C[G] of an algebraic group G in characteristic zero is reduced (see Section 3.h of [23] for a modern approach). But for SL2 (C) a more elementary argument is sufficient. Theorem 13. The algebra B(Fn ) is reduced. Proof. The polynomial x1 x4 − x2 x3 − 1 ∈ C[x1 , x2 , x3 , x4 ] is irreducible, and an irreducible element in a factorial ring is prime. Therefore the coordinate algebra C[SL2 (C)] = C[x1 , x2 , x3 , x4 ]/(x1 x4 − x2 x3 − 1) is a domain, and hence reduced. Moreover, we obtain that n A(Fn ) ≅ ⊗ C[SL2 (C)] k=1 is reduced [3, V, §15, Theorem 3]. It follows that B(Fn ) = A(Fn )SL2 (C) ⊂ A(Fn ) is also reduced.  An explicit presentation of B(Fn ) is given by Ashley, Burelle, and Lawton in [1]. For a free group Fn = ⟨γ1 , . . . , γn ∣ ∅⟩ of rank n the algebra B(Fn ) is generated by the following n(n2 + 5)/6 elements tγi , 1 ≤ i ≤ n, tγi γj , 1 ≤ i < j ≤ n, and tγi γj γk , 1 ≤ i < j < k ≤ n (see [14]). In the following examples we give the explicit presentation B(Fn ) in the rank two and three. Example 14. The rank two case goes back to Fricke and Klein; they proved that B(F2 ) is isomorphic to free polynomial ring C[tγ1 , tγ2 , tγ1 γ2 ] (see Section 2.2 in [12]). Example 15. The case of rank three is not isomorphic to a polynomial algebra. More precisely, for a free group on three generators F3 = ⟨a, b, c ∣ ∅⟩, the coordinate ring of X(F3 , SL2 (C)) has been computed for instance in [14, 20]: (7) B(F3 ) = C[ta , tb , tc , tab , tac , tbc , tabc ]/(F ) where (8) F = F (ta , tb , tc , tab , tac , tbc , tabc ) = t2abc − p tabc + q 8 with p, q ∈ C[ta , tb , tc , tab , tac , tbc ] given by (9) p = ta tbc + tb tac + tc tab − ta tb tc and (10) q = t2a + t2b + t2c + t2ab + t2ac + t2bc + tab tac tbc − ta tb tab − ta tc tac − tb tc tbc − 4 . The trace functions tabc , tacb are solutions of the quadratic equation over C[ta , tb , tc , tab , tac , tbc ] (11) X2 − p X + q = 0 . We have (12) tabc + tacb = p, tabc tacb = q and (tabc − tacb )2 = ∆ where (13) ∆ = p2 − 4q . (see Section 5.1 in [12]). 4. Local properties In this section we keep on reviewing properties of the scheme of representations and the scheme of characters, now focusing on local properties. For instance, in order to check that the scheme is reduced, it is sufficient to prove that it is locally reduced. 4.1. Zariski tangent space. For a representation ρ ∶ Γ → SL2 (C), Z 1 (Γ, Ad ρ) denotes the space of 1-cocycles or crossed morphisms twisted by Ad ρ, namely linear maps d ∶ Γ → sl2 (C) that satisfy d(γ1 γ2 ) = d(γ1 ) + Adρ(γ1 ) d(γ2 ), ∀γ1 , γ2 ∈ Γ. Weil’s construction [40] gives the following theorem (see also [19]): Theorem 16. The Zariski tangent space to the scheme R(Γ, SL2 (C)) at ρ is naturally isomorphic to Z 1 (Γ, Ad ρ). Proof. By Lemma 7, we have that a tangent vector of R(Γ, SL2 (C)) at ρ corresponds to a scheme morphism (α, α∗ ) ∶ ({∗}, A2 ) → ( Spm (A(Γ)), A(Γ)) such that α(∗) = ρ, and α∗ ∶ A(Γ) → A2 is an algebra homomorphism. In turn, this corresponds to a representation ρα ∶ Γ → SL2 (A2 ) such that for all γ ∈ Γ ρα (γ) = (id +  d(γ))ρ(γ) such that d ∶ Γ → sl2 (C). Notice that SL2 (A2 ) = {(id + X)A ∣ A ∈ SL2 (C) and tr(X) = 0}. Now, it is easy to check that ρα is a homomorphism if and only if d ∶ Γ → sl2 (C) is a cocycle.  The cohomology of Γ with coefficients in the Γ-module sl2 (C) twisted by Ad ρ is iso- morphic to H 1 (Γ, sl2 (C)) ≅ Z 1 (Γ, Ad ρ)/B 1 (Γ, Ad ρ), where B 1 (Γ, Ad ρ) denotes the subspace of inner crossed morphisms, namely crossed morphisms d ∶ Γ → sl2 (C) for which there exists a ∈ sl2 (C) so that d(γ) = a − Adρ(γ) (a), ∀γ ∈ Γ. 9 Theorem 17. If ρ is simple, then the Zariski tangent space to X(Γ, SL2 (C)) at χρ is naturally isomorphic to H 1 (Γ, Ad ρ). This is [19, Thm 2.13]. Recall from the previous section that the variety of representations R(Γ, SL2 (C))red is the union of affine varieties. We denote by dimρ R(Γ, SL2 (C))red the maximal dimension of all components of R(Γ, SL2 (C))red containing ρ. Lemma 18. For any representation ρ ∈ R(Γ, SL2 (C)), we have dimρ R(Γ, SL2 (C))red ≤ dim Z 1 (Γ; Ad ρ), with equality if, and only if, ρ is reduced and ρ is a smooth point of the representation variety. Proof. The proof is an easy consequence of the following inequalities: dimρ R(Γ, SL2 (C))red ≤ dim Tρ (R(Γ, SL2 (C))red ) ≤ dim Tρ R(Γ, SL2 (C)) = dim Z 1 (Γ, Ad ρ), where Tρ denotes the Zariski tangent space at ρ of both the variety or the scheme.  Definition 19. We call ρ ∈ R(Γ, SL2 (C)) scheme smooth if dimρ R(Γ, SL2 (C))red = dim Z 1 (Γ; Ad ρ), Example 20. If Γ is a finite group, then every ρ ∈ R(Γ, SL2 (C)) is scheme smooth, because H 1 (Γ, Ad ρ) = 0 and the following lemma. Lemma 21. If H 1 (Γ, Ad ρ) = 0, then ρ is scheme smooth. Proof. We use that every d ∈ Z 1 (Γ, Ad ρ) is inner, namely there is a ∈ sl2 (C) such that d(γ) = a−Adρ (γ)a for every γ ∈ Γ. It can be checked that d is a vector tangent to the orbit by conjugation, to the path of conjugation by exp(ta). This yields dimρ R(Γ, SL2 (C))red = dim Z 1 (Γ, Ad ρ).  In what follows we call a representation ρ ∶ Γ → SL2 (C) reduced and normal if the local ring A(Γ)mρ is reduced and normal, respectively. Here mρ denotes the maximal ideal associated to ρ. The tangent cone T Cρ R(Γ, SL2 (C)) is the spectrum of the graded C-algebra associated to the local ring A(Γ)mρ . Moreover, the tangent space Tρ R(Γ, SL2 (C)) is the smallest affine subspace which contains the tangent cone (see [27, III.§3]). From Proposition 11 we obtain: Lemma 22. Let ρ ∶ Γ → SL2 (C) be a representation. Then ρ is reduced and normal if the tangent cone T Cρ R(Γ, SL2 (C)) is reduced and normal, respectively. 5. Computing the character scheme of a finitely presented group In this section we prove Theorem 1, for that purpose we recall the skein algebra. Let CΓ denote the group ring. The tensor algebra over CΓ is denoted by T(CΓ). Definition 23. The skein algebra is SΓ = T(CΓ)/ (e − 2, α ⊗ β − β ⊗ α, α ⊗ β − αβ − αβ −1 ∣ α, β ∈ Γ) . 10 The definition of the skein algebra is motivated by the trace functions identities in Equation (6). Based on work of Brumfiel and Hilden [4], Przytycki and Sikora proved in [33] the following: Theorem 24 ([33]). There is an isomorphism of C-algebras Φ ∶ SΓ Ð → B(Γ) determined ≅ by Φ([γ]) = tγ . As explained by Marché in [21], the theorem also follows from a more general result of Procesi [31]. Recall we have a presentation Γ = ⟨γ1 , . . . , γn ∣ rl , l ∈ L⟩ as in (1). The natural projection π ∶ Fn → Γ induces a surjection π∗ ∶ SFn ↠ SΓ that factors to an epimorphism π̄ ∶ SFn /K ↠ SΓ , where K ⊂ SFn ≅ B(Fn ) is the ideal K = ([α] − [α′ ] ∣ α ∈ Fn , π(α) = π(α′ )) . Lemma 25. The map π̄ ∶ SFn /K ↠ SΓ is an isomorphism of C-algebras. Proof. To construct the inverse, start with a set-theoretic section s ∶ Γ → Fn of the projec- tion π ∶ Fn → Γ, extend it linearly to T(CΓ) → T(CFn ) and compose the extension with the projection to SFn /K. By construction of K, we have for all α, β ∈ Γ, s(αβ)−s(α)s(β) ∈ K, therefore s induces a morphism of algebras s̄ ∶ SΓ → SFn /K, that satisfies [α] = s̄(π̄([α])) for all α ∈ Fn and [β] = π̄(s̄([β])) for all β ∈ Γ. Since classes of the elements in the group span linearly the skein algebra, s̄ and π̄ are inverse from one another.  Corollary 26. Let I = (tα − tβ ∣ α, β ∈ Fn , π(α) = π(β)). Then B(Γ) ≅ B(Fn )/I. Proof of Theorem 1. Using Corollary 26, the proof amounts to find the suitable generat- ing set for the ideal I. Given α, β ∈ Fn , we introduce the element Θ(α, β) = tαβ − tα , so that I = (Θ(α, β) ∣ α, β ∈ Fn , β ∈ ker(π)) . The proof consists in finding a generating set for the ideal I according to the statement of the theorem. Firstly, as ker(π) is normally generated by the relations rl , l ∈ L, by using the equalities Θ(α, β1 β2 ) = Θ(αβ1 , β2 ) + Θ(α, β1 ), ∀α, β1 , β2 ∈ Fn , Θ(α, γβγ ) = Θ(γ αγ, β), −1 −1 ∀α, β, γ ∈ Fn , we get: I = (Θ(α, rl ) ∣ α ∈ Fn , l ∈ L) . For an element α ∈ Fn , let ∣α∣ denote its word length in the canonical generators. Define (14) Ik = (Θ(α, rl ) ∣ α ∈ Fn , ∣α∣ ≤ k, l ∈ L) . so that I = ⋃k Ik . We claim that (15) Ik+1 = Ik , for k ≥ 2. 11 For that purpose we use an equality due to Vogt [14, Lemma 4.1.1]: 2tabcd = ta tb tc td − tc td tab − tb tc tad − ta td tbc − ta tb tcd (16) + tad tbc − tac tbd + tab tcd + td tabc + tc tabd + tb tacd + ta tbcd . From this equation, we easily deduce (17) 2Θ(abc, d) = (tabc − ta tbc − tc tab + ta tb tc )Θ(1, d) + (tab − ta tb )Θ(c, d) − tac Θ(b, d) + (tbc − tb tc )Θ(a, d) + ta Θ(bc, d) + tb Θ(ac, d) + tc Θ(ab, d), which implies (15). As I = I2 , I is generated by Θ(1, r), Θ(γi±1 , r), and Θ(γi±1 γj±1 , r), so we just need to get rid of the powers −1 and to reduce to i < j. This is proved by using the equalities (18) Θ(α, r) = tα Θ(1, r) − Θ(α−1 , r), (19) Θ(αβ, r) = tα Θ(β, r) − Θ(α−1 β, r), (the first one is obviously a particular case of the second one), which in its turn follow from the trace identity ta tb = tab + ta−1 b .  Proof of the Addendum. To simplify notation, assume there is only one relation r and decompose it as r = u−1 v. We prove the equalities (20) (tar − ta ∣ a ∈ Fn ) = (tbv − tbu ∣ b ∈ Fn ) = (tbv − tbu ∣ b ∈ {1, γi , γi γj }) . The first equality is elementary by writing a = bu. The second equality follows from tbv − tbu = Θ(b, v) − Θ(b, u) by applying the same arguments as in the proof of Theorem 1, in particular equalities (17), (18), and (19), allow to reduce the word length of b. Then the addendum follows from (20).  Remark 27. The ideal I1 defined in (14) is the ideal considered by González-Acuña and Montesinos-Amilibia [14]. They proved that the coordinate ring of the representation variety (not the scheme) is isomorphic to (B(Fn )/I1 )red ≅ B(Fn )/ rad(I1 ) . In Section 8 we see that the ideal I1 may be non-radical even if the coordinate ring B(Γ) is reduced (whether the ideal I1 is a radical ideal or not may depend on the presentation of Γ). Thus, when we compute a scheme, I1 is not sufficient and we need to consider I2 . The following example illustrates that the ideals I2 and I1 may be different, even when B(Γ) is reduced. Example 28. Let Γ = ⟨a, b ∣ r⟩, r = baba−1 b−1 a−1 , be the trefoil group. The algebra B(Γ) is a quotient of B(F2 ) ≅ C[ta , tb , tab ]. As Γ is a one-relator group, we obtain: I1 = (tr − 2, ta r − ta , tb r − tb ) and I2 = (tr − 2, ta r − ta , tb r − tb , tab r − tab ) . Let’s compute I1 . By using the trace relations we obtain ta r = tb and hence ta r − ta = tb − ta . Moreover, we have tb r = tb tr − tr b−1 and tr b−1 = ta . Therefore, tb r − tb = tb tr − ta − tb = tb (tr − 2) + (tb − ta ), 12 and we obtain I1 = (tr − 2, ta − tb ). The trace relations (6) give: tbaba−1 b−1 a−1 = taba tbab − t(ba)3 taba = tab ta − tb tbab = tab tb − ta t(ba)3 = t3ab − 3 tab , and therefore tr − 2 = (tab ta − tb )(tab tb − ta ) − t3ab + 3 tab . We obtain the following primary decomposition of I1 : I1 = (ta − tb , t2b − tab − 2) ∩ (ta − tb , (tab − 1)2 ) and I1 is clearly not radical (see also [16]). Notice that the first ideal corresponds to the characters of non-simple representations tab−1 − 2 = ta tb − tab − 2. On the other hand, we have that ab r = (ab)baba−1 (ab)−1 , and hence tab = tbaba−1 modulo I2 . Now, tab r − tab = tbaba−1 − tab = tbab ta − t(ba)2 − tab = (tab tb − ta )ta − (t2ab − 2) − tab We obtain the following primary decomposition of I2 : I2 = (ta − tb , t2b − tab − 2) ∩ (ta − tb , tab − 1) and I2 is radical (see also the notebook [16]). Remark 29. We observe also that C[tγ1 , tγ2 , tγ1 γ2 ]/I1 depends on the presentation and not only on the group. An explicit example is given by an other presentation of the trefoil group (the group of Example 28): the presentation ⟨x, y ∣ x2 y −3 ⟩ produces a radical ideal I1 (see [16] for the computations). Remark 30. The ideal I3 is the ideal considered by Miura and Suzuki in [25]. They proved that the skein module of a group with three generators and two relations (and hence the coordinate ring of the representation scheme) is isomorphic to B(F3 )/I3 . As pointed out in the proof of Theorem 1, we have in general that I2 = I3 , see Equation (15), and therefore it is sufficient to consider less generators of the ideal. Remark 31. Our result was mainly motivated by Corollary 10.1.7 in [11]. This corollary states that any product of matrices (and their inverses) constructed from a given set {A1 , . . . , An } ⊂ SL2 (C) can be written as a linear combination of id, Ai , 1 ≤ i ≤ n, and Ai Aj , 1 ≤ i < j ≤ n. 6. The scheme of non-simple representations In concordance with [19] we use the term simple representation instead of the more common term irreducible representation in order to avoid confusion with irreducible al- gebraic varieties or schemes, as well as reduced or non-reduced schemes. Thus the term non-simple in the title of this section would be reducible in other papers. Definition 32 ([19]). Let ρ ∶ Γ → SL2 (C) be a representation. We say that ρ is simple if the image ρ(Γ) spans M2 (C) as a C-vector space. 13 Lemma 33. Let ρ ∶ Γ → SL2 (C) be a representation. Then ρ is simple if and only if there exist γ1 , γ2 , γ3 in Γ such that tr (ρ(γ1 )ρ(γ2 )ρ(γ3 )) − tr (ρ(γ1 )ρ(γ3 )ρ(γ2 )) ≠ 0 . Proof. Let id ∈ SL2 (C) denote the identity matrix. For a matrix M ∈ M2 (C) we put M0 = M − tr(M 2 id. Then id, A, B and C generate M2 (C) as a C-vector space if and only ) if (A0 , B0 , C0 ) is a C-basis for the subspace sl2 (C) of trace free matrices of M2 (C). Now observe that the map sl2 (C)3 → C given by (21) (X0 , Y0 , Z0 ) ↦ tr(X0 Y0 Z0 ) is a determinant map, e.g. multilinear and alternating (see Section V in [10]). A direct calculation gives 1 (22) tr(ABC) = tr(A0 B0 C0 ) − ( tr(A) tr(BC) 2 + tr(B) tr(AC) + tr(C) tr(AB) − tr(A) tr(B) tr(C)) , and hence tr(ABC) − tr(ACB) = 2 tr(A0 B0 C0 ) . Thus tr(ABC) ≠ tr(ACB) if and only if A0 , B0 , C0 are linearly independent (equivalently, they span sl2 (C)). As id ∈ ρ(Γ), the lemma follows.  Remark 34. Notice that Equation (22) and the fact that (21) is a determinant map imply the equality tabc + tacb = p in (12). Motivated by Lemma 33 we define: Definition 35. Let Γ be a group, and B(Γ) the universal algebra of SL2 (C)-characters. The ideal Jns ⊂ B(Γ) is defined by Jns = Jns (Γ) ∶= (tabc − tacb ∣ a, b, c ∈ Γ) , and the quotient Bns (Γ) = B(Γ)/Jns is called the universal algebra of non-simple SL2 (C)-characters. Remark 36. We have Jns ⊂ B(Γ) = A(Γ)SL2 (C) ⊂ A(Γ) , and Lemmata 3.4.1, 3.4.2 and Remark 3.4.3 in [28] give that Jns A(Γ) ⋂ B(Γ) = Jns and SL2 (C) (A(Γ)/Jns A(Γ)) ≅ A(Γ)SL2 (C) /Jns = Bns (Γ) . For an abelian group Γ0 we have Jns (Γ0 ) = (0) since abc = acb in Γ0 , and hence Bns (Γ0 ) = B(Γ0 ). Proposition 37. Let Γ be a finitely generated group. The abelianization morphism Γ → Γab induces an isomorphism of algebras B(Γab ) Ð → Bns (Γ). ≅ Proof. By writing c = b−1 a−1 d, we have an equality of sets {tabc − tbac ∣ a, b, c ∈ Γ} = {td − t[b,a]d ∣ a, b, d ∈ Γ} So both sets generate the same ideal in B(Γ). The left hand side set spans Jns , the right hand side set spans the ideal of the abelianization, by Theorem 1.  14 7. Abelian groups Throughout this section Γ denotes a finitely generated abelian group. Przytycki and Sikora proved that the skein algebra of an abelian group is reduced [32, 33], hence its character scheme is also reduced. In [36] Sikora described the variety of characters when Γ is torsion free. Here we discuss the case with torsion, and we show in particular that B(Γ) is reduced with completely different methods. Let β denote the first Betti number of Γ and T its torsion subgroup, so that we have a short exact sequence 1 → T → Γ → Zβ → 1 with T finite. The scheme of characters of T is a variety with finite cardinality. For each χ ∈ X(T, SL2 (C)), set X(Γ, SL2 (C))χ = res−1 (χ) where res∶ X(Γ, SL2 (C)) → X(T, SL2 (C)) is the map induced by restriction. Theorem 38. For Γ an abelian group as above, X(Γ, SL2 (C)) is reduced and X(Γ, SL2 (C)) = ⋃ X(Γ, SL2 (C))χ χ∈X(T,SL2 (C)) is its decomposition into irreducible components. Furthermore (C∗ )β if χ is not central X(Γ, SL2 (C))χ ≅ { ∗ β (C ) / ∼ if χ is central where (λ1 , . . . , λβ ) ∼ ( λ11 , . . . , λ1β ), for λ1 , . . . , λβ ∈ C∗ . When χ ∈ X(T, SL2 (C)) is central, a point in X(Γ, SL2 (C))χ is the character of a representation that maps the generators of Zβ to diagonal matrices with eigenvalues λ±1 i . Thus the action of the involution is the action of the Weyl group. In particular: Corollary 39. The singular locus of X(Γ, SL2 (C)) is the set of central characters of Γ. To prepare the proof of Theorem 38, we need a lemma on the Zariski tangent space. Lemma 40. If ρ ∈ R(Γ, SL2 (C)) is non-central, then dim Z 1 (Γ, Adρ) ≤ β + 2. Proof of Lemma 40. We may assume β ≥ 1, as the finite case has been considered in Example 20. Chose elements γ1 , . . . , γβ ∈ Γ so that they project to a generating set of Γ/T ≅ Zβ . This choice yields a splitting Γ ≅ Zβ × T . To bound the dimension of the space of 1-cocycles, consider the monomorphism of vector spaces: Z 1 (Γ, Adρ) → sl2 (C) × ⋯ × sl2 (C) × Z 1 (T, Adρ∣T ) (β) (23) d ↦ (d(γ1 ), . . . , d(γβ ), d∣T ) where ρ∣T and d∣T ∈ Z 1 (T, Adρ∣T ) denote the respective restrictions to T of ρ and d. Since T is finite, H 1 (T, Adρ∣T ) = 0 and therefore 0 if ρ∣T is central (24) dim Z 1 (T, Adρ∣T ) = B 1 (T, Adρ∣T ) = { 2 otherwise because dim B 1 (T, Adρ∣T ) = dim sl2 (C) − dim sl2 (C)Adρ(T ) . Next we distinguish cases, according to whether the restriction ρ∣T is central or not. When the restriction ρ∣T is central, we may assume that ρ(γ1 ) is non-central. Then for every d ∈ Z 1 (Γ, Adρ), from γ1 γi = γi γ1 we get (following the rules of crossed homomor- phism or Fox calculus): (25) (Adρ(γ1 ) − id)d(γi ) = (Adρ(γi ) − id)d(γ1 ) 15 As ρ(γ1 ) is non-central, rank(Adρ(γ1 ) − id) = 2 (because the kernel of (Adρ(γ1 ) − id) is the tangent line to the 1-parameter group containing ρ(γ1 )). Hence, from (25) applied to i = 2, . . . , β and using (24), the image of (23) is contained in the kernel of a linear map sl2 (C)β → sl2 (C)β−1 of rank 2(β −1), and we get the bound dim Z 1 (Γ, Adρ) ≤ 3β −2(β −1). When the restriction ρ∣T is non-central chose γ0 ∈ T such that ρ(γ0 ) is non-central. Then apply the same argument as above to the equalities γ0 γi = γi γ0 , for i = 1, . . . , β. Now the image of (23) is contained in the kernel of a linear map sl2 (C)β ⊕ C2 → sl2 (C)β of rank 2β, hence dim Z 1 (Γ, Adρ) ≤ 3β + 2 − 2β.  For a morphism τ ∶ T → Q/Z and a splitting Γ ≅ Zβ × T , define the subset of represen- tations Vτ ⊂ (R(Γ, SL2 (C))red as: (26) Vτ = {exp(θa) exp(2πτ a) ∣ θ ∈ hom(Γ, C), a ∈ Q}, where Q denotes the quadric Q = {a ∈ sl2 (C) ∣ det(a) = 1} = sl2 (C) ∩ SL2 (C). To see that the set Vτ consists of representations, notice that exp(2πa) = id for every a ∈ Q. The splitting Γ ≅ Zβ × T is used to extend τ ∶ T → Q/Z to the whole Γ, but the set Vτ does not depend on this extension or splitting. We shall use Vτ to describe the preimage of X(Γ, SL2 (C))χ by the natural projection R(Γ, SL2 (C)) → X(Γ, SL2 (C)), that we denote by R(Γ, SL2 (C))χ = {ρ ∈ R(Γ, SL2 (C)) ∣ χρ∣T = χ}. Consider χ ∈ X(T, SL2 (C)) non-central. Every representation of T with character χ is of the form exp(2πτ a), for some a ∈ Q and some τ ∶ T → Q/Z. Notice that for χ ∈ X(T, SL2 (C)) non-central, we can chose either τ or −τ , but this is the only ambiguity in the choice of τ . In fact Vτ = Vτ ′ if and only if τ = ±τ ′ . Lemma 41. For χ ∈ X(T, SL2 (C)) non-central, and τ as above, R(Γ, SL2 (C))χ is reduced and R(Γ, SL2 (C))χ = Vτ ≅ (C∗ )β × Q. In particular R(Γ, SL2 (C))χ is irreducible and smooth. Proof. As all abelian subgroups of SL2 (C) are contained in a one-parameter group, we have equality of sets (R(Γ, SL2 (C))χ )red = Vτ . Using Lemmas 18 and 40, since dim Vτ = β + 2 every point in R(Γ, SL2 (C)) is scheme smooth (hence reduced), and (R(Γ, SL2 (C))χ )red = R(Γ, SL2 (C))χ . Finally, notice that the natural map (C∗ )β ×Q → Vτ is a bijection because exp(2πτ a) is non-central. By construction this map is regular, and by smoothness and Zariski’s main theorem [26, 3.20] it is biregular.  When χ is central, R(Γ, SL2 (C))χ contains representations with parabolic elements, hence we need other tools to study it. Definition 42. The cone of group homomorphisms of rank at most one from Γ to sl2 (C) is the determinantal variety: hom1 (Γ, sl2 (C)) ≅ hom1 (Γ/T, sl2 (C)) ≅ hom1 (Cβ , sl2 (C)). Namely, hom1 (Γ, sl2 (C)) consists of all group homomorphisms in hom(Γ, sl2 (C)) whose image is contained in a linear space of dimension at most 1. This is the C-cone on the Segre variety Pβ−1 × P2 . Now consider a central character χ ∈ X(T, SL2 (C)). We chose ρ0 ∈ R(Γ, SL2 (C))χ a central representation of Γ whose restriction to T has character χ. 16 Lemma 43. For χ ∈ X(T, SL2 (C)) central, and given ρ0 ∈ R(Γ, SL2 (C))χ also central, (R(Γ, SL2 (C))χ )red = {exp(θ)ρ0 ∣ θ ∈ hom1 (Γ, sl2 (C))}. Furthermore R(Γ, SL2 (C))χ is scheme-smooth at non-central representations. Proof. As in Lemma 41, equality (R(Γ, SL2 (C))χ )red = {exp(θ)ρ ∣ θ ∈ hom1 (Γ, sl2 (C)) as sets is a consequence of the fact that every abelian subgroup of SL2 (C) is contained in a one-parameter group, which in its turn is the image by the exponential of a line in sl2 (C). Scheme smoothness at non-central representations follows again from Lemmas 18 and 40 and the dimension count.  Lemma 44. The cone hom1 (Γ, sl2 (C)) is isomorphic to the quadratic cone to R(Γ, SL2 (C)) at any central representation and it is the whole tangent cone. Proof. Let ρ be a central representation. In particular Ad ρ is trivial and Z 1 (Γ, Adρ) = hom(Γ, sl2 (C)) (namely, group homomorphisms with no restriction on the dimension of the image). To compute the quadratic cone, notice that, by Baker-Campbell-Hausdorff formula, for any θ ∈ hom(Γ, sl2 (C)): 2 [θ(γ ),θ(γ )]+O(t3 ) etθ(γ1 ) etθ(γ2 ) e−tθ(γ1 ) e−tθ(γ2 ) = et 1 2 , ∀γ1 , γ2 ∈ Γ. This yields that hom1 (Γ, sl2 (C)) is the quadratic cone at ρ. (This can be viewed equiv- alently with the obstruction theory of Goldman.) Not only hom1 (Γ, sl2 (C)) is the qua- dratic cone, by construction it is also the whole tangent cone, because all higher order terms in Baker-Campbell-Hausdorff formula vanish when [θ(γ1 ), θ(γ2 )] = 0.  Related to Lemma 44, notice that a theorem of Goldman and Millson [13, Thm. 9.3] guarantees that the singularities of R(Γ, SL2 (C)) are at most quadratic, as Γ is virtually a Bieberbach group. Since hom1 (Γ, sl2 (C)) is a determinantal scheme it is reduced and normal [5], and by Lemma 22, we get: Corollary 45. A central representation is a reduced and normal point of the scheme R(Γ, SL2 (C)). From the previous results in this section we deduce: Corollary 46. The scheme R(Γ, SL2 (C)) is reduced and normal. Its singular locus is precisely the set of central representations, that have quadratic singularities modeled in the C-cone on Pβ−1 × P2 . Proof of Theorem 38. As R(Γ, SL2 (C)) is reduced and normal, and its irreducible com- ponents are R(Γ, SL2 (C))χ , X(Γ, SL2 (C)) is also reduced and normal, and is components are the X(Γ, SL2 (C))χ . Fix a ∈ Q and consider ϕ̃a ∶ hom(Γ, C) ≅ Cβ → R(Γ, SL2 (C))χ θ ↦ exp(θa)ρ0 where ρ0 ∈ R(Γ, SL2 (C))χ is a central representation when χ is central as in Lemma 43, or ρ0 = exp(2πτ a) for some nontrivial τ ∶ T → Z/Q as in (26) when χ is not central. The map ϕ̃a factors through the exponential map on each factor C to ϕa ∶ (C∗ )β → R(Γ, SL2 (C))χ . 17 We compose it with the projection π∶ R(Γ, SL2 (C))χ → X(Γ, SL2 (C))χ and we claim that the composition (27) π ○ ϕa ∶ (C∗ )β → X(Γ, SL2 (C))χ is a surjection. When χ is not central, the claim follows from Lemma 41 and the de- scription of Vτ in Equation (26), as SL2 (C) acts transitively on Q (hence every orbit by conjugation in R(Γ, SL2 (C))χ meets the image of ϕa ). When χ is central, we use the description of Lemma 43 and in this case we similarly prove that every orbit by con- jugation in R(Γ, SL2 (C))χ of a semi-simple representation meets the image of ϕa . We conclude the surjectivity by recalling that every character is the character of a semi-simple representation. Next we discuss the inverse images of the surjection π ○ ϕa in (27). We use two elemen- tary properties: ● If two semisimple representations have the same character, then they are conju- gate. Notice that the image of ϕa consists of only semisimple representations. ● We write a diagonal representation of Γ as diag(θ, θ−1 ) for θ∶ Γ → C∗ a homomor- phism. If diag(θ1 , θ2−1 ) is conjugate to diag(θ2 , θ2−1 ), then θ1 = θ2±1 . To apply these remarks to our situation, we may assume that a ∈ Q is diagonal. Now, if τ is central, then it follows that π ○ ϕa in (27) factors to the quotient as in the statement of the theorem (28) (C∗ )β / ∼→ X(Γ, SL2 (C))χ which is a bijection. This does not hold anymore when τ is non central, as the restriction to T of a representation in the image of ϕa is fixed, it is a representation ρ0 = exp(2πτ a) with τ ∶ T → Z/Q nontrivial. We conclude that when τ is non central (27) is already a bijection. The maps (27) and (28) are regular bijections, and by Zariski’s main theorem [26, 3.20] they are both biregular, since X(Γ, SL2 (C))χ is normal.  Next we describe the algebra B(Zβ ) = C[X(Zβ , SL2 (C))] in terms of traces. Start with the presentation Zβ = ⟨a1 , . . . , aβ ∣ [ai , aj ] = 1⟩ Proposition 47. With the previous presentation, B(Zβ ) = C[tai , tai aj ]/(tai aj − taj ai , fai ,aj , gai ,aj ,ak , hai ,aj ,ak ,al )i,j,k,l∈{1,...,β} where the subindex are different on each fai ,aj , gai ,aj ,ak , hai ,aj ,ak ,al , and (29) fa,b =t2a + t2b + t2ab − ta tb tab − 4, (30) ga,b,c =ta (ta tbc + tb tac + tc tab − ta tb tc ) − 2tab tac − 4tbc + 2tb tc , (31) ha,b,c,d =(2tab − ta tb )(2tcd − tc td ) − (2tac − ta tc )(2tbd − tb td ). Furthermore, for computing trace functions of elements of Zβ , we use the standard trace identities (6) and (16) and add (32) 2tabc = ta tbc + tb tac + tc tab − ta tb tc . Remark 48. (a) Equation (32) does not need to be included in the presentation of the algebra B(Zβ ), but it is needed to describe the trace functions of group elements. It can be read as tabc = 21 p, where p is as in (9). Notice that from the relations (29), (30) and (31) it follows that p2 − 4q = 0, so the polynomial F in (8) reads as F = (tabc − 21 p)2 . Vanishing of F is a standard trace identity, but we include (32) to get rid of the square. 18 (b) Using (32), (30) reads as ga,b,c = 2(ta tabc − tab tac − 2tbc + tb tc ). Notice also that, for F as in (8): ∂F ∂tbc = −ta tabc + tab tac + 2tbc − tb tc = − 12 ga,b,c . (c) If we allow that some of the subindex are equal, then we can reduce to a single family of equations, because: 2fa,b = ga,b,b and 2ga,b,c = ha,a,b,c . Proof of Proposition 47. By Theorem 38, σ B(Zβ ) = (C[λ1 , . . . , λβ , µ1 , . . . , µβ ]/(λi µi − 1)i=1,...,β ) where σ(λ1 , . . . , λβ , µ1 , . . . , µβ ) = (µ1 , . . . , µβ , λ1 , . . . , λβ ). A point in X(Zβ , SL2 (C)) with coordinates (λ1 , . . . , λβ , µ1 , . . . , µβ ) is the character of the representation that maps each generator ai ∈ Zβ to the diagonal matrix diag(λi , λ1i ). To compute the invariant subalge- bra, we change coordinates tai = λi + µi = λi + λ1i , xi = λi − µi = λi − λ1i , so σ B(Zβ ) = (C[ta1 , . . . , taβ , x1 , . . . , xβ ]/(t2ai − x2i − 4)i=1,...,β ) and σ(ta1 , . . . , taβ , x1 , . . . , xβ ) = (ta1 , . . . , taβ , −x1 , . . . , −xβ ). The algebra of invariants by σ is generated by the tai and the quadratic monomials zij = xi xj , i, j = 1, . . . , β. Thus B(Zβ ) = C[tai , zij ]/I where I is the ideal generated by: t2ai − zii − 4, zij − zji , zij zkl − zil zkj , for i, j, k, l ∈ {1, . . . , β}, possibly equal. The zij can be written in terms of traces as follows: 1 λi λj zij = xi xj = λi λj + − − = tai aj − tai a−1 = 2tai aj − tai taj λi λj λj λi j and the presentation of B(Zβ ) as a quotient of C[tai , tai aj ] follows by writing the zij in terms of traces. Finally, notice that the unique relation we need to add to compute the trace of any element from these variables is (32), see the discussion in Remark 48, because the trace of elements of length larger that three is deduced from Vogt relation (16).  We show a couple of examples that illustrate Theorem 38 when Γ has torsion. Example 49. We compute the scheme of characters of Z ⊕ Z/4Z = ⟨a, b ∣ [a, b] = b4 = 1⟩. As Z ⊕ Z/4Z is a quotient of Z2 , by Proposition 47 the coordinates are (ta , tb , tab ) and they satisfy (33) t2a + t2b + t2ab − ta tb tab − 4 = 0 Furthermore, as b4 = 1, we get tb (tb + 2)(tb − 2) = 0. Thus there are three components: 19 ● When tb = 0, the character restricted to Z/4Z is non-central. By replacing tb = 0 in (33) we get t2a + t2ab − 4 = 0, which can be rewritten as (ta + itab )(ta − itab ) = 4, which is isomorphic to C∗ . ● When tb = ±2, (33) becomes (ta ∓ tab )2 = 0, and since we know it is reduced, we can get rid of the square and so ta = ±tab . Namely, two lines, because C∗ / ∼≅ C. Example 50. Next consider Z ⊕ Z ⊕ Z/4Z = ⟨a, b, c ∣ [a, b] = [a, c] = [b, c] = c4 = 1⟩. The same considerations as in Example 49 yield that the coordinates for the scheme of characters are (ta , tb , tc , tab , tac , tbc ) and it has three components: ● One component has equations: tc =0⎫ ⎪ ⎪ ⎪ ⎪ ta + tac − 4 =0⎪ 2 2 ⎪ ⎪ ⎬ tb + tbc − 4 =0⎪ 2 2 ⎪ ⎪ ⎪ ⎪ −2tab + ta tb − tac tbc =0⎪⎪ ⎭ So it is isomorphic to C × C . ∗ ∗ ● The remaining two components have equations tc ± 2 =0⎫ ⎪ ⎪ ⎪ tac ± ta =0⎪⎪ ⎪ ⎪ ⎬ tbc ± tb =0⎪ ⎪ ⎪ ⎪ ⎪ ta + tb + tab − ta tb tab − 4 =0⎪ 2 2 2 ⎪ ⎭ and are isomorphic to X(Z2 , SL2 (C)) ≅ (C∗ × C∗ )/ ∼. 8. Examples of non-reduced schemes of characters Kapovich and Millson have proved in [18] that there are no restrictions on the local geometry of SL2 (C)-character schemes of 3–manifold groups, in particular there are non- reduced character schemes of those groups. In this section we give explicit examples of non-reduced SL2 (C)-character schemes of 3–manifold and orbifold groups. Example 51. Let S 3 (K, 3) denote the three-dimensional orbifold with underlying space S 3 , branching locus the figure-eight knot K and branching index 3. It is a Euclidean orbifold [39, Ch. 13]. The Euclidean structure is relevant, because the translational part of the Euclidean holonomy is a non-integrable infinitesimal deformation of its rotational part. This is similar to the examples of Lubotzky and Magid in [19, pp. 40–43], for representations of Euclidean 2-orbifolds in GL2 (C) that are proved to be non-reduced. Furthermore in [13, §9.3] Goldman and Millson prove that those are double points, as the singularities are at most quadratic. We address triple points in Example 52 below, using Nil instead of Euclidean orbifolds. The fundamental group of the figure eight knot exterior has a presentation: π1 (S 3 ∖ K) = ⟨a, b ∣ ab−1 a−1 ba = bab−1 a−1 b⟩ where a and b are represented by meridian loops. Thus a presentation of the orbifold fundamental group can be obtained by adding the relation a3 = 1: Γ = π1orb (S 3 (K, 3)) = ⟨a, b ∣ ab−1 a−1 ba = bab−1 a−1 b, a3 = 1⟩. 20 The algebra B(Γ) is a quotient of B(F2 ) ≅ C[ta , tb , tab ] by and ideal I. A computer supported calculation yields [16]: I = (ta − 2, tb − 2, tab − 2) ∩ (ta + 1, tb + 1, tab + 1) ∩ (ta + 1, tb + 1, (tab − 1)2 ). So X(Γ, SL2 (C)) consists of three points, one of them non-reduced (double point). The point ta = tb = tab = 2 is the trivial character, ta = tb = tab = −1 is the non-trivial abelian character, and (ta , tb , tab ) = (−1, −1, 1) is the double point that corresponds to the char- acter of a simple representation in SU(2). This representation is a lift of the rotational part of the holonomy of the Euclidean structure in PSU(2) ≅ SO(3). We can also understand this double point as the result of a tangency. The variety of characters of the figure eight knot exterior is the plane curve given by (x2 y − 2x2 − y 2 + y + 1)(y − x2 + 2) = 0, where x = ta = tb and y = tab . The group Γ is obtained from the figure eight knot by adding the relation a3 = 1. A representation of an element of order three is either trivial (and has trace 2) or has trace −1. Therefore, the case where the image of a is non-trivial corresponds to x = −1. The line x = −1 intersects y − x2 − 2 = 0 transversely (at the abelian representation) and x2 y − 2x2 − y 2 + y + 1 = 0 tangentially, giving the double point. The fact that the intersection between x2 y − 2x2 − y 2 + y + 1 = 0 and x = −1 is not transverse corresponds to the fact that the trace of the meridian is not a local parameter at a Euclidean degeneration of hyperbolic cone manifolds [29]. Example 52. Let S 3 (Wh, (m, n)) be the orbifold with underlying space S 3 , branching lo- cus the Whitehead link Wh, and branching indexes m and n. The orbifold S 3 (Wh, (4, 2)) has Nil geometry [38, p. 112]. The isometry group of Nil surjects onto Isom(R2 ), which in its turn surjects onto O(2) ⊂ SO(3), but the rotational part of the Nil holonomy of S 3 (Wh, (4, 2)) in O(2) ⊂ SO(3) ≅ PSU(2) does not lift to SU(2), because of the elements of order two. Instead, it lifts to a representation of S 3 (Wh, (8, 4)) in SU(2), and this is the orbifold we consider. As in the previous example, we notice that the trace of the meridian is not a local parameter at a Nil degeneration [30], it is in fact a singularity of order three. Hence this representation will be a non reduced point of the character scheme of S 3 (Wh, (8, 4)). For the explicit computation, start with the fundamental group of the Whitehead link exterior π1 (S 3 ∖ Wh) = ⟨a, b ∣ aba−1 b−1 a−1 bab = baba−1 b−1 a−1 ba⟩. Its scheme of characters is the hypersurface of C3 with equation (z 3 − xyz 2 + (x2 + y 2 − 2)z − xy)(x2 + y 2 + z 2 − xyz − 4) = 0, where x = ta , y = tb and z = tab . The component x2 + y 2 + z 2 − xyz − 4 = 0 consists of abelian √ characters. The intersection of z 3 − xyz 2 + (x2 + y 2 − 2)z − xy = 0 with the lines x = ± 2 and y = 0 (corresponding to rotations of order 8 and 4 respectively) is z 3 = 0. Hence two triple points. The whole scheme of characters X(S 3 (Wh, (8, 4))) can be computed using the note- book [16]: besides the two triple points, it contains 21 simple points (3 simple characters, 4 central characters and 14 abelian non-central characters). Example 53. In this example and the next one, we consider a 3-manifold Mn with boundary a torus obtained by attaching a (2, 2n)-torus link exterior Cn (a cable space) and K, the orientable I-bundle over the Klein bottle, along a boundary component. The I-bundle K has the homotopy type of the Klein bottle, hence ΓK = π1 (K) ≅ ⟨γ, µ ∣ µγµ−1 = γ −1 ⟩ , 21 and the peripheral subgroup is π1 (∂K) = ⟨γ, µ2 ⟩. Let Cn be the exterior of the (2, 2n)-torus link, with n ≥ 2. It is a Seifert-fibered space with orbifold surface an annulus with one cone point of order n. Hence ΓCn = π1 (Cn ) ≅ ⟨a, c ∣ [a, cn ] = 1⟩, where cn is a generator of the center. Up to conjugation π1 (Cn ) has two peripheral subgroups: ⟨ac, cn ⟩ and ⟨a, cn ⟩. We obtain a 3-manifold Mn = Cn ∪h K by identifying the boundary component of K with a boundary component of Cn via a homeomorphism h ∶ ∂K → ∂2 Cn , where ∂2 Cn is the component of ∂Cn with peripheral subgroup ⟨a, cn ⟩ (thus ⟨ac, cn ⟩ is the peripheral subgroup of Mn ). The homeomorphism h∶ ∂K → ∂2 Cn is chosen so that h∗ (γ) = a, and h∗ (γµ2 ) = cn . A presentation of the fundamental group of Mn is Γn ∶= π1 (Mn ) ≅ ΓCn ∗Z⊕Z ΓK ≅ ⟨a, c, γ, µ ∣ µγµ−1 γ = [a, cn ] = 1, γµ2 = cn , a = γ⟩ . We simplify the presentation: Γn ≅ ⟨c, µ, γ ∣ µγµ−1 = γ −1 , γµ2 = cn ⟩ ≅ ⟨c, µ ∣ µcn µ−2 µ−1 cn µ−2 = 1⟩ ≅ ⟨c, µ ∣ cn µ−3 cn µ−1 = 1⟩ . The peripheral subgroup of π1 (Mn ) is ⟨µ−2 c, cn ⟩. Let us consider the case n = 2. The scheme X(M2 , SL2 (C)) can be computed from the notebook [16], we describe the components. Firstly, since the abelianization of π1 (M2 ) is Z × Z/4Z, there are three components of non-simple (or abelian) characters, described in Example 49. In addition there are two components containing simple characters, given by the ideals I = (tc tcµ + 2tµ , t2µ , tc tµ , t2c ) and J = (tµ , tcµ ) . The ideal J is radical but I is not: rad(I) = (tc , tµ ). We check that C[tc , tµ , tcµ ]/I is the coordinate ring of a double line, following Section II.3.5 in [9], by considering new coordinates: x ∶= tc tcµ + 2 tµ , y ∶= tc , z ∶= tcµ . With these coordinates I ≅ (x, y 2 ) and therefore C[tc , tµ , tcµ ]/I ≅ C[x, y, z]/(x, y 2 ) ≅ C[y, z]/(y 2 ) . The representation variety (not the scheme) of M2 is studied by K. Baker and K. Petersen in [2]. In fact M2 is a once-punctured torus bundle with tunnel number one. This torus bundle is also named M2 in [2], and in [2, Section 2.1] the following presentation for π1 (M2 ) is given: π1 (M2 ) ≅ ⟨α, β ∣ β −2 = α−1 βα2 βα−1 ⟩ by putting µ = α−1 and c = βα−1 . Example 54. We continue Example 53: now we show that for general n ≥ 2, the scheme X(Mn , SL2 (C)) is non-reduced, by generalizing the double line when n = 2. As π1 (Mn ) is generated by c and µ, X(Mn , SL2 (C))red is a subvariety of C3 with coordinates (tc , tµ , tcµ ). Consider n )} ⊂ X(Mn , SL2 (C))red . Yk = {(tc , tµ , tcµ ) ∈ C3 ∣ tµ = 0, tc = 2 cos( πk for k = 1, . . . , n − 1. Lemma 55. For k = 1, . . . , n − 1, Yk is a component of X(Mn , SL2 (C))red containing simple characters. Lemma 56. The scheme X(Mn , SL2 (C)) is non-reduced at any point of Yk , for k = 1, . . . , n − 1. 22 Proof of Lemma 55. First we describe a one-parameter family of characters and next we prove that this family forms a component. Consider characters χ ∈ X(Mn , SL2 (C)) such that: ● χ(µ) = χ(µγ) = 0 and χ(γ) = 2. Namely χ restricted to π1 (K) is the character of a representation of π1 (K) that maps µ to a rotation of angle π in H3 and γ to the identity. ● χ(c) = χ(ac) = cos( πkn ) and χ(a) = 2. Therefore χ restricted to π1 (Cn ) is the character of a representation of π1 (Cn ) that maps c to a rotation of angle 2πk/n in H3 and a to the identity. Thus the restriction of χ to both π1 (K) and π1 (Cn ) is the character of a representa- tion with image a cyclic group, and the restriction to the attaching torus is central, i.e. contained in {±id}. One can deform such a character by conjugating separately the image of π1 (K) and the image of π1 (Cn ), yielding a one-parameter of conjugacy classes of representations of π1 (Mn ). To prove that this family of characters is a component, we first look at representations of π1 (K) = ⟨γ, µ ∣ µγµ−1 = γ −1 ⟩. They lie in two families: (a) Representations that map γ to ±id, and µ to any value. (b) Representations that preserve an unoriented hyperbolic geodesic l ⊂ H3 , that map γ to an isometry preserving the orientation of l, and that map µ and µγ to π- rotations with axis perpendicular to l. Given ρ a representation of π1 M , we make two assertions: ● If the restriction ρ∣π1 K is in case (a) and ρ(µ2 ) ≠ −id, then ρ(π1 M ) is abelian. ● If the restriction ρ∣π1 K is in case (b) and ρ(γ) ≠ ±id, then ρ(π1 M ) preserves an unoriented geodesic in hyperbolic space. Both assertions follow from the attaching relations γµ2 = cn and a = γ by elementary considerations, and they yield that Yk is a component.  Proof of Lemma 56. For any χ ∈ Yk simple, χ restricted to Cn is constant (see the proof of Lemma 55). In particular its restriction to ∂Mn is also constant. By writing χ = χρ , as we can construct deformations of χ = χρ that remain constant in ∂Mn , dim ( ker (H 1 (Mn , Ad ρ) → H 1 (∂Mn , Ad ρ))) ≥ 1. On the other hand, by a standard argument on Poincaré duality [17, 35], rank (H 1 (Mn , Ad ρ) → H 1 (∂Mn , Ad ρ)) = 1 Thus dim H 1 (Mn ; Ad ρ) ≥ 2. It follows that the Zariski tangent space to the scheme at a simple χ ∈ Yk has dimension ≥ 2 by Theorem 17. As generic characters in Yk are simple and dim Yk = 1, the scheme is non-reduced at generic points of Yk . By Lemma 8, the set of non-reduced points is the Zariski-closed and it contains Yk .  Remark 57. The line Yk restricts to a point in X(∂Mn , SL2 (C)). The image of the restriction is always an isotropic subspace, and when the restriction is non-singular the image is a Lagrangian submanifold, cf. [17, 35]. This example proves that non-singularity is needed to have a Lagrangian submanifold. Remark 58. The manifold Mn of Example 54 should be compared to an example of Schnauel and Zhang [34], who attach a cable space to a torus knot exterior. In this way they obtain a boundary slope not detected by valuations on the variety of characters. 23 9. The character scheme of the Borromean rings Let us start with a presentation for the group ΓBor of the Borromean rings. a′ b a c b′ c′ Figure 1. The Borromean rings We take the over presentation and we obtain: ⎧ ⎪ a′ = c′ a(c′ )−1 , ⎧ ⎪ a′ = cac−1 , ⎪ ⎪ ′ ⎪ ⎪ ′ ⎨ b = a′ b(a′ )−1 , and ⎨ b = aba−1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ c = b c(b ) , ⎩ c = bcb . ′ ′ ′ −1 ′ −1 This gives the presentation (34) ΓBor ≅ ⟨a, b, c ∣ [a, [b, c−1 ]], [b, [c, a−1 ]], [c, [a, b−1 ]]⟩ and as usual one of the relation is a consequence of the other two relations. The generators a, b and c are meridians, and the corresponding longitudes are `a = [b, c−1 ], `b = [c, a−1 ] and `c = [a, b−1 ] . In what follows we consider the free group F(a, b, c) and the surjection F(a, b, c) ↠ ΓBor . Notice also that there is a surjective homomorphism ΓBor ↠ Z3 (the abelianization). Moreover, there are three obvious surjections ΓBor ↠ F2 , one of the three meridians is mapped onto the trivial element the other meridians are mapped to a generating pair of F2 . From these surjections, we obtain an closed immersions X(Z3 , SL2 (C)), X(F2 , SL2 (C)) ↪ XBor ↪ X(F(a, b, c), SL2 (C)) . Here we write XBor ∶= X(ΓBor , SL2 (C)) for short. Recall that the coordinate ring of X(F (a, b, c), SL2 (C)) ⊂ C7 is reduced and isomorphic to the quotient C[ta , tb , tc , tab , tac , tbc , tabc ]/(t2abc − p tabc + q) where p and q are the polynomials defined in equations (9) and (10) respectively. The scheme XBor has several components. These components arise from surjections of ΓBor onto Z3 (the abelianization), and the free group F2 (one of the meridians is mapped onto the trivial element). Let us first identify these components: ● First of all there is at least one the distinguished component X0 which contains the character of a lift of the holonomy representation. We are looking for the ideal I0 . 24 ● There are the characters of the non-simple representations Xns ⊂ XBor . Since the Abelianization is a surjection ΓBor ↠ Z3 we obtain an inclusion X(Z3 , SL2 (C)) ↪ XBor . The ideal Ins corresponding to this component was investigated in Section 7. According to Proposition 47, the ideal Ins is generated by (35) t2a + t2b + t2ab − ta tb tab − 4, t2a + t2c + t2ac − ta tc tac − 4, t2b + t2c + t2bc − tb tc tbc − 4, ta (ta tbc + tb tac + tc tab − ta tb tc ) − 2tab tac − 4tbc + 2tb tc , tb (ta tbc + tb tac + tc tab − ta tb tc ) − 2tab tbc − 4tac + 2ta tc , tc (ta tbc + tb tac + tc tab − ta tb tc ) − 2tac tbc − 4tab + 2ta tb , 2tabc − ta tbc − tb tac − tc tab + ta tb tc . The ideal Ins is a prime, and hence radical, since C[ta , tb , tc , tab , tac , tbc , tabc ]/Ins the coordinate algebra of the irreducible variety (C∗ × C∗ × C∗ )/ ∼ (see Theorem 38). ● Next there are representations which map one of the generators to a central ele- ment ±id ∈ SL2 (C). This gives rise to six 3-dimensional components Xa± , Xb± and Xc± in XBor . Each of those components is isomorphic to C3 and has therefore no singular points. The ideal corresponding to Xa± is Ia± ∶= (ta ∓ 2, tab ∓ tb , tac ∓ tc , tabc ∓ tbc ) and C[ta , tb , tc , tab , tac , tbc , tabc ]/Ia± ≅ C[tb , tc , tbc ] . Similar for the other generators. All these ideals are prime ideals, and hence they are radical. Theorem 59. The coordinate algebra C[XBor ] = C[XBor ]red is reduced. More precisely, we have C[XBor ] ≅ C[ta , tb , tc , tab , tac , tbc , tabc ]/IBor where IBor = Ins ∩ Ia+ ∩ Ia− ∩ Ib+ ∩ Ib− ∩ Ic+ ∩ Ic− ∩ I0 is the intersection of prime ideals in C[ta , tb , tc , tab , tac , tbc , tabc ]. Therefore, the representation scheme has eight irreducible components XBor = Xns ∪ Xa+ ∪ Xa− ∪ Xb+ ∪ Xb− ∪ Xc+ ∪ Xc− ∪ X0 . Proof. Computer supported calculations [16] give us an ideal I7 = Ins ∩ Ia+ ∩ Ia− ∩ Ib+ ∩ Ib− ∩ Ic+ ∩ Ic− ⊂ C[ta , tb , tc , tab , tac , tbc , tabc ], such that IBor ⊂ I7 . If I, J are ideals in C[x1 , . . . , xn ] then the ideal quotient is (I ∶ J) = {f ∈ C[x1 , . . . , xn ] ∣ gf ∈ I for all g ∈ J} (see Chapter 4 in [7]). It turns out that (I ∶ J) is an ideal in C[x1 , . . . , xn ] containing I. Moreover we have that V (I ∶ J) ⊃ V (I) ∖ V (J) where V (I) denotes the vanishing set of the ideal in Cn and S the Zariski-closure of S ⊂ Cn . Now, we use ideal division to define the ideal I0 ∶= (IBor ∶ I7 ). The notebook calculations give us that I0 is radical in Q[ta , tb , tc , tab , tac , tbc , tabc ] , and IBor = I0 ∩ I7 . Hence IBor = Ins ∩ Ia+ ∩ Ia− ∩ Ib+ ∩ Ib− ∩ Ic+ ∩ Ic− ∩ I0 . 25 The ideal I0 is generated by t2abc − tabc p + q, ta tbc − tb tac , tb tac − tc tab , (36) ta tabc − tab tac , tb tabc − tab tbc , tc tabc − tac tbc . Lemma 60. The ideal I0 is radical in C[ta , tb , tc , tab , tac , tbc , tabc ]. Proof. Computer supported calculations show that I0 ⊂ Q[ta , tb , tc , tab , tac , tbc , tabc ] is a radical ideal (see [16]). Now, Lemma 3.7 in [15] shows that I0 ⋅ C[ta , tb , tc , tab , tac , tbc , tabc ] ⊂ C[ta , tb , tc , tab , tac , tbc , tabc ] is also radical.  We let X0 denote the corresponding subvariety of XBor . This equations had already be studied by Sparaco [37]. Lemma 61 (Sparaco). The variety X0 is irreducible. Proof. The projection on to C7 → C4 onto the subspace generated by the tab , tac , tbc and tabc coordinates induces a birational map between X0 and the hypersurface with equation t4abc − t2abc (2tab tac tbc − t2ab − t2ac − t2bc + 4) + t2ab t2ac t2bc − t3ab tac tbc − tab t3ac tbc − tab tac t3bc + t2ab t2ac + t2ab t2bc + t2ac t2bc It follows now that this hypersurface is irreducible and hence X0 also (see [37]).  It follows that the ideal I0 is a prime ideal.  9.1. The distinguished component X0 . Notice that X0 is the unique component con- taining faithful representations. Hence X0 is the canonical component of the character variety of the Borromean rings. The singular locus X0sing is 1-dimensional. More precisely, the X0sing is the union of twelve lines and six points [16]. The lines are formed by characters of non-simple representations. The characters of the central representations are part of X0sing . At each character of a central representation three of the lines intersect. So the lines form a cube with corners the characters of a central representations. Hence, the characters of the central representations are singular points of the singular set. The characters of two central representations given by ⎧ ⎪ a ↦ a id ⎧ ⎪a ↦ ′a id ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨b ↦ b id and ⎨b ↦ ′b id ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩c ↦ c id ⎪ ⎩c ↦ c id ′ are connected by a line in X0sing if and only if two of the x and ′x are equal. The line con- sists of characters of non-simple representations. For example the character of the trivial representation (id, id, id), and the character of the central representation (id, id, −id), are connected by the line L = {(2, 2, z, 2, z, z, z) ∣ z ∈ C} ⊂ X0 . The six isolated points in X0sing are characters of SU(2)-representations. More pre- cisely, the points are characters of binary-dihedral representations which map one of the 26 generators to ±id and the other two to half-turns about two orthogonal lines. For example: ⎧ ⎪ a ↦ ±id ⎪ ⎪ ⎪ (±2, 0, 0, 0, 0, 0, 0) is the character of ρ ∶ ⎨b ↦ ( 0i −i 0 ) . ⎪ ⎪ ⎪ ⎪ ⎩c ↦ ( i 0 ) 0 i These characters are the midpoints of the faces of the cube. 9.2. Intersections between X0 and the other components. All components do intersect X0 . More precisely: ● X0 ∩ Xa± : These intersection form the faces of the cube. The intersection X0 ∩ Xa± is two-dimensional and isomorphic to C2 . In fact, I0 + Ia± = (ta ∓ 2, tab ∓ tb , tbc ∓ tc , 2tbc − tb tc , 2tabc ∓ tb tc ) and hence for the coordinate ring C[X0 ∩ Xa± ] ≅ C[ta , tb , tc , tab , tac , tbc , tabc ]/(I0 + Ia± ) ≅ C[tb , tc ] . Similar argument applies for Xb± and Xc± . Notice that the lines in X0sing are contained in these intersection. For example: X0 ∩ Xa+ contains the following four characters of central representations ⎧ ⎪ a ↦ id ⎪ ⎪ ⎪ ⎨b ↦ b id where a , b ∈ {±1}, ⎪ ⎪ ⎪ ⎪ ⎩c ↦ c id and the following four lines of characters of non-simple representations L1 = {(2, 2, z, 2, z, z, z) ∣ z ∈ C}, L2 = {(2, −2, z, −2, z, −z, −z) ∣ z ∈ C}, L3 = {(2, z, 2, z, 2, z, z) ∣ z ∈ C}, L4 = {(2, z, −2, z, −2, −z, −z) ∣ z ∈ C} . form a square. ● X0 ∩Xred : This intersection is formed by the 12 lines of non-simple representations. It is contained in X0sing . References [1] Caleb Ashley, Jean-Philippe Burelle, and Sean Lawton. Rank 1 character varieties of finitely pre- sented groups. Geom. Dedicata, 192:1–19, 2018. [2] Kenneth L. Baker and Kathleen L. Petersen. Character varieties of once-punctured torus bundles with tunnel number one. Internat. J. Math., 24(6):1350048, 57, 2013. [3] Nicolas Bourbaki. Algebra II. Chapters 4–7. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2003. [4] G. W. Brumfiel and H. M. Hilden. SL(2) representations of finitely presented groups, volume 187 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 1995. [5] Winfried Bruns and Udo Vetter. Determinantal rings, volume 1327 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1988. [6] P. Cartier. Groupes algébriques et groupes formels. In Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962), pages 87–111. Librairie Universitaire, Louvain; GauthierVillars, Paris, 1962. [7] David Cox, John Little, and Donal O’Shea. Ideals, varieties, and algorithms. Undergraduate Texts in Mathematics. Springer, New York, third edition, 2007. [8] David Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathematics. Springer- Verlag, New York, 1995. [9] David Eisenbud and Joe Harris. The geometry of schemes, volume 197 of Graduate Texts in Math- ematics. Springer-Verlag, New York, 2000. [10] Werner Fenchel. Elementary geometry in hyperbolic space, volume 11 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1989. 27 [11] N. Pytheas Fogg. Substitutions in dynamics, arithmetics and combinatorics, volume 1794 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002. Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. [12] W. Goldman. Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. In Handbook of Teichmüller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pages 611–684. Eur. Math. Soc., Zürich, 2009. [13] William M. Goldman and John J. Millson. The deformation theory of representations of fundamental groups of compact Kähler manifolds. Inst. Hautes Études Sci. Publ. Math., (67):43–96, 1988. [14] F. González-Acuña and José Marı́a Montesinos-Amilibia. On the character variety of group repre- sentations in SL(2, C) and PSL(2, C). Math. Z., 214(4):627–652, 1993. [15] Michael Heusener, Vicente Muñoz, and Joan Porti. The SL(3, C)-character variety of the figure eight knot. Illinois J. Math., 60(1):55–98, 2016. [16] Michael Heusener and Joan Porti. The scheme of characters in SL2 : Sage worksheet, 2022. http: //math.univ-bpclermont.fr/~heusener/traces/SL(2,C)-character_variety.html. [17] C. Hodgson. Degeneration and regeneration of hyperbolic structures on three-manifolds. Thesis, Princeton University, 1986. [18] Michael Kapovich and John J. Millson. On representation varieties of 3-manifold groups. Geom. Topol., 21(4):1931–1968, 2017. [19] Alexander Lubotzky and Andy R Magid. Varieties of representations of finitely generated groups. Mem. Amer. Math. Soc., 58(336):xi+117, 1985. [20] Wilhelm Magnus. Rings of Fricke characters and automorphism groups of free groups. Math. Z., 170(1):91–103, 1980. [21] Julien Marché. Character varieties in SL2 and Kauffman skein algebras. In Topology, Geometry and Algebra of low-dimensional manifolds, number 1991 in RIMS Kôkyûroku, pages 27–42, Apr 2016. [22] Hideyuki Matsumura. Commutative algebra, volume 56 of Mathematics Lecture Note Series. Ben- jamin/Cummings Publishing Co., Inc., Reading, Mass., second edition, 1980. [23] J. S. Milne. Algebraic groups, volume 170 of Cambridge Studies in Advanced Mathematics. Cam- bridge University Press, Cambridge, 2017. [24] J.S. Milne. A Primer of Commutative Algebra, expository notes edition, March 2020. https://www.jmilne.org. [25] Go Miura and Sakie Suzuki. The skein algebra of the Borromean rings complement. Internat. J. Math., 33(8):Paper No. 2250049, 28, 2022. [26] David Mumford. Algebraic geometry. I. Classics in Mathematics. Springer-Verlag, Berlin, 1995. [27] David Mumford. The red book of varieties and schemes, volume 1358 of Lecture Notes in Mathe- matics. Springer-Verlag, Berlin, 1999. [28] P. E. Newstead. Introduction to moduli problems and orbit spaces, volume 51 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi, 1978. [29] Joan Porti. Regenerating hyperbolic and spherical cone structures from Euclidean ones. Topology, 37(2):365–392, 1998. [30] Joan Porti. Regenerating hyperbolic cone structures from Nil. Geom. Topol., 6:815–852, 2002. [31] Claudio Procesi. A formal inverse to the Cayley-Hamilton theorem. J. Algebra, 107(1):63–74, 1987. [32] Józef H. Przytycki and Adam S. Sikora. Skein algebra of a group. In Knot theory (Warsaw, 1995), volume 42 of Banach Center Publ., pages 297–306. Polish Acad. Sci. Inst. Math., Warsaw, 1998. [33] Józef H. Przytycki and Adam S. Sikora. On skein algebras and SL2 (C)-character varieties. Topology, 39(1):115–148, 2000. [34] S. Schanuel and X. Zhang. Detection of essential surfaces in 3-manifolds with SL2 -trees. Math. Ann., 320(1):149–165, 2001. [35] Adam S. Sikora. Character varieties. Trans. Amer. Math. Soc., 364(10):5173–5208, 2012. [36] Adam S. Sikora. Character varieties of abelian groups. Math. Z., 277(1-2):241–256, 2014. [37] Leona Sparaco. Character Varieties Of Knots And Links With Symmetries. ProQuest LLC, Ann Arbor, MI, 2017. Thesis (Ph.D.)–The Florida State University. [38] Eva Suárez Peiró. Poliedros de Dirichlet de 3-variedades cónicas y sus deformaciones. PhD thesis, Madrid, 2003. Tesis de la Universidad Complutense de Madrid, Facultad Matemáticas, Departa- mento de Geometrı́a y Topologı́a, leı́da el 22-05-1998. [39] William P. Thurston. The Geometry and Topology of Three-Manifolds. (Electronic version from 2002 available at http://www.msri.org/publications/books/gt3m/), 1980. 28 [40] André Weil. Remarks on the cohomology of groups. Ann. of Math. (2), 80:149–157, 1964. Université Clermont Auvergne, CNRS, Laboratoire de Mathématiques Blaise Pascal, F-63000 Clermont-Ferrand, France E-mail address: michael.heusener@uca.fr. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Cerdany- ola del Vallès, Spain, and Centre de Recerca Matemàtica (CRM) E-mail address: porti@mat.uab.cat 29