On discriminants, Tjurina modifications and the geometry of determinantal singularities
aa r X i v : . [ m a t h . AG ] M a y On discriminants, Tjurina modifications and thegeometry of determinantal singularities
Anne Fr¨uhbis–Kr¨uger ∗ Institut f. Alg. Geometrie,Leibniz Universt¨at Hannover, GermanyMay 5, 2017
Abstract
We describe a method for computing discriminants for a large class offamilies of isolated determinantal singularities – more precisely, for sub-families of G -versal families. The approach intrinsically provides a decom-position of the discriminant into two parts and allows the computation ofthe determinantal and the non-determinantal loci of the family withoutextra effort; only the latter manifests itself in the Tjurina transform. Thisknowledge is then applied to the case of Cohen-Macaulay codimension 2singularities putting several known, but previously unexplained observa-tions into context and explicitly constructing a counterexample to Wahl’sconjecture on the relation of Milnor and Tjurina numbers for surface sin-gularities. Isolated hypersurface and complete intersection singularities are well studiedobjects and there are many classical results about different ascpects such astopology, deformation behaviour, invariants, classification and even metric prop-erties (see any textbook on singularities, e.g. [21], [18], [20]). Beyond completeintersections, however, knowledge is rather scarce and unexpected phenomenaarise. In this artcle, we focus on the class of determinantal singularities to passbeyond ICIS, as the properties already differ significantly, but classical resultson determinantal varieties and free resolutions provide strong tools to treat thiscase. Recently, significant progress has been made for this class, e.g. in [26],[2], [24], [8], [17]. In [15] the use of Tjurina modifications made it possible torelate a given determinantal singularity to an often singular variety, which hap-pens to be an ICIS under rather mild conditions. This could e.g. be exploitedin [15] and [32] to determine the topology of the Milnor fibre of an isolated ∗ partially supported by DFG priority program SPP 1489 ’Algorithmic and Experimen-tal Methods in Algebra, Geometry and Number Theory’ and by NTH-grant ’ExperimentalMethods in Computer Algebra’ V ( I t ) is based on the Jacobian criterion. It involves the elimination ofthe original variables from the ideal generated by I t and the ideal of minors ofappropriate size of the relative Jacobian matrix of I t . However, the complexityof this approach, which originates from the sensitivity of Gr¨obner basis com-putations to the number of occurring variables, makes it impractical for manyexamples. It is hence important to understand the structure of the discriminanttheoretically and to be able to decompose it appropriately by a priori argu-ments. Making use of Hironaka’s smoothness criterion [19], the structure of theperturbed matrix can be used to split the problem into two smaller problems,one dealing with the locus of determinantal singularities, the other one closelyrelated to the Tjurina transform as it describes the locus above which there aresingularities adjacent to an A singularity.In section 2, we first recall known facts about determinantal singularitiesand then proceed to revisit Hironaka’s smoothness criterion. In the followingsection 3, we consider the discriminant of versal families of determinantal sin-gularities of type ( m, n, t ) starting with the simplest case (2 , k, Singular [9].
Before focussing on the discriminant, we shall first recall the definition of prop-erties of the class of singularities on which we focus in the following sections:essentially isolated determinantal singularities (EIDS) as first introduced byEbeling and Gusein-Zade in [10]. Although certain classes of such singularitieshad been studied before (e.g. [13], [14] and [5]), simultaneous studies of cer-tain properties of EIDS of all matrix sizes and types only appear recently e.g.in [2] and [24]. In this section, we cover well-known facts about EIDS to givethe reader the background knowledge for the subsequent considerations on thediscriminant and the Tjurina transoform.
Definition 2.1
Let M m,n denote the set of all m × n -matrices with entries in C and let ≤ t ≤ min { m, n } . Then M tm,n := { A ∈ M m,n | rk ( A ) < t } is the generic determinantal variety. Remark 2.2 M tm,n can be understood as the variety in M m,n ∼ = C mn which isthe vanishing locus of the ideal of t -minors of the matrix x . . . x n ... ... x ( m − n +1 . . . x mn ∈ Mat( m, n, C [ x , . . . , x mn ]) .M tm,n has codimension ( n − t + 1)( m − t + 1) in M m,n and Sing( M tm,n ) = M t − m,n .Moreover, it is known that the sets M im,n \ M i − m,n for ≤ i ≤ t form a Whitneystratification for M tm,n . Remark 2.3
By a result of Ma [23] extending several previous results for max-imal, for submaximal and for -minors (see e.g. [6], [1]), the first syzygy moduleof the ideal of M tm,n is generated by linear relations. As we are interested in minors of matrices with arbitrary power series asentries, the generic determinantal varieties are not the objects of our primaryfocus. They are only a tool to formulate the following definition:
Definition 2.4
Let F : C k −→ M m,n ∼ = C mn be a polynomial map. Then X = F − ( M tm,n ) is a determinantal variety of type ( m, n, t ) , if codim( X ) =( m − t + 1)( n − t + 1) in C k . Passing to the convergent power series ring, we can also define and use these notions inthe setting of analytic space germs. Analogously all of the subsequent notions can be carriedover to space germs and the analytic setting. emark 2.5 By a well-known result of Eagon and Hochster [11], the conditionon the codimension ensures that the local ring of a germ of a determinantalvariety is Cohen-Macaulay.The same condition on the codimension also implies that all syzygies arise fromthe linear ones of the generic matrix (by the extension of scalars via the mapof local rings induced by F ). As a consequence, the entries of the syzygy matrixare linear combinations of the components F i,j of the map F . Definition 2.6 ([10])
A germ ( X, ⊂ ( C k , of a determinantal variety oftype ( m, n, t ) is called an EIDS at , if the corresponding F is transverse to allstrata M im,n \ M i − m,n of M tm,n outside the origin. Remark 2.7
For an EIDS ( X, defined by F − ( M tm,n ) the singular locus isprecisely the preimage of the singular locus of M tm,n , i.e. it is F − ( M t − m,n ) . Thus ( X, has an isolated singularity at the origin iff k ≤ ( m − t + 2)( n − t + 2) ;moreover, a smoothing of it exists if and only if the previous inequality is strict. As invertible row and column operations applied to a matrix do not changethe ideal of its minors, this holds for any matrix of an EIDS. Moreover, twosingularities should be considered equivalent, if one arises from the other bymeans of a coordinate change. These observations explain the structure of thegroup which describes the most suitable equivalence relation for determinantalsingularities:
Definition 2.8
Let G denote the group ( Gl m ( C { x } ) × Gl n ( C { x } )) ⋊ Aut( C { x } ) .Two determinantal singularities ( X , , ( X , ⊂ ( C k , of the same type ( m, n, t ) and defined by F and F respectively are called G -equivalent, if thereis a tuple ( R, L, Φ) ∈ G such that F = L − (Φ ∗ F ) R . Recall that a map germ is s -determined, if any other map which coincideswith it in all terms up to degree s is equivalent to it. If we want to stress theexistence of such an s without specifying the value of s , the map is referred toas finitely determined. Theorem 2.9 [25] An EIDS ( X, corresponding to a map F , defined by theideal of minors of the corresponding matrix (which we also denote by F byabuse of notation), is finitely G -determined if and only if it has finite G -Tjurinanumber τ G = dim C (Mat( m, n ; C { x } ) / ( J F + J op )) where J F denotes the submodule generated by the k matrices, each holding thepartial derivatives of the entries of F w.r.t. one of the variables, and J op = h AF + F B | A ∈ Mat( m, m ; C { x } ) , Mat( n, n ; C { x } ) i Remark 2.10
The group G is a subgroup of the group K of Mather and coin-cides with it e.g. for complete intersections and Cohen-Macaulay codimension singularities. It also appears as the subgroup K V of K in the literature, where is the generic determinantal singularity of the appropriate type (defined by M tm,n ). In the cases, where G -equivalence coincides with analytic equivalence, τ G is precisely the usual Tjurina number. In general, however, G is a propersubgroup of K as it respects the underlying matrix size. Thus there can e.g. beno element of G leading from one representation of Pinkham’s famous example[27] to the other, i.e. leading from a determinantal singularity of type (2 , , toone of type (3 , , (with the additional constraint of the matrix to be symmet-ric) or vice versa. It is important to observe that a restriction to G -equivalencedoes not fix the minimal size of the matrix, it only fixes some size, as any de-terminantal singularity of type ( m, n, t ) can easily be considered as one of type ( m + 1 , n + 1 , t + 1) by simply adding an extra line and an extra column ofwhich all entries are zero except the one where the row and column meet, whichshould then be chosen to be . If, on the other hand, a G -equivalence class of ( m + 1) × ( n + 1) -matrices contains a matrix of this particular structure , theclass will be referred to as essentially of type ( m, n, t ) . Keeping in mind, that the chosen equivalence can only be used to comparedeterminantal singularities of compatible types, we shall restrict our considera-tions to families for which each fibre is a determinantal singularity of appropriatetype. A similar restriction has already been used by Schaps in [28], where sheconsidered the notion of an M -deformation (or more generally an f -deformation)by deforming the entries of a given matrix M . There she gives a criterion whenan M -deformation is versal and provides examples of situations, in which itis not. However, the M -deformation of Schaps does not even provide a versalunfolding of the F as her example 1 shows, which she attributes to D. S. Rimwithout further reference: Example 2.11
Consider the determinantal singularity given by the -minorsof the matrix (cid:18) x αx βx γx x x x x (cid:19) ∼ G (cid:18) x sx tx x x x (cid:19) , where α, β, γ ∈ C sufficiently general (according to Schaps) or more precisely ( s : t ) ∈ P \ { (0 : 1) , (1 : 0) , (1 : 1) } , which immediately allows us to pass to theaffine chart t = 0 writing S for st . Note that two such matrices corresponding todifferent points in P \ { (0 : 1) , (1 : 0) , (1 : 1) } lead to non- G -equivalent matrices,as this would alter the cross-ratio of the four points (0 : 1) , (1 : 0) , (1 : 1) , ( s : t ) in P . The corresponding space germs, on the other hand, are isomorphic ina trivial way, as a change of e only manifests itself by multiplying some of theminors by invertible constants. τ G is in this example and a versal unfolding of the corresponding morphism F is given by (cid:18) x a Sx + b x + c + ex d x x x (cid:19) , where Schaps only accepts the deformation parameters a, b, c, d , but refuses e ,as it alters the original matrix. An interesting property of this family is that hanges to the parameter e only have an effect on the K -equivalence class ofthe corresponding space germ, if at least one of the other four parameters isnon-zero. Choosing a monomial C -basis m , . . . , m τ of the Mat( m, n ; C { x } ) / ( J F + J op ), it is then easy to write down a semiuniversal unfolding of the morphism F , by simply perturbing the corresponding matrix as follows: F t ,...,t τ = F + τ X i =1 t i m i . The corresponding family of space germs is also versal for determinantal de-formations of determinantal singularities in the following sense: Any familyof space germs with given determinantal singularity in the zero-fibre and onlydeterminantal singularities of appropriate type as fibres can be induced fromthe family of space germs described by F t ,...,t τ . To make notation a bit shorter,we call such a family of space germs G -versal.But the relation between the base of a G -versal family and a versal familycan be quite subtle and has not yet been studied in generality. In Pinkham’sexample, we only see one of the components of the base of the versal familyas the base of the G -versal family; in Rim’s example cited above the relation issignificantly less obvious: Example 2.12 (2.11 continued) A G -versal family with base ( C , has alreadybeen constructed above.A versal family of space germs with the given special fibre is I X = h ( S − x x + Ax + Dx , x x − Bx − Gx , x x + Cx − CG,Sx x + Ex + Hx , x x − F x − S F H, x x + 1 S EF i over a base ( V, which is the germ of the cone of the Segre embedding of P × P in P . More precisely, I V = h SAB + AE − ( S − BH, SAC + SAF − ( S − F H, SBC − EF,SBD + DE + ( S − EG, CD + DF + ( S − CG,SAG − DH − ( S − GH i (see example 3.4 in the Th`ese of Buchweitz [7] for the construction). The rela-tion between these seemingly completely unrelated base spaces becomes apparent, Following tradition in the standard basis community we also refer to a module element,of which the only non-zero entry is a monomial, as a monomial. For simplicity of notation, we can consider a non-singular codimension k germ as deter-minantal singularity essentially of type (1 , k, s soon as we consider them as the images of ( V ( B ) , under the projectionsto the first and second factor of ( C , × ( C , , where B = h ( S − b − ( S − − e ) A, a − (1 + e ) B, (1 + e ) d + C, ( S − c + ( S − − e ) D, a + E, Sd − F, c + (1 + e ) G, b − H i . (Note that the coefficients S , S − , S − − e and e are all units in the localring). The non-trivial comparison illustrated above also makes the question of semi-universality difficult to answer in generality; in some cases as, e.g. the Hilbert-Burch case, it is obvious, in others it may boil down to a case by case checkfor different matrix structures. In view of the possibility of semiuniversality infurther cases, we continue with the considerations in full generality and leavethis question to be answered at the time of application to specific matrix sizes(or even matrices).At this point, it is important to observe that the G -versal families obtainedby the above construction are indeed flat, as any relation lifts to a relation ofthe family by the second part of remark 2.5.Having restricted our interest to G -versal deformations, we can now state theobject, which we want to determine algorithmically in the subsequent section:the locus in the base C τ G of the G -versal family, above which the fibres possesssingularities, i.e. the G -discriminant locus of the family.For later use, we also need one further construction concerning determinantalsingularities: a Tjurina modification as introduced in [30] and used e.g. in [29]and [15]. This construction relies on the fact that the rows of an m × n matrix A representing a point of M tm,n span a ( n − t +1)-dimensional subspace in C n . Thisgives rise to a rational map P : M tm,n Grass( n − t + 1 , n ), of which we canresolve indeterminacies to obtain a map ˆ P : W = Γ P ( M tm,n \ M t − m,n ) ⊂ C m · n × Grass( n − t +1 , n ). Combining this with a map F defining a determinatal variety X of type ( m, n, t ) as in definition 2.3, we obtain the following commutativediagram: Y = X × M tm,n W ˆ F / / π (cid:15) (cid:15) W ρ (cid:15) (cid:15) ˆ P ! ! ❈❈❈❈❈❈❈❈❈ X F / / M tm,n P / / ❴❴❴ P r (1)If the dimension of X is large enough to allow the exceptional locus to be aproper subset of Y , this is indeed a modification. Explicit equations for Y aregiven in [15], in the simplest case, t = n ≤ m , the equations are given by F · s = 0 , where s = ( s , . . . , s n ) denotes the tuple of variables of Grass( n − , n ) =Grass(1 , n ) = P n − . 7nalogously the columns can be used for the same construction, as theyspan a ( m − t + 1)-dimensional subspace in C m . To study the discriminant, we need to detect the singularities of the fibres. Asalready mentioned in the introduction, we shall decompose the discriminantand compute the contributions separately. To this end, we shall exploit thesmoothness criterion of Hironaka in a similar way as in [3], but with a slightlymore involved train of thought. For readers’ convenience, we postpone thegeneral case and work out the key ideas in the smallest non-trivial case first:2-minors of 2 × (2 + k ) matrices. Definition 3.1 [19] Let ( X, ⊂ ( C n , be a germ with defining ideal I X, generated by f , . . . , f s ∈ C { x } := C { x , . . . , x n } , and assume that these powerseries form a standard basis of the ideal with respect to some local degree or-dering. Assume further that the power series f i are sorted by increasing order.The tuple ν ∗ ( X, ∈ N s then denotes the sequence of orders of the f i . The tuple ν ∗ detects singularities, as the following lemma states, which isimplicitly already present in Hironaka’s work: Lemma 3.2
The germ ( X, ⊂ ( C n , is singular at p if and only if ν ∗ ( X, > lex ( 1 , . . . , | {z } codim( X ) ) with respect to the lexicographical ordering > lex . Of course, the above definition and lemma also make sense for the germ atany other point p on X , as can be seen by moving the point p to 0 by a coordi-nate transformation and then passing to the germ.Based on these considerations, we now want to decide whether for a given2 × (2 + k )-matrix M , defining a determinantal variety X of type (2 , k, ν ∗ ( X, p ) > lex (1 , . . . ,
1) at some point p . If ν ∗ ( X, p ) starts with 1 as first entry,the singularity needs to be essentially of type (1 , k,
1) at p , as M has tocontain one entry which is of order zero and hence a unit in the local ring at p . In other words: if all entries of M are of order at least 1, the frist entry of ν ∗ ( X, p ) cannot be lower than 2. We hence know that X is singular at p , if andonly if one of the following two alternatives holds:(A) the ideal I A generated by the entries of M has order at least 1 at p ,(B) X is essentially of type (1 , k,
1) and singular.8n case (A), it suffices to determine where the ideal of 1-minors of M is of orderat least 1 to describe this contribution to the singular locus.The second case is significantly more subtle: Such a unit might be sitting at anyposition in the matrix, which implies a priori that the respective contributionsneed to be computed for each matrix entry. We know, however, that in case (B)the matrix is of rank precisely 1 at p , i.e. all column vectors are collinear andhence correspond to a point in P . At such points, the Tjurina transformation isan isomorphism, which allows us to pass to the Tjurina transform, determine itssingular locus outside V ( I A ) and take the closure thereof as the contribution (B).Considering a larger matrix size, say a singularity of type ( m, n, m ) with n ≥ m , and the corresponding maximal minors, we can proceed analogously, butmay a priori encounter m cases corresponding to the singularity being essentiallyof type ( m − i, n − i, m − i ) with 0 ≤ i < m . As larger minors also vanish,whenever all minors of a smaller size vanish, it suffices to consider the vanishinglocus of the ( m − × ( m −
1) minors to determine contribution (A).For contribution (B), we need to assume that the rank of the matrix is precisely m −
1. Hence its columns span a hyperplane in C m and we can again make useof the fact that the Tjurina transform is an isomorphism at such points. So wecan simply determine the singular locus of the Tjurina transform outside V ( I A )and take the closure thereof as we did for contribution (B) in the previous case.To give a concise overview of the necessary compuations, this is summarized inalgorithm 1. There the input is restricted to polynomials for purely practicalreasons: it should consist of finitely many terms. Algorithm 1
Discriminant for EIDS of type ( m, n, m ) (sequential)
Require: M ⊂ Mat( m, n ; C [ x , . . . , x r ]), m ≤ n defining EIDS at 0 Ensure: ideals I A , I B describing the discriminant of the versal family of thegiven EIDS as follows: • I A describes contribution (A) • I B describes contribution (B) • I A ∩ I B describes the discriminant matrix N := versalG ( M ) ideal I A := h (m-1)-minors of N i I A = eliminate ( I A ; x , . . . , x r ) ideal I T j := ( s , . . . , s m ) · N , with generators denoted as f , . . . , f n ideal I B := I T j + minor (cid:18)(cid:16) ∂f i ∂v j (cid:17) i,j , n − m + 1 (cid:19) where v = ( x, s ) I B = ( I B : h s i ∞ ) I B = eliminate ( I B ; x , . . . , x r , s , . . . , s n ) I B = ( I B : I ∞ A ) return ( I A , I B ) 9he algorithm 1 requires a saturation in step 6 to remove any contribution ofthe irrelevant ideal. As this can be a significant bottelneck, a parallel approachcan be helpful: replace step 6 by running step 7 in all charts D ( s i ) of theprojective space and intersect the resulting ideals I B,i to obtain I B . Remark 3.3
In lines 3 and 7 of 1, we use elimination which means that weendow the resulting complex space with the annihilator structure (cf. [18], Def.1.45), which is not compatible with base change. We might as well have chosenthe Fitting structure relying on resultant methods instead of elimination, as thisis compatible with base change.The choice of elimination over resultants is mostly based on the purely practicalfact that the implementation of elimination in
Singular is significantly morerefined than the one of resultants.
Example 3.4 (a) Consider the determinantal singularity defined by the -minors of a matrix of the form M = (cid:18) x . . . x r − x r x r +1 . . . x r − f ( x , . . . , x r − ) (cid:19) . The G -versal family with this special fiber can be written as M t = (cid:18) x + a . . . x r − + a r − x r x r +1 . . . x r − F ( x , . . . , x r − , a r , . . . , a s ) + a (cid:19) , where F ( x , . . . , x r − , a r , . . . , a s ) corresponds to a versal deformation withsection of f ( x ) . For determining the contributions to the discriminant, wenow apply our algorithm and obtain: I A = h x + a , . . . , x r − + a r − , x r , . . . , x r − , F ( a , . . . , a s ) + a i ∩ C [[ a ]= h F ( a ) + a i I B = discriminant of F(x , . . . , x r − , a r , . . . , a s ) + a The locus, above which we see determinantal singularities, is the smoothhypersurface V ( F ( a , . . . , a s ) + a ) and the remaining part of the discrim-inant is precisely the discriminant of the versal family with section in theright hand lower entry.(b) As the next example, we consider 3 families of ICMC2 singularities: M = (cid:18) x y zw x y + f ( v ) (cid:19) M = (cid:18) x y zw u x + f ( v ) (cid:19) M = (cid:18) x y zw x + g ( u, v ) y + h ( u, v ) (cid:19) , where f ( v ) = v k for some k ∈ N and ( g ( u, v ) , h ( u, v )) describes a fat pointin the plane. Then, M is a -fold in ( C , and the other two are -folds n ( C , . Direct computation yields the following versal families: M a,b = (cid:18) x y zw x + P k − i =0 a i v i y + v k + P k − i =0 b i v i (cid:19) M b = (cid:18) x y zw u x + v k + P k − i =0 b i v i (cid:19) M t = (cid:18) x y zw x + G ( u, v, t ) y + H ( u, v, t ) (cid:19) , with suitably chosen G ( u, v, t ) and H ( u, v, t ) (cf. [14]). The Tjurina trans-forms for the three cases are: I T j, = h sx + tw, sy + t ( x + k − X i =0 a i v i ) , sz + t ( y + v k + k − X i =0 b i v i ) i I T j, = h sx + tw, sy + tu, sz + t ( x + v k + k − X i =0 b i v i ) i I T j, = h sx + tw, sy + t ( x + G ( u, v )) , sz + t ( y + H ( u, v )) Passing to the two affine charts of P , we immediately see that all the V ( I T j,i ) are non-singular. Hence, I B = h i and I A describes the wholediscriminant in these cases.(c) To illustrate contributions (A) and (B) not only in the extremal casesshown above, we give two surface and two 3-fold examples from the listof simple ICMC2 singularities [14], which for the surface case conincideswith Tjurina’s list of rational triple point singularities in [30].As first example of a surface singularity, we consider Tjurina’s A , , singularity. Its versal family is: (cid:18) x x + a x + a x + x a + a x + a x (cid:19) for which the two contributions to the discriminant are: I A = h a + a , a + a a − a i I B = h a (4 a + 27 a ) i . So contribution (A) is a -dimensional smooth subvariety of the base andcontribution (B) consists of two hypersurfaces, a smooth one and a cylin-der over a plane cusp.As next example, we consider Tjurina’s D singularity: (cid:18) x + a x + a x x + x a + x a + a x + a x + x x + a (cid:19) , for which a direct computation yields I A = h a a + a a − a , a + a a + a i , hich again happens to be a smooth subvariety of codimension . Contri-bution (B) is an irreducible hypersurface of degree , of which we do notgive the explicit equations here.The first 3-fold example has the versal family (cid:18) x + x x + x a + a x + x a + a x x x + x a + a x + x x + a (cid:19) . Here contribution (A) is an irreducible hypersurface of degree 7, whereascontribution (B) is the hypersurface defined by I B = h a ( a − a ) i . The versal family in the final example is: (cid:18) x + a x + x x + x a + a x + x a + a x + x x + x a + a x (cid:19) . In this case, contribution (A) is an irreducible hypersurface of degree 5and contribution (B) is an irreducible hypersurface of degree 14.
Up to now, we had restricted our considerations to ideals of maximal mi-nors to allow a clearer exposition of the material. For considering non-maximalminors, we first observe that the P m − which was used in case (B) above isjust a manifestation of a Grassmannian in the simplest case, hyperplanes in C m . Passing to non-maximal minors, however, the Grassmannian has morestructure which we need to recall before continuing with our study of the dis-criminant.Classically the Grassmannian describing the set of r -dimensional linear sub-spaces of an n -dimensional vector space V or equivalently of ( r − P r − ⊂ P n − can be embedded into projective space by thePl¨ucker embedding: Grass( r, n ) −→ P r ^ V ! ∼ = P ( nr ) − span( v , . . . , v r ) v ∧ · · · ∧ v r The image of this embedding is closed; the equations of the image are quadraticin the variables of P ( nr ) − : we denote the variables as x i ,...,i r for any givensequence of indices 1 ≤ i < i < · · · < i r ≤ n . Purely for convenience ofnotation, we extend this to any subset of { , . . . , n } with r elements. To thisend, we set x i ,...,i r = 0, if two elements of the index conincide and postulatethat permutations of indices change the sign by the sign of the permutation.Then each Pl¨ucker relation is of the form r X j =0 ( − j x i ,...,i r − ,k j · x k ,..., ˆ k j ,...,k r = 012here i , . . . , i r − and k , . . . , k r are subsets of { , . . . , n } , i.e. the ideal of theimage of the Pl¨ucker embedding is generated by quadratic polynomials. At thispoint it is important to stress that a Pl¨ucker coordinate x i ,...,i r can be inter-preted as the r -minor of the matrix with columns v , . . . , v r involving the rows i , . . . , i r .Now we are ready to consider the general case of an EIDS of any type ( m, n, t )with 1 < t ≤ m ≤ n . The case (A) does not present additional difficulties here,as the ideal I A can be computed directly as ( t − M . As we have seen before, case (B) comprises all singular points at which thematrix M has rank precisely t −
1. Appending t − n ( t −
1) new variables, and imposing the condition that at least one ( t − X bytaking the t minors of the new matrix. With our previous considerations aboutthe Grassmannian, this can also be more conveniently expressed by introducingPl¨ucker coordinates instead of the additional columns and then leads to a setof equations of the form t X l =1 ( − l y i ,..., ˆ i l ,...,i t m i l ,j = 0for all strictly increasing t -tuples { i , . . . , i t } ⊂ { , . . . , m } and for all j ∈{ , . . . , n } , where m l,j denotes the entry of M at the position ( l, j ). We nowconsider the ideal generated by these polynomials and by the generators of theimage of the Pl¨ucker embedding. After saturating out the irrelevant ideal, thenew ideal describes the part of X which is relevant for contribution (B). Wecan then compute the singular locus thereof, saturate out the maximal ideal inthe y i ,...,i t and then eliminate the orignal variables x and all variables y i ,...,i t as before to obtain I B . For practical purposes, a parallel approach using acovering of the Grassmannian with affine charts should again be the choice inimplementations due to the extremely high number of variables and the partic-ularly simple structure of the ideal of the Grassmannian in each chart.Comparing the construction above with the general construction of the Tju-rina transform in [15], we see that the use of the Grassmannian in both settingsis the same and that I B captures precisely the singular locus of the Tjurinatransform as before. Therefore, we have decomposed the discriminant of a de-terminantal singularity in the following way: Proposition 3.5
Let ( X, ⊂ C N be a determinantal singularity of type ( m, n, t ) , m > n , defined by F − ( M tm,n ) and let X be its G -versal family. Further assumethat dim ( X ) ≥ m . Then the discriminant of X decomposes naturally into twocontributions:(A) points in the base space, above which there are determinantal singularities(B) points in the base space, above which there are singularities leading tosingular points in the Tjurina transform. X in the preceding proposition ensuresthat the exceptional locus of the Tjurina modification is a lowerdimensionalclosed subset of the Tjurina transform. In the above decomposition, the contri-bution related to the Tjurina transform may be empty in some cases, whereasthe other one always contains at least the origin. The above considerations not only yield a decomposition of the discriminant.They show that determinantal singularities ( F − ( M tm,n ) ,
0) possess in generaltwo kinds of contributions to the singular locus: The structural contributionarising from F − ( M t − m,n ), to which the Tjurina transform is partly blind, anda contribution arising from the map F itself, which manifests itself in the Tju-rina transform. The well-studied special case of ICMC2 singularities, in which G -versality and versality are known to conincide, provides a good setting toconsider this in more detail and illustrate the consequences. Lemma 4.1
Let ( X, ⊂ ( C N , be an ICMC2 singularity, i.e. of type ( t, t +1 , t ) , with generic linear entries.1. It has a smooth Tjurina transform, if and only if N ≥ t .2. The singular locus of the Tjurina transform is a determinantal variety oftype ( t + 1 , N, t + 1) of dimension t − N − in P t − for t + 1 ≤ N < t . Proof:
In the case of generic linear entries in the matrix M of X , the ideal of theTjurina transform Y is generated by bi-homogeneous polynomials of bidegree,(1 , (cid:0) A | M T (cid:1) , where the first columns hold the derivatives w.r.t. the original variables andthe remaining ones the derivatives w.r.t. the variables of the P t − . Then A isa ( t + 1) × N matrix with homogeneous entries of degree 1, which only involvethe variables of the P t − and which are generic, because the entries of M weregeneric. As the singular locus of X is just the origin, the Tjurina modificationis an isomorphism outside the origin and we therefore only need to evaluate theJacobian criterion above V ( x ). This causes the last t columns of the Jacobianmatrix and all generators of the ideal of the Tjurina transform to vanish. Hencethe singular locus of the Tjurina transform is precisely the vanishing locus ofthe maximal minors of A .For the first claim, it suffices to observe that the codimension of the singularlocus of Y is N − t in 0 × P t − , which needs to exceed t − Y to be smooth,i.e. we obtain the condition N > t −
1. These arguments also prove the secondclaim. (cid:3)
14s the matrix describing the singular locus of the Tjurina transform is asquare matrix for N = t + 1, we immediately get the following corollary: Corollary 4.2
The singular locus of the Tjurina transform of an ICMC2 sin-gularity ( X, ⊂ ( C t +1 , of type ( t, t + 1 , t ) with generic linear entries is ahypersurface of degree t + 1 in P t − . For the other extreme of N = 2 t −
1, i.e. for isolated singular points in theTjurina transform, it is also possible to determine the number of points as itcoincides with the value of the only non-zero term in the Hilbert Polynomial,the constant term, in this case. This polynomial itself can be obtained from agraded free resolution (given by the Eagon-Northcott complex for determinantalvarieties of type ( m, n, m )). We only give an example of such a computation:
Corollary 4.3
The singular locus of the Tjurina transform of an ICMC2 sin-gularity ( X, ⊂ ( C , of type (3 , , with generic linear entries consists of10 points in general position in P . Proof:
It is well-known that the Hilbert polynomial can be read off from theBetti diagram of a minimal free resolution (see e.g. [16] or [12]). Here thesituation is particularly simple: the choice of N = 5 and t = 3 leads to asingular locus Σ of the Tjurina transform which only consists of points in P and can be described by the vanishing of the 4-minors of a 5 × −→ t M i =1 O P ( − l i ) −→ t +1 M i =1 O P ( − k i ) −→ I Σ −→ , the Hilbert polynomial and hence the number of points is P ti =1 l i − P t +1 i =1 k i . In our setting, all l i are have the value − k i are − (4 · − ·
16) = 10 points. (cid:3)
As the Tjurina transform can be non-singular, there are cases in which thecontribution ( B ) of the discriminant of the versal family is empty as e.g. forthe generic ICMC2 of type (2 , ,
2) in ( C k ,
0) for k ≥
4. But there are, ofcourse, many non-generic matrices with singular Tjurina transform even in thesedimensions. The smoothness of the Tjurina transform is actually a statementabout the adjacencies of an EIDS, as the following lemmata show:
Lemma 4.4
Let ( X, ⊂ ( C k , , be an EIDS for which contribution (B) tothe discriminant of the versal family is not empty, then ( X, is adjacent to an A singularity and has a smoothing passing through A singularities. roof: If p is a point in the base of the versal deformation belonging to contri-bution (B), then the Tjurina transform of the fibre above this point is singularand has only ICIS singularities, which are themselves adjacent to an A . More-over, as contribution (B) is non-empty, it contains by construction an open setwhich does not meet contribution (A). Above this open set, there are no fibreswith determinantal singularities, and the Tjurina modification is already an iso-morphism for these fibres. Hence the original singularity is also adjacent to an A and possesses a smoothing which passes through A singularities. (cid:3) Remark 4.5
There are smoothable EIDS for which no smoothing passes throughan A singularity as can be seen from the results in [15] and [32]: For surfacesingularities of type (2 , , in ( C , this is precisely the determinantal sin-gularity with generic linear entries. For 3-fold singularities of type (2 , , in ( C , these are precisely the singularities with b − b = − . As we always have b = 1 , this difference implies b = 0 , whence the Tjurina transform is smooth,no adjacency to an A is possible and the contribution (B) to the discriminant isempty. However, these singularities are smoothable through a different mecha-nism: They pass through the EIDS with generic linear entries of the appropriateambient dimesion. For the latter, any non-trvial deformation is a smoothing.In dimensions, in which the determinantal singularity with generic linear entriesis a rigid EIDS, the contribution (B) will always be empty, as the terminal ob-ject in the adjacency diagramm is a rigid determinantal singularity which causescontribution (A) to be the whole base of the versal family. Therefore, we cannotdecide in general whether a singularity is adjacent to an A based solely on thefact that contribution (B) is empty. But even in this case, it can make sense toconsider the locus above which there are ICIS singularities, by omitting the finalsaturation by I A in the computation of contribution (B). These last observations also indicate that passing to the Tjurina transformprovides valuable information about the original singularity, but this informa-tion also relies on knowledge about the contribution (A). The cases of surfacesin ( C ,
0) and 3-folds in ( C ,
0) show how different the behaviour can be. Toillustrate this, we first discuss Wahl’s conjecture about the relation betweenMilnor and Tjurina number in the surface case: we reprove the easier directionthat quasihomogeneity implies µ = τ − µ = τ − Lemma 4.6
Let ( X, ⊂ ( C , be a quasihomogeneous isolated determinantalsingularitiy of type (2 , , with at most isolated singularities in the Tjurina ransform. Then µ = τ − . Proof:
In this proof we denote the Tjurina transform of X by Y and weassume that the presentation matrix of X is chosen to have quasihomogeneousentries and respect row and column weights as in [13]. This implies that Y isquasihomogeneous w.r.t. the same weights.From [15], we know that µ ( X ) = 1 + X p ∈ Sing( Y ) µ ( Y, p ) , i.e. it differs from the sum over the Milnor numbers of the singularities ( Y, p )of the Tjurina transform by 1. The singularities of the Tjurina transform areat most ICIS singularities and hence satisfy µ ( Y, p ) ≥ τ ( Y, p ) with equalityprecisely in the case of quasihomogeneous singularities [22]. So it remains toestablish the relation τ ( X ) = X p ∈ Sing( Y ) τ ( Y, p ) + 2to prove the claim. However, after a few preliminary considerations this leadsto a Gr¨obner basis computation which we will sketch for a general matrix of thegiven properties in the rest of the proof.To this end, we first recall from [15] that T ( X ) ∼ = N ′ = H ( Y , T Y ) ⊕ M p ∈ Sing( Y ) T ( Y, p )implying for the corresponding dimensions τ ( X ) = dim C H ( Y, T Y ) + X p ∈ Sing( Y ) τ ( Y, p ) . As all of these C { x } -modules are finite dimensional C -vector spaces, this alsoinduces an isomorphism of C -vector spaces which can be expressed in terms ofa monomial basis of T X . To complete the proof, we therefore need to identifythose basis elements in T X which do not contribute to ⊕ p ∈ Sing( Y ) T Y,p .To keep the presentation of the rest of the proof as simple as possible, wedenote the variables by x, y, z, w and denote the tuple of these four variables by x in the following. Since the Tjurina transform only contains isolated singularities, X can be expressed in terms of a matrix A = (cid:18) x y za b c (cid:19) a, b, c ∈ m ⊂ C { x } and no term of a is divisible by x .The Tjurina transform is described by the ideal I T j = h sx + ta, sy + tb, sz + tc i with Jacobian matrix s ta y ta z ta w x atb x s + tb y tb z tb w y btc x tc y s + tc z tc w z c , where a subscript stands for the partial derivative by the respective variable.At t = 0 there is a 3-minor s , whence (0 , , , × (1 : 0) cannot be a singularpoint of the Tjurina transform. Therefore it suffices to consider the chart D ( t ).We know from [15] that N ′ = (( C [ s, t ] { x } ) /K ) (1) where the subscript (1) denotes the degree 1 part in s and t and the module K is generated by the columns of the Jacobian matrix of above and I T j · C { x } . Aswe are interested in K (1) and possibly slices of higher degree, but not in K (0) ,we now replace the columns 5 and 6 of the above matrix by their multiples with s and t . To fix a numbering of the generators of K (1) , we keep the resultingnumbering of the columns: starting with the partial derivatives by the x, y, z and w and continung with the s and t multiples we just introduced, the firsteight generators are the columns of the following matrix: s ta y ta z ta w sx sa tx tatb x s + tb y tb z tb w sy sb ty tbtc x tc y s + tc z tc w sz sc tz tc . The last 9 generators are then ordered as in the following matrix: sx + ta sy + tb sz + tc sx + ta . . .
00 0 0 0 sz + tc . We also know that the sum over the T ( Y, p ), which are all sitting above theorigin of C , can be computed as the ( x -local, but s -global) Tjurina module inthe respective chart of P by considering the module T = ( C [ s ] { x } ) /K obtained by dehomogenizing the previous module w.r.t. the variable t . Ourtask will now be a comparison of K (1) and K and of the respective quotientsby means of the corresponding leading ideals, which we obtain from a standardbasis computation.By the assumption that the entries of the original matrix are contained in m , only the first 4 generators of K and K can possibly contain x -degree zeroentries. Using a mixed ordering which first compares w.r.t. a global ordering18 > lex t , then a negative lexicographical ordering in x < y < z < w and finallya module ordering, it is now easy to see that the first three generators have s asentry in their leading monomial and that the leading monomials are in pairwisedifferent entries not allowing any non-vanishing s-polynomial among these. Wecan thus directly eliminate all s -terms from the other generators by reducingwith these three generators and assume from now on that generator 4 and allfurther ones do not involve any s – both in K and in K . From now on, we usethese reduced columns, not involving s , instead of the original columns 4 to 17denoting the i -th columns thereof by C i .We immediately see that any s-polynomial computation arising from C i and C j with i, j ≥ K (1) and t · K ; moreover, it does not involve any s as we had chosen an elimination ordering for s . Therefore the standard bases for h C , . . . , C i and its dehomogenization w.r.t. t are in 1:1 correspondence. Con-sidering an s-polynomial between C i and C j with 5 ≤ i ≤
17 and the respective1 ≤ j ≤
3, a direct computation shows that its normal form w.r.t. { C , C , C } already lies in the module t · h C , . . . , C i C { x } and hence reduces to zero w.r.t.a standard basis generated by C , . . . , C . The respective relations are statedin the table below. There the abbreviation jacob denotes the jacobian matrixand j ∈ { , , } stands for the component in which the leading term is found,i.e. the index of the appropriate C j for the s-polynomial.NF(spoly( C , C j ) , { C , C , C } ) = t ( C + a x C + a y C + a z C + b x C + b y C + b z C + c x C + c y C + c z C )NF(spoly( C , C j ) , { C , C , C } ) = t (( a x + b y + c z ) C + det(jacob( a, b, c )) C + ( a y b x − a x b y + a z c x + b z c y − a x c z − b y c z ) C + ( b z c y − b y c z ) C + ( a y c z − a z c y ) C + ( a z b y − a y b z ) C + ( b x c z − b z c x ) C + ( a z c x − a x c z ) C + ( a x b z − a z b x ) C + ( b y c x − b x c y ) C + ( a x c y − a y c x ) C + ( a y b x − a x b y ) C )NF(spoly( C , C j ) , { C , C , C } ) = t ( C + C + C + C )NF(spoly( C , C j ) , { C , C , C } ) = tC NF(spoly( C , C ) , { C , C , C } ) = t ( a x C + b x C + c x C )NF(spoly( C , C ) , { C , C , C } ) = t ( a x C + b x C + c x C )NF(spoly( C , C ) , { C , C , C } ) = t (( a x C + b x C + c x C )NF(spoly( C , C ) , { C , C , C } ) = t ( a x C + b x C + c x C )NF(spoly( C , C ) , { C , C , C } ) = t ( a x C + b x C + c x C )NF(spoly( C , C ) , { C , C , C } ) = t (( a x C + b x C + c x C )NF(spoly( C , C ) , { C , C , C } ) = t ( a x C + b x C + c x C )NF(spoly( C , C ) , { C , C , C } ) = t ( a x C + b x C + c x C )NF(spoly( C , C ) , { C , C , C } ) = t (( a x C + b x C + c x C )This only leaves potential differences between standard bases of K (1) and K ins-polynomials arising from C and C i for 1 ≤ i ≤ K (1) , we can see no contributing s-polynomial which arises between C i C with 1 ≤ i ≤
3, because its leading monomial would be in ( s, t )-degree2. In K on the other hand, such an s-polynomial is relevant, as C is non-zero(due to the need for a pure power in w to appear to allow finite dimension),is of s -degree zero and has lower w -order in each entry than the correspondingentry of the second row of M . Considering this more closely, we see that sC and s C can both contribute to relevant s-polynomials with C , C , C . Onthe other hand, using the linear combinations of C , C , C indicated by thecolumns of the right adjoint of the 3 × C , C , C ,the minor of the Jacobian matrix corresponding to C , C , C reduces to zeroin all entries. Hence s C does not provide any new contribution but reducesto lower degree terms in s . Therefore we obtain precisely two ( s, t )-degrees forwhich the standard basis of K contains elements not necessarily appearing inthe one of K (1) . This implies that their leading monomials can be part of thecomputed monomial basis of N ′ , but reduce to zero in T = ( C [ x ] { x ) /K . Ifthey are in the monomial basis of N ′ , they contribute to dim C H ( Y, T Y ).Showing that no x -multiple of two leading monomials of the still remainings-polynomials is non-zero in N ′ , we obtain dim C H ( Y, T Y ) ≤ x · NF(spoly( sC , { C , C , C } )) = a w tC + b w tC + c w tC − atC y · NF(spoly( sC , { C , C , C } )) = a w tC + b w tC + c w tC − btC z · NF(spoly( sC , { C , C , C } )) = a w tC + b w tC + c w tC − ctC Of course, such relations continue to hold after multiplication with s and subse-quent reduction by C , C , C , which now leaves only one case to be considered: w · NF(spoly( sC , { C , C , C } )) , but by the Euler relation this expression reduces to zero, as we are consideringthe quasihomogeneous case.For proving the equality part of the statement, it suffices to establish τ ( X ) = X p ∈ Sing( Y ) τ ( Y, p ) + 2 . Thus we need to show that dim C H ( Y, T Y ) is at least 2 for all quasihomoge-neous ICMC2 surface singularities of type (2 , ,
2) which have at most isolatedsingularities in their Tjurina transform. This certainly holds for the simplestsuch singularity given by the matrix (cid:18) x x x x x x (cid:19) , as an explicit Gr¨obner basis computation for K provides the leading monomials(0 , t ,
0) and (0 , , t ) arising from the s-polynomials of the pairs ( sC , C ) and20 s C , C ). As we know that all ICMC2 surface singularities of type (2 , ,
2) areadjacent to this singularitiy (see [14]), the principle of conservation of numberthen ensures the upper semicontinuity of dim C H ( Y, T Y ) concluding the proof. (cid:3) Remark 4.7
The Milnor number of a semiquasihomogeneous ICIS is knownto coincide with the Milnor number of its quasihomogeneous initial part, theTjurina number of a semiquasihomogeneous ICIS is bounded from above by theTjurina number of its quasihomogeneous initial part. Therefore the precedinglemma also establishes the inequality µ ≥ τ − for semiquasihomogeneous isolated determinantal surface singularities of type (2 , , with at most isolated singularities in the Tjurina transform. Remark 4.8
The considerations in the proof also show that dim C H ( Y, T Y ) can be computed explicitly in the non-quasihomogeneous case:Considering the module K defined above, we first observe that already N ′ =(( C [ s, t ] { x } ) /K ) (1) has to be a finite dimensional vector space due to finitedeterminacy of the given singularity. Then the desired dimension of H ( Y, T Y ) is precisely dim C ( N ′ ) − dim C ( T ) where T = ( C [ s ] { x } ) /K . Note that the valueof dim C H ( Y, T Y ) can exceed , as the following example shows: (cid:18) x x x x + 2 x − x − x + x − x + x + x (cid:19) In this example, τ ( X,
0) = 34 and P p ∈ Sing( Y ) τ ( Y, p ) = 31 which yields a dif-ference of dim C H ( Y, T Y ) = 3 . However, µ ( Y ) = 39 and hence this does notprovide a counterexample to Wahl’s conjecture. Remark 4.9
Looking at the above proof more closely, we can even determinethe basis elements contributing to N ′ /T . Considering a standard basis of K w.r.t. the ordering chosen in the proof, they are precisely the monomials in thebasis of N ′ which are divisible by monomials in the leading module L ( K ) arisingfrom reduction of elements of the form s j x i ta x tb x tc x where j ∈ { , } as we know from the proof and i ∈ N takes all values which aresufficiently low not to push the whole element beyond the determinacy bound. We now construct a counterexample to Wahl’s conjecture, i.e. a non-quasi-homogeneous determinantal surface singularity for which µ = τ − X, Y . If we can choose all of thesesingularities as quasihomogeneous, but w.r.t. different weights for the respectivesingularities, and if we can furthermore ensure that dim C H ( Y, T Y ) = 2, this isprecisely the desired counterexample. All of these constraints are e.g. satisfiedfor the singularity ( X,
0) defined by the maximal minors of the matrix (cid:18) z + x y x k + w w l z y (cid:19) , where the value of k, l ∈ N is at least 3. Its Tjurina transform Y has twoquasihomogeneous singularities, an A l − at (0 , (0 : 1)) and a D k +1 at (0 , (1 : 0)): A l − : h w l + tx + tz, z + ty, y + tx k + tw i ∼ C h w l + tx + t x k + t w , y, z i with monomial basis of the Tjurina algebra(1 , , , ( w, , , . . . , ( w l − , , D k +1 : h sw l + z + x, sz + y, sy + x k + w i ∼ C h z, y, s x + x k + w + s w l i with monomial basis(0 , , , (0 , , s ) , (0 , , s ) , (0 , , x ) , . . . , (0 , , x k − ) . As both of these are quasihomogeneous, the Tjurina and Milnor number coincidefor each of the two singularities and we have a total Milnor number of of Y of k + l which implies that µ ( X,
0) = k + l + 1. On the other hand, it is aneasy computation to see that a C -vector space basis of T ( X ) is given by themonomials: (cid:18) (cid:19) , . . . , (cid:18) x k − (cid:19) , (cid:18) (cid:19) , . . . , (cid:18) w l − (cid:19) , (cid:18) (cid:19) , (cid:18) w (cid:19) . Hence, the Tjurina number is k + l + 2 and we can even discern the three con-tributions in the monomial basis: the first k − T of the D k +1 singularity , the ( k + 1)-stto the ( k + l − T of the A l − singularity leavingprecisely two elements for H ( Y, T Y ).To see that the determinantal singularity is not quasihomogeneous, we considertwo hyperplane sections: with V ( w ) and with V ( x ). Both lead to quasihomo-geneous space curve singularities, but w.r.t. different weights: (cid:18) z + x y x k z x (cid:19) of Type A k ∨ L and More precisely, the last 2 elements correspond to those basis elements involving s . z y w w l z y (cid:19) of Type E l +6 (2) (for l J l +46 , ( l + 46 )which are quasihomogeneous w.r.t. weights (2 , k + 1 , , − ) in the first case and( − , l + 4 , l + 2 ,
3) in the second. By an easy calculation, we see that it is im-possible to choose weights for y and z satisfying both conditions at the sametime for k, l ≥ b − b and the Tjurina number τ X using their invariant γ := τ − ( b − b ) : Observation 4.10 ([8]) a) γ ≥ for all simple ICMC2-singularities of di-mension and increases in value as we move higher in the classification.b) b − b ≥ − , with equality for the generic linear section and one infinitefamily.c) b − b is constant for certain infinite families with values − (one family), (two families), and (two families).d) γ is constant in all other considered infinite families in the table of simplesingularities with only one exception where both b − b and γ increasewith τ .e) For singularities of the form (cid:18) x y zw v g ( x, y ) (cid:19) with g a simple hypersur-face singularity, γ = 3 and b − b = µ ( g ) − . To explain these observations, which differ greatly from the rather rigidstructure observed in the surface case, we use again the Tjurina modification.In all cases in question, the Tjurina transform has at most quasihomogeneoushypersurface singularities, whence we know that its Milnor and Tjurina numberscoincide. From [15], we know that b = 1 and b coincides with the Tjurina num-ber of the Tjurina transform. So observation a) simply states that τ X − τ Y ≥ τ Y = 0 and τ X = 1 for the generic linear section, the A +0 -singularity, implying that this singularity is not adjacent to an A singularity.As all other singularities in the list are adjacent to it, this explains the lowerbound for the difference and hence observation a).The first part of observation b) follows immediately from the fact that b = 1.The second part is concerned with the family (cid:18) x y zw x y + v k (cid:19) . b = 0. TheTjurina number of X , however, increases in the family, τ X = 2 k −
1, and isclosely related to the maximal number of A +0 -singularities which can appearin a fibre of the versal family. This maximal number is achieved e.g. by theperturbation (cid:18) x y zw x y + v k + α (cid:19) for any 0 = α ∈ C , where we see precisely k such singularities. Observation c)then simply states that similar behaviour with constant topological type of theTjurina transform also occurs for other families which have singularities in theirTjurina transform.Observation d), on the other hand, singles out families in which the increaseof Tjurina number originates from the increase in Milnor/Tjurina number of theTjurina transform and the maximal number of A +0 -singularities appearing in afibre of the versal family does not change. In the last considered family, where b − b and γ increase with τ X , we see a first example of increasing contributionsto both the Tjurina transform and the purely determinantal part.The only part of the last observation, which still remains to be explained, isthe statement γ = 3. As already observed in [14], T X is isomorphic to T of the plane curve singularity defined by the right hand lower entry. Hence τ X − τ Y is the difference arising from deformations with section as opposed tousual deformations for the respective plane curve. In all cases in question, thisdifference is precisely 2 giving rise to γ = ( τ X − τ Y ) + b = 3. Remark 4.11
Although this article explains many of the recent surprising ob-servations about ICMC2 singularities, this is merely a glimpse into the new phe-nomena we are seeing in determinantal singularities. Extending exisiting toolsto the determinantal setting and combining methods from the theory of syzy-gies, from classical singularity theory and topology, we seem to have reached apoint now, where we can start thinking about a more systematic study of generaldeterminantal singularities.
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