See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/367361297 SL(2, C) Scheme Processing of Singularities in Quantum Computing and Genetics Preprint · January 2023 DOI: 10.20944/preprints202301.0529.v1 CITATIONS 0 READS 23 4 authors: Some of the authors of this publication are also working on these related projects: Quantum information and symmetries in the biology codes View project MARCONI 2 View project Michel Planat Institut FEMTO-ST (Besançon) and Quantum Gravity Research (Los Angeles USA) 280 PUBLICATIONS   2,120 CITATIONS    SEE PROFILE Marcelo M. Amaral Quantum Gravity Research 63 PUBLICATIONS   248 CITATIONS    SEE PROFILE David Chester Quantum Gravity Research 42 PUBLICATIONS   217 CITATIONS    SEE PROFILE Klee Irwin Quantum Gravity Research 82 PUBLICATIONS   405 CITATIONS    SEE PROFILE All content following this page was uploaded by Michel Planat on 24 January 2023. The user has requested enhancement of the downloaded file. Citation: Planat, M.; Amaral, M.M.; Chester, D.; Irwin, K. SL(2,C) scheme processing of singularities in quantum computing and genetics. Axioms 2022, 1, 0. https://doi.org/ Received: Accepted: Published: Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2023 by the authors. Submitted to Axioms for possible open access publication under the terms and conditions of the Creative Commons Attri- bution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Article SL(2,C) scheme processing of singularities in quantum computing and genetics Michel Planat 1,*,† , Marcelo M. Amaral 2,† , David Chester 2,† and Klee Irwin 2,† 1 Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne-Franche-Comté, F-25044 Besançon, France 2 Quantum Gravity Research, Los Angeles, CA 90290, USA; Marcelo@QuantumGravityResearch.org (M.M.A.); DavidC@QuantumGravityResearch.org (M.M.A.); Klee@QuantumGravityResearch.org (K.I.) * Correspondence: michel.planat@femto-st.fr † These authors contributed equally to this work. Abstract: Revealing the time structure of physical or biological objects is usually performed thanks 1 to the tools of signal processing like the fast Fourier transform, Ramanujan sum signal processing 2 and many other techniques. For space-time topological objects in physics and biology, we propose a 3 type of algebraic processing based on schemes in which the discrimination of singularities within 4 objects is based on the space-time-spin group SL(2,C). Such topological objects possess an homotopy 5 structure encoded in their fundamental group and the related SL(2,C) multivariate polynomial 6 character variety contains a plethora of singularities somehow analogous to the frequency spectrum 7 in time structures. Our approach is applied to an Akbulut cork in exotic R4, to an hyperbolic model 8 of topological quantum computing based on algebraic surfaces and to microRNAs in genetics. Such 9 diverse topics reveal the manifold of possibilities of using the concept of a scheme spectrum. 10 Keywords: Finitely generated group; SL(2,C) character variety; algebraic surfaces; schemes; exotic 11 R4; topological quantum computing; microRNAs. 12 0. Introduction 13 In signal processing of a time series, the lines of the Fourier spectrum are described 14 by discontinuities. The approach may be generalized with number theory by taking 15 Ramanujan sums as the building blocks of the signal expansion, e.g. see [1]. 16 A more ambitious approach is to use algebraic geometry to reveal the singularities of 17 an object. In recent papers, we explored topics of quantum computing and DNA biology 18 with a common algebraic geometrical tool that we now call SL(2,C) scheme processing. 19 The necessary ingredient is a finitely generated group π expressing the symmetries of 20 the investigated object. The SL(2,C) character variety associated to π is determined and 21 summarized by its Groebner basis G. The multivariate polynomials in G may contain 22 isolated (or non-isolated) singularities that characterize the richness of the object. In this 23 essay, the polynomials are reduced to surfaces living in the 3-dimensional projective space 24 P3(Q) over the rationals. In this way, we may use tools of schemes for the resolution of 25 singularities [2] that are implemented in the software Magma [3]. 26 Scheme theory was created by A. Grothendieck to generalize smooth manifolds to 27 algebraic varieties possibly decorated with singularities. For our purpose, it is enough to 28 see a scheme as a geometrical object defined by the vanishing of polynomials defined over 29 an affine (or a projective space) like those living in G. 30 In Section 1, we briefly describe the mathematical formalism used in our paper. It 31 includes the definition of the character variety V representing the finitely generated group 32 π over the group SL(2,C) and how a Groebner basis G is obtained from V in practice. 33 The section mentions the distinction between simple singularities and singularities whose 34 support is not zero dimensional. We then investigate the algebraic geometry of three 35 types of complex objects in physics and biology. The first two objects rely on quantum 36 Version January 20, 2023 submitted to Axioms https://www.mdpi.com/journal/axioms Version January 20, 2023 submitted to Axioms 2 of 11 computing following our previous papers [4,5]. Section 2 discusses the symmetries and 37 related representations of the Akbulut cork W, a fundamental object in the theory of exotic 38 4-manifolds. Section 3 refers to the symmetries and the related representations of an 39 hyperbolic 3-manifold found in the context of magic states in quantum computing. Section 40 4 investigates a third class of objects, which are a family of biological molecules called 41 microRNAs that regulate the types and amounts of proteins [6]. In Section 5, we provide 42 commentary on our results. 43 1. Theory 44 Details about the theory are described below. The corresponding implementation is 45 available on the Magma software [3]. 46 1.1. The SL(2,C) character variety of a finitely generated group and a Groebner basis 47 Let fp be a finitely generated group, we describe the representations of fp in the 48 (double cover alias the group extension of order two of the Lorentz group) SL(2,C), the 49 group of (2× 2) matrices with complex entries and determinant 1. The group SL(2,C) may 50 be seen simultaneously as a ‘space-time’ (a Lorentz group) and a ‘quantum’ (a spin) group. 51 Representations of fp in SL(2,C) are homomorphisms ρ : fp → SL(2,C) with char- 52 acter κρ(g) = tr(ρ(g)), g ∈ fp. The set of characters allows to define an algebraic set by 53 taking the quotient of the set of representations ρ by the group SL(2,C), which acts by 54 conjugation on representations [7]. 55 Such an algebraic set is called the SL(2,C) character variety of fp. It is made of a 56 sequence of multivariate polynomials called a scheme X. The vanishing of polynomials 57 defines the ideal I(X) of the scheme X. A Groebner basis G(X) is a particular set of the 58 polynomial ring I(X) that has to follow algorithmic rules (similar to the Euclidean division 59 for univariate polynomials). 60 For the effective calculations of the character variety, we make use of a software on 61 Sage [8]. We also need Magma [3] for the calculation of a Groebner basis, at least for 3- and 62 4-letter sequences. 63 1.2. Singularities of an algebraic surface 64 Simple singularities 65 The surfaces S of interest in this case are said to be almost not singular in the sense 66 that they have at worst simple singularities. In Magma, it is referred to a simple or A-D-E 67 singularity if it is an isolated singularity on S which is analytically of the type An, n ≥ 1,Dn, 68 n ≥ 4, E6, E7 or E8. 69 The A-D-E type and the number of simple singularities is reflected in our notation. E. 70 g., S(lA1) means a surface with l singularities of type A1, S(A2) means a surface with a single 71 singularity of type A2 and S(D4) means a surface with a single singularity of type D4 . The 72 Cayley cubic encountered in our previous paper is κ(4A1) 4 (x, y, z) = xyz + x2 + y2 + z2 − 4 73 [5]. The Fricke surface of type D4 is S(D4) = xyz + x2 + y2 + z2 − 8(x + y + z) + 28 [9, 74 Figure 16]. Many other examples can be found in [6]. 75 The relevant Magma command for such simply singular surfaces is HasOnlySimpleSin- 76 gularities(S). 77 1.3. Arbitrary singularities 78 Let X ∈ P3(Q) be a singular surface (a surface containing a non zero dimensional 79 singular subset of points). Let Y ∈ P3(Q) be a regular surface (devoid of singularities) 80 above X that is, a representation (in a sense to be qualified) ρ : Y → X. The set of such 81 morphisms ρ belongs to the so-called spectrum SpecX(Y) of Y in X. 82 In our context, the formal desingularization of an (hyper)surface X in P3(Q) is realized 83 with a proper birational map Y → X, with Y is regular. The formal desingularization is 84 realized with the introduction of a formal prime divisor Spec ÔX,p → X, where p ∈ X is a 85 Version January 20, 2023 submitted to Axioms 3 of 11 regular point of codomension 1, OX,p is the structure sheaf at p of X seen as a scheme and 86 the hat means the completion [10]. 87 Taking the affine surface S(x, y, z), the related homogeneous polynomial defines a 88 hypersurface X(x, y, z, w) ∈ P3(Q) so that a formal prime divisor is actually a morphism 89 ϕ : Specϕ[[t]] → X defined by the Q-algebra homomorphism ϕ♯ : Q([x, y, z, w]/ < X >→ 90 Fϕ[[t]] where Fϕ = Q(s)[α], s a parameter and α is defined by the vanishing of a minimal 91 polynomial. 92 The relevant Magma command for this case is FormallyResolveProjectiveHyperSurface(S: 93 AdjComp := true). Thanks to the setting AdjComp := true, the command returns the number 94 of essential singularities allowing to obtain the formal divisors needed to only compute 95 birational invariants or adjoint spaces [10,11]. Figure 1. The affine singular surface S2(x, y, z) = z4 + 2yz3 + x2 − 6yz− 2x − 8 found in the Groebner basis for the transcription factor Prdm1 [6, Section 3.1]. 96 1.4. Kodaira-Enriques classification 97 Given an ordinary projective surface S in the projective space P3 over a number field, 98 if S is birationally equivalent to a rational surface, the software Magma [3] determines the 99 map to such a rational surface and returns its type within five categories. The returned 100 type of S is P2 for the projective plane, a quadric surface (for a degree 2 surface in P3), a 101 rational ruled surface, a conic bundle or a degree p Del Pezzo surface where 1 ≤ p ≤ 9. 102 A further classification may be obtained for S in P3 if S has at most point singularities 103 (unless the singularities may be formally resolved as we described in the previous sub- 104 section for the case of characteristic zero). Magma computes the type of S (or rather, the 105 type of the non-singular projective surfaces in its birational equivalence class) according 106 to the classification of Kodaira and Enriques [12]. The first returned value is the Kodaira 107 dimension of S, which is −∞, 0, 1 or 2. The second returned value further specifies the type 108 within the Kodaira dimension −∞ or 0 cases (and is irrelevant in the other two cases). 109 Kodaira dimension −∞ corresponds to birationally ruled surfaces. The second return 110 in this case is the irregularity q ≥ 0 of S. So S is birationally equivalent to a ruled surface 111 over a smooth curve of genus q and is a rational surface if and only if q is zero. 112 Kodaira dimension 0 corresponds to surfaces which are birationally equivalent to a K3 113 surface, an Enriques surface, a torus or a bi-elliptic surface. 114 Every surface of Kodaira dimension 1 is an elliptic surface (or a quasi-elliptic surface 115 in characteristics 2 or 3), but the converse is not true: an elliptic surface can have Kodaira 116 dimension −∞, 0 or 1. 117 Surfaces of Kodaira dimension 2 are algebraic surfaces of general type. 118 A singular surface 119 Let us illustrate our approach with a selected singular surface encountered in the 120 context of the transcription factor Prdm1 in our recent paper [6, Section 3.1]. The affine 121 surface under question is S2(x, y, z) = z4 + 2yz3 + x2 − 6yz − 2x − 8. It contains 9 essential 122 Version January 20, 2023 submitted to Axioms 4 of 11 singularities in the desingularization. A 3-dimensional plot of S2(x, y, z) is in Figure 1. The 123 surface is a conic bundle of K3 type. 124 The projective surface is S2(x, y, z, w) = z4 + 2yz3 +w2(x2 − 6yz)− 2xw3 − 8w4. There 125 exists 5 distinct types of minimal polynomial. For one of the divisors, the minimal poly- 126 nomial is α2 + 2s + 1 and the prime divisor in the desingularization is the morphism 127 ϕ : Specϕ[[t]] → X defined by 128 x → 1, y → st, z → t, w → αt2 + 0(t4). The 9 formal morphisms are used to compute the global sections (or adjoints) Homi,j. 129 For i, j < 3, one obtains Hom1,1 = Hom2,1 = Hom2,2 = Hom2,3 = Hom3,1 = Hom3,2 = 130 Hom3,3 = 0, Hom1,2 = [z, w], Hom1,3 = [xz, yz, z2, xw, yw, zw, w2]. 131 2. SL(2,C) scheme processing of an Akbulut cork 132 This example belongs to the theory of 4-manifolds and was investigated by the authors 133 in the context of magic states of quantum computing [4]. To our earlier work we add the 134 investigation of SL(2,C) character variety of the relevant fundamental groups. In particular, 135 we put the central object of the exotic R4 manifolds –the Akbulut cork– in a new perspective, 136 by revealing its singularity spectrum. 137 The theory of 4-manifolds is described in books [13–15]. Here we are interested in the 138 decomposition of a 4-manifold into one- and two-dimensional handles as shown in Fig. 2 139 [13, Fig. 1.1 and Fig. 1.2]. Let Bn and Sn be the n-dimensional ball and the n-dimensional 140 sphere, respectively. An observer is placed at the boundary ∂B4 = S3 of the 0-handle B4 141 and watch the attaching regions of the 1- and 2-handles. The attaching region of 1-handle 142 is a pair of balls B3 (the yellow balls), and the attaching region of 2-handles is a framed 143 knot (the red knotted circle) or a knot going over the 1-handle (shown in blue). For closed 144 4-manifolds, there is no need of visualizing a 3-handle since it can be directly attached to 145 the 0-handle. The 1-handle can also be figured out as a dotted circle S1 × B3 obtained by 146 squeezing together the two three-dimensional balls B3 so that they become flat and close 147 together [14, p. 169] as shown in Fig. 2b. 148 For the attaching region of a 2- and a 3-handle one needs to enrich our knowledge by 149 introducing the concept of an Akbulut cork to be described in the next paragraph [4, Figure 150 3]. 151 Figure 2. (a) Handlebody of a 4-manifold with the structure of 1- and 2-handles over the 0-handle B4, (b) the structure of a 1-handle as a dotted circle S1 × B3, (c) an Akbulut cork W = 946(−1, 1). 2.1. Akbulut cork 152 A Mazur manifold is a contractible, compact, smooth 4-manifold (with boundary) not 153 diffeomorphic to the standard 4-ball B4 [13]. Its boundary is a homology 3-sphere. If we 154 restrict to Mazur manifolds that have a handle decomposition into a single 0-handle, a 155 Version January 20, 2023 submitted to Axioms 5 of 11 single 1-handle and a single 2-handle then the manifold has to be of the form of the dotted 156 circle S1 × B3 (as in Fig. 2b) (right) union a 2-handle. The simplest object of this type is the 157 Akbulut cork shown in Fig. 2c [16,17]. 158 Given p, q, r (with p ≤ q ≤ r), the Brieskorn 3-manifold Σ(p, q, r) is the intersection 159 in the complex 3-space C3 of the 5-dimensional sphere S5 with the surface of equation 160 zp1 + z q 2 + z r 3 = 0. The smallest known Mazur manifold is the Akbulut cork W and its 161 boundary is the Brieskorn homology sphere Σ(2, 5, 7). The Akbulut cork has a simple 162 definition in terms of the framings ±1 of (−3, 3,−3) pretzel knot also called K = 946 [18, 163 Fig. 3]. It has been shown that ∂W = Σ(2, 5, 7) = K(1, 1) and W = K(−1, 1). 164 An exotic R4 is a differentiable manifold that is homeomorphic but not diffeomorphic 165 to the Euclidean space R4. An exotic R4 is called small if it can be smoothly embedded as 166 an open subset of the standard R4 and is called large otherwise. Here we are concerned 167 with an example of a small exotic R4. 168 According to [18], there exists an involution f : ∂W → ∂W that surgers the dotted 169 1-handle S1 × B3 to the 0-framed 2-handle S2 × B2 and back, in the interior of W. There 170 exists a smooth contractible 4-manifold V with ∂V = ∂W, such that V is homeomorphic 171 but not diffeomorphic to W relative to the boundary [16, Theorem 1]. This leads us to our 172 next paragraph. 173 2.2. The manifold W̄ mediating the Akbulut cobordism between exotic manifolds V and W 174 A cobordism between two oriented m-manifolds M and N is any oriented (m + 1)- 175 manifold W0 such that the boundary is ∂W0 = M̄ ∪ N, where M appears with the reverse 176 orientation. The cobordism M × [0, 1] is called the trivial cobordism. Next, a cobordism W0 177 between M and N is called an h-cobordism if W0 is homotopically like the trivial cobordism. 178 The h-cobordism due to S. Smale in 1960, states that if Mm and Nm are compact simply- 179 connected oriented M-manifolds that are h-cobordant through the simply-connected (m + 180 1)-manifold Wm+1 0 , then M and N are diffeomorphic [15, p. 29]. 181 However this theorem fails in dimension 4. If M and N are cobordant 4-manifolds, 182 then N can be obtained from M by cutting out a compact contractible submanifold W and 183 gluing it back in by using an involution of ∂W. 184 The h-cobordism under question in our example may be described by attaching an 185 algebraic cancelling pair of 2- and 3-handles to the interior of Akbulut cork W. The 4- 186 manifold W̄ mediating V and W resembles the Akbulut cork with the dot replaced by a 187 0-surgery The manifold under question is nothing but L7a6(0, 1)(0, 1)] (see [16, p. 355] or 188 [4, Figure 3c]). 189 2.3. The character variety for an Akbulut cork W 190 With Snappy, we find that the fundamental group ruling the Akbulut cork W = 191 946(−1, 1) is the two-generator group 192 π1(W) = 〈 a, b|aBAb2 ABabaBABab, a3BAb3 AB 〉 , A = a−1, B = b−1. (1) With Sage software in Reference [8], we compute the corresponding character variety. 193 Then, from Magma [3], the Groebner basis is found in the form 194 Version January 20, 2023 submitted to Axioms 6 of 11 Figure 3. The surface SW(x, y, z) found within the Groebner basis of the SL(2,C) character variety for the Akbulut cork W. GW(x, y, z) = z(z − 2)[x + y + f1(z)][y2 − y + f2(z][y + f3(z)] f4(z), f1(z) = −171/34 z8 + 913/34 z7 − 2429/34 z6 + 4733/34 z5 − 2443/17 z4 −224/17 z3 + 1847/17 z2 − 1103/34 z − 1, f2(z) = −95/34 z8 + 477/34 z7 − 1221/34 z6 + 2331/34 z5 − 1089/17 ∗ z4 −285/17 z3 + 858/17 z2 − 337/34 z − 1 f3(z) = −71/17 z7 + 376/17 z6 − 997/17 z5 + 1942/17 z4 − 1993/17 z3 −188/17 z2 + 1477/17 z − 436/17, f4(z) = z7 − 5 z6 + 13 z5 − 25 z4 + 24 z3 + 4 z2 − 18 z + 5. (2) The factor containing f1(z) in Equation 3 is the singular surface SW(x, y, z) shown in 195 Figure 3. It is a rational scroll and a surface of a general type. The factor containing f2(z) 196 is an hyperelliptic function of genus 7, discriminant ≈ 1.327502101 and points at infinity 197 (1, 0, 0) and (1,−95/34, 0). The factor containing f3(z) is an ordinary curve. The factor 198 containing f4(z) is a seventh-order polynomial. 199 Formal desingularization of the surface SW(x, y, z) 200 The formal desingularization of a hypersurface X in the three-dimensional projective 201 space P3Q over the rationals Q is described in [10] and can be explicitely given with Magma 202 [3, Section 122.5.3]. 203 To the surface X = SW(x, y, z) we associate the degree 8 homogeneous polyno- 204 mial SW(x, y, z, w) = x + y + f1(z, w) ∈ Q(x, y, z, w) , with f1(z, w) = −171/34z8 + 205 913/34 wz7 + · · · − 1103/343 w7z − w8. 206 For the projective surface X = SW(x, y, z, w), the two essential (over the three) singular 207 morphisms are 208 x → 1 y → st − 1 z → t w → αt + O(t8), (3) Version January 20, 2023 submitted to Axioms 7 of 11 209 x → 1 y → s z → 34/171 (s + 1)t7 + O(t8) w → O(t8), (4) where α has minimal polynomial −s − α8 f1(α−1). 210 These formal morphisms are used to compute the global sections (or adjoints) Homi,j. 211 For the surface SW(x, y, z) one gets 212 Hom1,1 = Hom2,1 = Hom2,2 = 0, Hom1,2 = [z6, z5w, z4w2, z3w3, z2w4, zw5, w6] · · · 213 2.4. The character variety for the mediating manifold W̄ 214 The fundamental group π1(W̄) of the h-cobordism W̄ is as follows [4] 215 π1(W̄) = 〈 a, b|a3b2 AB3 Ab2, (ab)2aB2 Ab2 AB2 〉 . (5) The cardinality structure of subgroups of this fundamental group is 216 ηd[π1(W̄)] = ηd[π1(W)] = [1, 0, 0, 0, 1, 1, 2, 0, 0, 1, 0, 5, 4, 9, 7, 1 · · · ], where the bold digits refers to our investigation in [4, Table 1] about the existence 217 of a finite geometry obtained with a subgroup of the corresponding index. The smallest 218 case occurs at index 10 and the geometry is the Mermin pentagram, a type of contextual 219 geometry. 220 As before, with Sage software in Reference [8], we compute the corresponding charac- 221 ter variety. Then, from Magma, the Groebner basis is found in the form 222 GW̄ = (z − 2)(z2 − z − 1)[x + g1(z)][y + g2(z)]g3(z), g1(z) = −z9 + 3z8 + 5z7 − 18z6 − 10z5 + 43z4 − 2z3 − 27z2 + 3z + 4, g2(z) = −z8 + 2z7 + 6z6 − 11z5 − 15z4 + 24z3 + 9z2 − 10z − 2, g3(z) = z7 + z6 − 5z5 − 6z4 + 6z3 + 5z2 − 2z − 1. (6) Although the card seq for manifolds W and W̄ are the same, we clearly see that 223 the character variety are distinct. In the later case, it contains two ordinary curves and 224 polynomials of degree 1, 2 and 7. 225 3. SL(2,C) scheme processing in topological quantum computing 226 In this section, we are interested in the SL(2,C) character variety of the fundamental 227 group of (hyperbolic) 3-manifold L10n46 (alias otet0800002). This manifold describes the 228 4-fold (irregular) covering of the figure-eight knot 41 = K4a1 [19, Table 2]. It is connected 229 to the magic state describing the contextual geometry of two-qubits (e.g. [19, Figure 1] or 230 [20, Table 1]). The basic idea has been to put magic states and informationally complete 231 POVMs under the same hat [21]. Thanks to the character variety and the related algebraic 232 surfaces , we can add the topological aspect to the previous description. 233 The Groebner basis of the character variety for the fundamental group of 3-manifold 234 L10n46 may be obtained with Magma. The 3-generator fundamental group is 235 π1(L10n46) = ⟨a, b, c|abAcBac, abbCBccBBBACbcbC⟩. (7) Version January 20, 2023 submitted to Axioms 8 of 11 The character variety GL10n46(e, f , g, h)(x, y, z) is now seven-dimensional as in [6] 236 and selected choices of parameters e, f , g and h allow us to determine the algebraic surfaces 237 within the Groebner basis. 238 For instance, we find 239 GL10n46(0, 0, 0, 0) = yS1(x, y, z)S2(x, y, z)S3(x, y, z) · · · S1(x, y, z) = f (A1) 2,{} (x, y, z) = xyz + xz 2 + x2 + y2 + z2 + yz − x − 6, S2(x, y, z) = xz2 + yz − x − 2, S3(x, y, z) = −x2z2 + 2x2 + y2 + z2 + 2x − 4. (8) Figure 4. Left: the (degree 3) del Pezzo surface S1(x, y, z) = f (A1) 2,{} (x, y, z) = xyz + xz 2 + x2 + y2 + z2 + yz − x − 6 . Middle: the (rational scroll) surface S2(x, y, z) = xz2 + yz − x − 2 . Right: the (del Pezzo degree 4) surface S3(x, y, z) = −x2z2 + 2x2 + y2 + z2 + 2x − 4. The first surface S1(x, y, z) = f (A1) 2,{} in the product (8) has a single simple singularity of 240 type A1 whose reduced singular subscheme is of degree 2 with a vanishing support. 241 The second surface S2(x, y, z) is a singular rational scroll with a single essential singu- 242 larity. The corresponding singular morphism Spec Fϕ[[t]] → X = S2(x, y, z, w) is 243 x → 1 y → s z → t w → ((−2s3 − 6s)/(s2 + 4)α+ (−2s2 − 4)/(s2 + 4))αt + O(t3), (9) where α has minimal polynomial α2 − sα− 1. 244 The third surface S3(x, y, z) = −x2z2 + 2x2 + 2x + y2 + z2 − 4 is a singular surface of 245 the degree 4 del Pezzo type. There are 4 essential singularities. We retain one of the two 246 singularities with a nontrivial minimal polynomial. The corresponding singular morphism 247 Spec Fϕ[[t]] → X = S1(x, y, z, w) is 248 x → 1 y → s z → t w → αt − 1/(s4 + 4s2 + 4)t2 + (−1/2 s4 + 9/2)/(s6 + 6s4 + 12s2 + 8)αt3) + O(t5) (10) where α has minimal polynomial α2 − 1/(s2 + 2). 249 All three affine surfaces are shown in Figure 4. 250 Version January 20, 2023 submitted to Axioms 9 of 11 4. SL(2,C) scheme processing in microRNAs 251 In this section, we focus on the SL(2,C) character varieties attached to microRNAs 252 (miRNAs for short). This case was already tackled in our recent paper [6]. 253 The miRNAs are short (approximately 22 nt long) single-stranded RNA molecules 254 playing a fundamental role in the expression and regulation of genes by targeting specific 255 messenger RNAs (mRNAs) for degradation or translational repression. The genes encoding 256 miRNAs are much longer than the processed mature miRNA molecule. There are pre- 257 miRNAs comprising of approximately 70-nucleotides in length which are folded into 258 imperfect stem-loop structures. 259 Each miRNA is synthesized as an miRNA duplex comprised of two strands (-5p and - 260 3p). However, only one of the two strands becomes active, which is selectively incorporated 261 into the RNA-induced silencing complex in a process known as miRNA strand selection 262 [22,23]. For details about the miRNA sequences, we use the Mir database [24,25]. 263 Disregulation of miRNAs may lead to a disease like cancer. A key microRNA known 264 as an oncomir (involved in immunity and cancer) is mir-155 [6]. 265 Here, we select examples of human miRNAs from the perspective of evolution. The 266 generator of the group π to be considered is a short (about 8-letter long) seed made of two to 267 four distinct bases in the set {A,U,G,C}. Most of the time, π is close to a free group of rank 268 equal the number of distinct bases in the seed minus one. This point can be checked by the 269 cardinality structure of conjugacy classes subgroups of π (denoted card seq). Exceptions to 270 this rule and the occurrence of singularities (isolated or not) in the corresponding character 271 variety built from π, is a witness of a potential disease. Unlike the case of transcription 272 factors, until now, singularities possibly found with microRNAs are isolated singularities. 273 According to Reference [26, Table 3], the slowest evolving miRNA gene is hsa-mir-503 274 (the notation hsa is for the human specie). It is known that mir-503 regulates gene expression 275 from different aspects of pathological processes of diseases, including carcinogenesis, 276 angiogenesis, tissue fibrosis and oxidative stress [27]. The seed region for mir-503-5p is 277 AGCAGCGG and the corresponding Groebner basis for parameters (e, f , g, h) = (0, 0, 0, 0) 278 is very simple: Gmir−503−5p(0, 0, 0, 0) = κ (4A1) 4 (x, y, z), as shown in Figure 5 (Left). 279 For (e, f , g, h) = (1, 1, 0, 0), Gmir−503−5p(1, 1, 0, 0) = −3xyzκ3(x, y, z), with κ3(x, y, z) is 280 the Fricke surface found in [5, Section 3.3]. For (e, f , g, h) = (1, 1, 1, 1), there are many more 281 polynomials. One of them defines the Fricke surface xyz + x2 + y2 + z2 − 2x − y − 2. The 282 considered seed region for mir-503-3p is GGGUAUU. The surfaces in the Groebner basis 283 are very simple in this case and not even simple singularities lie in them. 284 One of the fastest evolving microRNA is mir-214 [26, Table 3]. First mir-214 was 285 reported to promote apoptosis in HeLa cells. Presently, mi-214 is implicated in an extensive 286 range of conditions such as cardiovascular diseases, cancers, bone formation and cell 287 differentiation [28]. For mir-214-5p and the seed sequence GCCUGU, one finds the surface 288 f (A2) 1,{1:0:0:0}(x, y, z) = xyz + y 2 + z2 − 4 within Gmir−214−5p(0, 0, 0, 0), as shown in Figure 289 5. A surface of the same type f (A2) 1,{1:0:0:0}(x, y, z) = xyz + y 2 + z2y − 2z − 1 is found in 290 Gmir−214−5p(1, 1, 1, 1). For a longer seed, surfaces are not found to contain singularities. 291 5. Conclusion 292 Over the last few years, the authors of this article have found that some mathematical 293 techniques employed for quantum information processing and quantum computing may 294 also apply to biology at the genome scale. More precisely, group theory and representations 295 of symmetries with characters of finite groups have been used for topological quantum 296 computing (TQC) [19] or elementary particles [29], and the the encoding of proteins [30]. 297 Methods for dealing with infinite groups and SL(2,C) representations of such groups in 298 TQC papers [5,31] were similarly employed for transcription factors [6] and miRNAs [6]. 299 Our efforts in this paper are belonging to the field of scheme processing, where 300 the desingularization of discontinuities of algebraic surfaces is a spectrum. The scheme 301 spectrum is a well known concept in commutative algebra. The prime spectrum of a ring 302 Version January 20, 2023 submitted to Axioms 10 of 11 Figure 5. Left: the Cayley cubic κ4(x, y, z) found in the character variety for the slowest evolving miRNA gene hsa-mir-503. The surface f (A2) 1,{1:0:0:0}(x, y, z) found in the character variety Gmir−214−5p of the fast evolving gene hsa-mir-214. is the set of prime ideals of the ring R and denoted by Spec(R). The sheaf of rings O 303 is the relevant algebraic geometrical notion introduced by Grothendieck to develop this 304 field [2,32]. We touched this important concept of schemes while investigating non-zero 305 dimensional sets of singularities in surfaces belonging to the SL(2,C) character variety of 306 an infinite group. We are fortunate that the software Magma is designed to implement 307 schemes in a variety of applications (curves, surfaces and more). Another computer algebra 308 system with similar facilities is Singular [33]. 309 We have no doubt that scheme theory will play and increasing role in our future efforts 310 of relating quantum concepts and biology. In the future, this may be applicable to the field 311 of ‘quantum consciousness’ [34]. 312 Author Contributions: “Conceptualization, M.P.; methodology, M.P and M.A.; software, M.P.; 313 validation, M.A., D.C. and K.I.; formal analysis, D.C., M.P. ; investigation, D.C., M.A., M.P. ; resources, 314 K.I.; data curation, M.P.; writing—original draft preparation, M.P.; writing—review and editing, 315 M.P. and M.A.; visualization, M.A.; supervision, M.P.; project administration, M.P., K.I.; funding 316 acquisition, K.I. All authors have read and agreed to the published version of the manuscript”. 317 Funding: “This research received no external funding” 318 Informed Consent Statement: “Not applicable” 319 Data Availability Statement: “Data are available from the authors after a reasonable demand” 320 Conflicts of Interest: “The authors declare no conflict of interest” 321 MDPI Multidisciplinary Digital Publishing Institute 322 References 323 1. Planat, M.; Minarovjech, M.; Saniga, M. 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