Straight equisingular deformations and punctual Hilbert schemes

Straight Equisingular Deformations and Punctual arXiv:1810.06444v2 [math.AG] 12 Jun 2019 Hilbert Schemes ∗ Gert-Martin Greuel Dedicated to Lê Dũng Tráng on the occasion of his 70th Birthday Abstract We study ”straight equisingular deformations”, a linear subfunctor of all equisingular deformations and describe their seminuniversal de- formation by an ideal containing the fixed Tjurina ideal. Moreover, we show that the base space of the seminuniversal straight equisingular deformation appears as the fibre of a morphism from the µ-constant stratum onto a punctual Hilbert scheme parametrizing certain zero- dimensional schemes concentrated in the singular point. Although eq- uisingular deformations of plane curve singularities are very well un- derstood, we believe that this aspect may give a new insight in their inner structure. Introduction Let z be a singular point of the reduced curve C contained in the smooth complex surface Σ. The topological type of the germ (C, z) can be described by the resolution graph of a good embedded resolution of (C, z), and an eq- uisingular deformation of (C, z) is one with constant resolution graph. In the first section we fix the notations and recall the classical concepts of con- stellation, clusters and proximities, resulting in the notions of the essential tree T ∗ (C, z) and the cluster graph G(C, z), which is also a complete invari- ant of the toplogical type of (C, z). These notions go back to Enriques and Chisini, for a comprehensive and modern treatment of we refer to [Cas00]. ∗ Department of Mathematics, University of Kaiserslautern 2010 Mathematics Subject Classification: 14B05, 14B07, 32S05, 32S15 1  A deformation of (C, z) is equisingular or an es-deformation, if there exist equimultiple sections through the points of the essential tree (i.e. through the non-nodes of a minimal good resolution) of (C, z) such that the fam- ily can be blown up successively along these sections. Those deformations for which these sections can be simultaneously trivialized are called straight equisingular or straight es. These deformations have been considered by J. Wahl in [Wah74] (not under this name) and they were further characterized in [GLS07], showing, e.g., that for semiquasi-homogeneous and Newton non- degenerate singularities every equisingular deformation is straight equisin- gular. In Section 2 we introduce, in addition, for any plane curve singularity the topological singularity ideal I s (C, z) such that I s (C, z)/Ifeaix (C, z), with Ifeaix (C, z) the fixed Tjurina ideal, is the tangent space to straight equisingu- lar deformations. This provides an explicit description of the semiuniversal straight equisingular deformation as a linear subspace of all equisingular deformations. We illustrate this by an example using Singular [DGPS]. In Section 3 we consider the punctual Hilbert scheme, parametrizing zero-dimensional schemes Z ⊂ Σ with support in one point. We are in- terested in the rooted Hilbert scheme, a subscheme of the punctual Hilbert scheme consisting of the topological singularity scheme Z s (C, z) defined by the ideal I s (C, z), with fixed cluster graph G = G(C, z). Mapping an eq- uisingular family of plane curve singularities to the induced family of topo- logical singularity schemes, provides a morphism from the functor of all es- deformations to the rooted Hilbert functor. We prove that the fibre of the induced morphism from the base space of the semiuniversal es-deformation of (C, z) to the rooted Hilbert scheme is exactly the base space of the semi- universal straight equisingular deformations. This gives in addition a nice formula for the rooted Hilbert scheme in terms of the number of free vertices in the essential tree T ∗ of (C, z). Let us finish the introduction by commenting on the positive charac- teristic case. Though we work here over the complex numbers, the results should hold also for algebraically closed ground fields of characteristic zero. Equisingular deformations of plane curve singularities in arbitrary charac- teristic have been studied in [CGL07a] and [CGL07b]. In positive char- acteristic one has to distinguish between “good” and “bad” characteristic and we conjecture that the results of this paper can be extended to good characteristic as well. However, in bad characteristic one has to distinguish between equisingular deformations and weakly equisingular deformations, i.e., those that become equisingular after a finite base change, with very dif- ferent semiuniversal deformations. It would be interesting to study straight es-deformations in positive characteristic, in particular in bad characteristic. 2  Acknowledgement: The article is a revised excerpt of parts from Chapter I.1 of the forthcoming book [GLS18] by Chistoph Lossen, Eugenii Shustin and the author, where more details can be found. I like to thank C. Lossen and E. Shustin for the many pleasant years of joint work on the book. 1 Equisingularity for Plane Curve Singularities Before we define in the next section straight equisingular deformations, recall the definition of an equisingular deformation of a reduced plane complex curve singularity (C, z) ⊂ (Σ, z). It is well-known that we can resolve the singularity (C, z) by finitely many blowing ups of points. More precisely, there exists a good embedded resolution of the singularity of C at z, that is, a sequence of morphisms of smooth surfaces πn+1 n π 1 π π : Σn+1 −−−→ Σn −→ · · · → Σ1 −→ Σ0 = Σ, such that πi is the blowing up of a point qi−1 ∈ Σi−1 infinitely near to q0 = z and such that in a neighborhood of E = π −1 (z), the reduced total transform of C n+1 X ∗ e π (C)red = C + Ei i=1 is a divisor with normal crossings, that is, a hypersurface having only nodes as singularities (see [GLS07, Section I.3.3] for details). Here C e is the (smooth) ∗ strict transform of C, π (C) (with its scheme structure) is the total trans- form and Ei = πi−1 (qi−1 ) ∼= P1 , i = 1, . . . , n + 1, is the exceptional divisor of πi in Σi (identified with its image in Σj , j ≥ i). A good embedded resolution π is called a minimal (good) embedded resolution, if only non-nodal singular points of the reduced total transform of (C, z) are blown up in the resolution process (if (C, z) is smooth, we do not blow up anything, i.e. π is the empty map). It is well-known that such a minimal good resolution is unique up to isomorphism over Σ. For q ∈ π −1 (z) we denote by C(q) , C b(q) and C e(q) the respective germs at q of the strict, the total and the reduced total transform of C and and by mt C(q) , mt Cb(q) and mt C e(q) their multiplicities. If the germ C(q) is non- empty, we say that the curve C goes through the infinitely near point q (or that q belongs to C). 3  Recall that classically E1 ⊂ Σ1 and its strict transforms in Σi , i > 1, together with z, is called the first infinitesimal neighbourhood of z ∈ Σ. For i > 1 the i-th infinitesimal neighbourhood of z consists of points in the first infinitesimal neighbourhood of some point in the i − 1-st infinitesimal neigh- bourhood of z. Any point belonging to some infinitesimal neighbourhood of qi , i ≥ 0, is called an infinitely near point of qi ∈ Σi or a point infinitely near to qi . Definition 1.1. (Constellation) If (q0′ , π1′ , q1′ , . . . , πm ′ , q ′ ) with q ′ ∈ Σ′ m i i ′ ′ ′ ′ and πi : Σi → Σi−1 blowing up qi−1 is another sequence of a good embedded resolution of (C, z), then (q0 , π1 , q1 , . . . , πn , qn ) and (q0′ , π1′ , q1′ , . . . , πm ′ , q′ ) m are called equivalent if m = n = −1 or m = n ≥ 0 and if there is a Σ- isomorphism Σn+1 → Σ′n+1 . An equivalence class of such sequences is called a constellation (of (C, z)) and denoted by T (C, z). It is easy to see that the Σ-isomorphism preserves infinitely nearness and that successively blowing up two different points in the two possible orders gives rise to Σ-isomorphic surfaces, and hence the same constellation. The concept of a constellation has been introduced to describe isomorphism classes of good embedded resolutions. Definition 1.2. (Constellation graph, essential tree, proximity) Let T = T (C, z) = (q0 , π1 , q1 , . . . , πn , qn ) be a constellation of (C, z). (1) We introduce a partial ordering on the points q0 , . . . , qn by qi ≤ qj :⇐⇒ qj is infinitely near to qi .   For each 0 ≤ i ≤ n, we set the level of qi as ℓ(qi ) := # j  qi ≥ qj − 1 . The point z = q0 , the origin of T , is the only point of level 0. (2) The graph of the constellation T is the oriented tree ΓT (with root z = q0 ) whose • points are in 1-1 correspondence with q0 , . . . , qn and • edges with pairs (qj , qi ) s.t. ℓ(qj ) = ℓ(qi ) + 1 and qj ≥ qi . Hence p ≥ q iff there is an oriented path in ΓT from p to q. (3) A point z 6= q ∈ T (C, z) is called essential for C iff the reduced total transform of C does not have a node at q. The origin z of T (C, z) is called essential for C, iff the germ (C, z) is not smooth. We call a point q ∈ T (C, z) a singular essential point for C if the strict transform of C at 4 q is singular. The maximal finite sub-constellation of T (C, z) such that all of its points are essential is called the essential constellation of (C, z) and denoted by T ∗ (C, z). By abuse of notation, a constellation T is also called a tree and T ∗ (C, z) is called the essential tree of (C, z). It describes the minimal embedded resolution of (C, z). (4) Finally, we call qj proximate to qi (notation qj 99K qi ), if qj is a point in Eqi , the exceptional divisor of πi+1 , or in any of its strict transforms. qj is a satellite point if it is proximate to (at least) two points qi , 0 ≤ i ≤ j − 1, otherwise it is free. It is clear that a point qj cannot be proximate to more than two points since the exceptional divisors have normal crossings. Notice that each satel- lite point in T (C, z) is an essential point. It may well be a non-singular essential point for C. It is classically known that the topological type of (C, z) can be charac- terized by several different sets of discrete data (see [GLS07, Section I.3.4] for a short overview). One characterization is by the graph ΓT , T = T (C, z) a constellation of (C, z), together with the multiplicities mt C b(q) , q ∈ T , of the total transform (or, equivalently, the multiplicities mt C e(q) of the re- duced total transform). This weighted oriented graph is sometimes called a resolution graph of (C, z). Remark 1.3. We have by [GLS18, Proposition I.1.11] the following proximity equality: X mt C(p) = mt C(q) . q∈T q99Kp The difference between the multiplicities of the total and strict transform of C at a point q ∈ T is X b(q) − mt C(q) = mt C mt Cb(p) p∈T q99Kp (see [GLS18, Remark I.1.11.1]). Moreover mt C e(q) −mt C(q) = 1 or 2, depend- ing if one or two points are proximate to q. In particular, the multiplicities of the strict transforms of C together with the proximities (q 99K p) determine the multiplicities of the total transforms and hence can be used to describe the topological type of C. This is used in the following definition. 5 Definition 1.4. (Cluster, Cluster graph) Let T ∗ = T ∗ (C, z) = (z = q0 , π1 , q1 , . . . , πn , qn ) be the essential tree of (C, z) and m = (m0 , . . . , mn ) the vector of multiplicities with m0 := mt(C, z) and mi := mt C(qi ) , the multiplicity of the strict transform of C at qi , 1 ≤ i ≤ n. We call the tuple (T ∗ , m) the cluster of (C, z). The triple G(C, z) = G(C, T ∗ ) := (ΓT ∗ , 99K, m), consisting of ΓT ∗ , the graph of T ∗ , the binary relation defined by q 99K p if q is proximate to p and the vector m of multiplicities of the strict transforms is called the cluster graph of (C, z). Notice that the cluster graph G(C, z) determines (and is determined by) the topological type the curve singularity (C, z). We consider now an embedded deformation (i, Φ, σ) of (C, z) ⊂ (Σ, z), with section over a complex germ (T, 0). This is a commutative diagram i (C, z)  / (C, z)   / (M, z)  I ① σ ①① Φ ①①①   {①① {0}  / (T, 0) of morphisms of complex space germs with (C, z) ⊂ (M, z) a hypersurface germ, σ a section of Φ. Φ is assumed to be flat as well as (M, z) → (T, 0) which has (Σ, z) a special fibre. An embedded deformation (without sec- tion), denoted by (i, Φ), is given by a diagram as above but with σ deleted. We usually choose small representatives for the germs and denote them with the same letters, omitting the base points. Note that, for small represen- i Φ tatives, we have M ≃ Σ × T over T , and we have morphism C ֒→ C → T with fibres the reduced curves Ct = Φ−1 (t), where we identify C with C0 and write z instead of (z, 0) ∈ Σ × T . If M = Σ × T with Φ the projection and if σ(t) = (z, t), then σ is called the trivial section. By [GLS07, Proposition II.2.2] every section can be locally trivialized by an isomorphism M ≃ Σ × T over T . The following definition of equisingularity makes sense for arbitrary com- plex germs (T, 0), even for Artinian ones. Let (C, z) be defined by f ∈ OΣ,z with multiplicity m = mt(f ). Definition 1.5. (Equisingular deformation) If (C, z) is smooth, any deformation of (C, z) is called equisingular. If (C, z) is singular let T ∗ = 6  e(q) T ∗ (C, z) be the essential tree of (C, z) and for each q ∈ T ∗ , q 6= z, let C be the germ at q of the reduced total transform of (C, z). An embedded deformation (i, Φ) of (C, z) over (T, 0) is then called equisingular, or an es–deformation, if the following three conditions hold: (1) There exists a section σ of Φ, called equimultiple section, such that Φ is equimultiple along σ. If (C, z) is defined by F ∈ OM,z this means that F ∈ Iσm , where Iσ denotes the ideal of σ(T, 0) ⊂ (M, z). (2) For each q ∈ T ∗ , q 6= z, there is a sequence of morphisms of germs (or of small representatives) π(q) Φ(q) : Ce(q) ֒→ M(q) → M → T, where Φ(q) : Ce(q) → T is an embedded deformation of C e(q) ⊂ Σ(q) along an equimultiple section σ(q) : T → Ce(q) of Φ(q) . (3) Each M(q) , q 6= z, is obtained by blowing up some M(p) along the section σ(p) , p of smaller level than q, with Ce(q) the reduced total transform of Ce(p) (for p = z we take M(p) = M, σ(p) = σ and Ce(p) = C). See [GLS07, Definition II.2.6] for a more detailed description. Remark 1.6. (1) Condition (1) of Definition 1.5 implies that for t ∈ T suffi- m , where F defines the germ (C , σ(t)) in ciently close to 0 we have Ft ∈ Iσ(t) t t (Mt , σ(t)) and we have Iσ(t) = mCt ,σ(t) . Hence, the multiplicity of (Ct , σ(t)) is constant for t near 0. For reduced base spaces this is equivalent to the given definition of equimultiplicity. (2) Recall that, for an equisingular deformation, the equimultiple sec- tions σ(q) through all essential points are unique (cf.[GLS07, Proposition II.2.8]). That is, after blowing up an equimultiple section, there is a unique section along which the blown up family is equimultiple. (3) Let (i, Φ, σ) be an embedded deformation of (C, z) ⊂ (Σ, z) along the section σ over a reduced complex germ (T, 0). Then Φ is equisingular iff the cluster graph G(Ct , σ(t)) is constant for t ∈ T . 2 Straight Equisingular Deformations We consider now a special class of equisingular deformations, originally in- troduced by Wahl, which we call straight. 7  Assume that the (unique) equimultiple sections σ(q) : T → Ce(q) from Definition 1.5 for an es-deformation of (C, z) over a complex germ (T, 0) are all trivial. We know that they can be trivialized for each q by a local isomorphism of germs M(q) ≃ Σ(q) × T over T at q, but in general not simul- taneously for all q by an isomorphism of (M, z) over T (e.g. the cross-ratio of more than three sections through one exceptional component is an invari- ant). Those deformations for which these sections can be simultaneously trivialized are called straight equisingular. Definition 2.1. (Straight equisingular deformation) A deformation with section of the reduced plane curve singularity (C, z) is called straight equisingular or a straight es-deformation if it is an equisingular deformation of (C, z) along the trivial section, such that the equimultiple sections σ(q) , q ∈ T ∗ (C, z), through the non-nodes of the reduced total transform of (C, z) from Definition 1.5 are all trivial. We denote by Def s(C,z) the category of straight es-deformations, the full subcategory of the category Def sec (C,z) of deformations with section, and by Def s(C,z) the functor of isomorphism classes of straight es-deformations. Let us give a concrete description of straight equisingular deformations, using the notations from Definition 1.5: Denote by Fe(q) ∈ OM(q) a generator of the ideal of Ce(q) ⊂ M(q) = Σ(q) ×T (the reduced total transform) and by m(q) ⊂ OΣ(q) the maximal ideal. The condition that σ(q) is the trivial equimultiple section of Φ(q) is equivalent to e m(q) Fe(q) ∈ m(q) · OΣ(q) × T , where m e(q) . e (q) is the multiplicity of C If Fb(q) ∈ OM(q) defines the total transform Cb(q) ⊂ Σ(q) × T and if m b (q) is the multiplicity of the total transform C b(q) of the curve germ (C, z), then this is also equivalent to b m(g) Fb(q) ∈ m(q) · OΣ(q) ×T . This can be seen easily by induction on the number of blowing ups to resolve (C, z), using Remark 1.3. For q = z, we understand Ce(q) = Cb(q) = C, defined by F ∈ OΣ×T , and both conditions mean F ∈ mm z ·OΣ×T with m = mt(C, z). So far everything works for an arbitrary complex germ (T, 0). If (T, 0) is reduced then the equimultiplicity condition for the trivial sections σ(q) , 8 q ∈ T ∗ , is equivalent to mt(C, z) = mt(Ct , z) for q=z and for q 6= z either to e(q) = mt(Ce(q),t , q) mt C or (equivalently) to b(q) = mt(Cb(q),t , q) mt C for all t ∈ T sufficiently close to 0. Ce(q),t resp. Cb(q),t denotes the reduced total, resp. the total transform of the fibre Ct of Φ over t. For (T, 0) = (C, 0) we can describe the straight equisingularity condition even more explicitly. We do this for the total transform Fb(q) ∈ OΣ(q) ×T of F ∈ OΣ×T = OΣ,z {t}. Then F can be written as F = f + tg1 + t2 g2 + ... with f, gi ∈ OΣ,z and f defining (C, z). Then Fb(q) reads as Fb(q) = fb(q) + tb g1,(q) + t2 gb2,(q) + ... where fb(q) , gbi,(q) ∈ OΣ(q) denote the total transforms of f , gi .   If we fix t = t0 ∈ T we write Ft0 = F t=t0 ∈ OΣ,z and Fb(q),t0 = Fb(q) t=t0 ∈ OΣ(q),q . Then the equisingularity condition is equivalent to mt(f ) = mt(Ft ) , mt(fb(q) ) = mt(Fb(q),t ) for all t ∈ T sufficiently close to 0 and for all q ∈ T ∗ , q 6= z. Thus we get: Lemma 2.2. Let f ∈ C{x, y} define a reduced plane P curve singularity (C, 0) ∗ with essential tree T and let F (x, y, t) = f (x, y) + i≥1 ti gi (x, y) define a one-parametric deformation of (C, 0). Then the following are equivalent: (i) The deformation of (C, 0) defined by F is equisingular and the (unique) equimultiple sections through the infinitely near points q ∈ T ∗ are trivial, i.e., the deformation is straight equisingular. 9  (ii) For all i ≥ 1, mt(f ) ≤ mt(gi ), mt(fb(q) ) ≤ mt(b gi,(q) ) f or q ∈ T ∗ , q 6= 0. We define now an ideal defining the topological singularity scheme, that will be identified in Corollary 2.8 with the tangent ideal to straight es- deformations. Definition 2.3. (Topological singularity ideal and scheme) Let (C, z) ⊂ (Σ, z) be a reduced plane curve singularity and let T ∗ = T ∗ (C, z) be the essential tree of (C, z). The ideal I s (C, z) := I(C, T ∗ (C, z)) ⊂ OΣ,z   := g ∈ OΣ,z  mt b b(q) for each q ∈ T ∗ g(q) ≥ mt C is called the topological singularity ideal of (C, z). It defines the topological singularity scheme Z s (C, z) := V (I s (C, z)), a zero-dimensional subscheme of Σ supported at {z}. Remark 2.4. Let T = T (C, z) be any constellation of (C, z). Then we can define more generally the cluster ideal of (C, z) w.r.t. T ,   I(C, T ) = g ∈ OΣ,z  mt gb(q) ≥ mt C b(q) for each q ∈ T , and Z(C, T ) = V (I(C, T )) the cluster scheme of (C, z) w.r.t. T , which is supported at {z}. The following lemma can be proved by induction on the cardinality of T ∗ (C, z), see [GLS18, Lemma I.1.22]. Lemma 2.5. For (C, z) ⊂ (Σ, z) a reduced plane curve singularity we have X mq (mq + 1) deg Z s (C, z) = dimC OΣ,z /I s (C, z) = , 2 ∗ q∈T (C,z) with mq = mt C(q) , the multiplicity of the strict transform of C at q. Example 2.6. (a) Let (C, z) be smooth. For the empty constellation T we obtain I(C, T ) = OΣ,z , that is, the scheme Z(C, T ) is the empty scheme. If (C, z) has the local equation y = 0 and if T = (z = q0 , q1 , . . . , qn ) is the constellation obtained by blowing up (C, z) n times, then I(C, T ) = hy, xn+1 i ⊂ C{x, y}. (b) If (C, z) is an ordinary m-fold singularity (i.e. m smooth branches with different tangents) then T ∗ (C, z) = (z) and I s (C, z) = mm Σ,z . 10 The following lemma shows the relation of cluster schemes to equisingular deformations of curve germs: Lemma 2.7. Let Z s (C, z) be the topological singularity scheme of (C, z) defined by I s (C, z). (a) A generic element g ∈ I s (C, z) defines Z s (C, z), in the sense that Z s (C, z) = Z s (C ′ , z) for (C ′ , z) the plane curve singulariy defined by g. (b) The elements g ∈ I s (C, z) defining Z s (C, z) have no common (in- finitely near) base point outside of T ∗ (C, z). (c) Two generic elements g, g ′ ∈ I s (C, z) are topologically equivalent. Note that “generic” in (a) means: there exists a polynomial defining Z s (C, z), and if d0 is the minimal degree of such a polynomial, then, for each d ≥ d0 , the set of polynomials g ∈ I s (C, z) of degree at most d defining Z s (C, z) is a Zariski-open, dense subset in the vector space of all polynomials of degree at most d and contained in I s (C, z). Proof. (a) Let f ∈ OΣ,z be a defining equation for (C, z). As f is (analyt- ically) finitely determined, the essential tree T ∗ (C, z) depends only on a sufficiently high jet of f . We may therefore assume that f is a polynomial of degree d. Then the polynomials g ∈ I s (C, z) of degree ≤ d are parameterized by a finite dimensional vector space of positive dimension. The conditions mt gb(q) = mt C b ′ , q ∈ T ∗ (C ′ , z), define a Zariski-open subset of it. The (q) density follows, since for almost all t ∈ C the germ f + tg has precisely the multiplicities mb q = mt C b′ at each q ∈ T (C ′ , z). (q) For the proof of (b) we refer to Proposition [GLS18, Proposition I.1.17]. (c) follows from (a), since Z s determines the topological type. We relate below the topological singularity ideal I s (C, z) to the equisin- gularity ideal     f + εg is an es -deformation of (C, z) es es I (C, z) = I (f ) := g ∈ OΣ,z   over Tε (along some section) resp., the fixed equisingularity ideal     f + εg is an es -deformation of (C, z) es es  Ifix (C, z) = Ifix (f ) := g ∈ OΣ,z  . over Tε along the trivial section 11 They satisfy I es (C, z) = j(f ) + Ifesix (C, z). Moreover, if j(f ) denote the Jacobian ideal we call I ea (C, z) := hf, j(f )i resp. Ifeaix (C, z) := hf, mz j(f )i the Tjurina ideal resp. the fixed Tjurina ideal. Tε denotes the fat point with structure sheaf C[ε], ε2 = 0, and Def − (C,z) (Tε ) is the tangent space to the − deformation functor Def (C,z) , i.e., Def − − − (C,z) (Tε ) = {g ∈ OΣ,z |f + εg ∈ Def (C,z) (Tε )}/isomorphism in Def (C,z) . Corollary 2.8. Let (C, z) ⊂ (Σ, z) be a reduced plane curve singularity, defined by f ∈ OΣ,z . (1) If g ∈ I s (C, z)f , then f + tg are topologically equivalent for all but finitely many t ∈ C and f + tg defines an equisingular deformation of (C, z) over (C, 0) along the trivial section. (2) We get     f + εg is a straight es-deformation of (C, z) s s I (C, z) = I (f ) = g ∈ OΣ,z   over Tε (along the trivial section) and I s (C, z)/Ifeaix (C, z) is the tangent space to the functor of (isomorphism classes of ) straight es-deformations. (3) We have the inclusions Ifeaix (C, z) ⊂ I ea (C, z) ⊂ I es (C, z) and Ifeaix (C, z) ⊂ I s (C, z) ⊂ Ifesix (C, z) ⊂ I es (C, z). es,f ix Def es es ea es ea (C,z) (Tε ) = I (C, z)/I (C, z) resp. Def (C,z) (Tε ) = If ix (C, z)/If ix (C, z) is the tangent space to the functor of isomorphism classes of es-deformations resp. of es-deformations with (trivial) section. The dimensions satisfy dimC I es (C, z)/Ifesix (C, z) = dimC I ea (C, z)/Ifeaix (C, z) = 2, dimC Ifesix (C, z)/Ifeaix (C, z) = dimC I es (C, z)/I ea (C, z). Moreover, the forgetful morphism Ifesix (C, z)/Ifeaix (C, z) → I es (C, z)/I ea (C, z) is an isomorphism. 12 Proof. (1) follows from the proof of Lemma 2.7, (2) from the proof of Lemma 2.2 applied to f + tg, t2 = 0. (3) The inclusion Ifeaix (C, z) ⊂ I s (C, z) follows form the fact that for g ∈ Ifeaix (C, z) the deformation f + εg is trivial along the trivial section, hence straight, over Tε . The other inclusions follow from the definitions. For the first dimension statements see [GLS18, Lemma I.2.13]. The second follows from the first and the following exact sequences, 0 → Ifesix (C, z)/Ifeaix (C, z) → I es (C, z)/Ifeaix (C, z) → I es (C, z)/Ifesix (C, z) → 0, 0 → I ea (C, z)/Ifeaix (C, z) → I es (C, z)/Ifeaix (C, z) → I es (C, z)/I ea (C, z) → 0. The statement about the forgetful morphism follows, since both spaces have the same dimension and the cokernel is I es (C, z)/j(f ) + Ifesix (C, z) = 0. Remark 2.9. If a deformation functor has a semiuniversal object, then the tangent space to the functor (see [GLS07, App. C]) coincides with the Zariski tangent space of the semiuniversal deformation. We set τ s (C, z) := dimC OΣ,z /I s (C, z), τ es (C, z) := dimC OΣ,z /I es (C, z), τfesix (C, z) := es (C, z), dimC OΣ,z /Ifix τ ea (C, z) := dimC OΣ,z /I ea (C, z), τfeaix (C, z) := ea (C, z), dimC OΣ,z /Ifix with τ ea (C, z) = dimC Def (C,z) (Tε ) (usual deformations), τ ea (C, z)f ix = dimC Def sec (C,z) (Tε ) (deformations with section), τ ea (C, z) − τ es (C, z) = dimC Def es (C,z) (Tε ) (es-deformations), τ ea (C, z) − τfesix (C, z) = dimC Def es,f ix (C,z) (Tε ) (es-deformations with section), τ ea (C, z) − τ s (C, z) = dimC Def s(C,z) (Tε ) (straight es-deformations). By Corollary 2.8 we have τ es (C, z) − τfesix (C, z) = τ ea (C, z) − τfeaix (C, z) = 2. Moreover, by Lemma 2.5, X mq (mq + 1) τ s (C, z) = , 2 q∈T ∗ (C,z) with mq the multiplicity of the strict transform of (C, z) at q. 13  Let us recall the following result about straight es-deformations from [GLS07, Proposition II.2.69], basically due to Wahl [Wah74]. Proposition 2.10. Let (C, z) ⊂ (Σ, z) be a reduced plane curve singularity defined by f ∈ OΣ,z . Then the following are equivalent: (a) There are τ ′ = τ (C, z) − τ es (C, z) elements g1 , . . . , gτ ′ ∈ I es (C, z) such that  X  ′ pr ′ ϕes : V f + ti gi ⊂ (Σ× Cτ , (z, 0)) −−−→ (Cτ , 0) i is a semiuniversal equisingular deformation of (C, z). (b) There exist g1 , . . . , gτ ′ ∈ I es (C, z) inducing a basis of I es (C, z)/hf, j(f )i such that  X  ′ pr ′ ϕes : V f + ti gi ⊂ (Σ× Cτ , (z, 0)) −−−→ (Cτ , 0) i is a semiuniversal equisingular deformation of (C, z). (c) Each locally trivial deformation of the reduced exceptional divisor E of a minimal embedded resolution of (C, z) ⊂ (Σ, z) is trivial. (d) I es (C, z) = hf, j(f ), I s (C, z)i. (e) Each equisingular deformation of (C, z) is straight equisingular. Note that straight equisingular deformations are deformations with sec- tion while equisingular deformations are deformations without section. For- getting the section we get a morphism from the category Def s(C,z) of straight es-deformations to the category Def es(C,z) of es-deformations. Saying that an equisingular deformation is straight equisingular means that it is isomor- phic (as deformation without section) to the image in Def es (C,z) of a straight equisingular deformation. The image Def ws s es (C,z) of Def (C,z) in Def (C,z) is called the category of straight es-deformation without section. It follows that Def ws s (C,z) (Tε ) = hj(f ), I (C, z)i / hf, j(f )i is the tangent space to the functor of isomorphism classes of straight equi- singular deformation of (C, z) without section. We mention also the following consequence for semiquasi-homogeneous and Newton nondegenerate singularities from [GLS07, Proposition II.2.17 and Corollary II.2.71]. 14 Corollary 2.11. Let (C, z) ⊂ (Σ, z) be a reduced plane curve singularity defined by f ∈ OΣ,z , and let τ ′ = τ (C, z) − τ es (C, z). (a) If f = f0 + f ′ is semiquasi-homogeneous with principal part f0 being quasi-homogeneous of type (w1 , w2 ; d), then I es (C, z) = hj(f ), I s (C, z)i = hj(f ), xα y β | w1 α + w2 β ≥ di and a semiuniversal equisingular deformation for (C, z) is given by  Xτ ′  es ′ pr ′ ϕ :V f+ ti gi ⊂ (Σ× Cτ , (z, 0)) −−−→ (Cτ , 0) , i=1 for suitable g1 , . . . , gτ ′ representing a C-basis for the quotient  hj(f ), xα y β | w1 α + w2 β ≥ di hf, j(f )i . (b) If f is Newton non-degenerate with Newton diagram Γ(f ) at the origin, then I es (C, z) = hj(f ), I s (C, z)i = hj(f ), xα y β | xα y β has Newton order ≥ 1i and a semiuniversal equisingular deformation for (C, z) is given by  Xτ ′  es ′ pr ′ ϕ :V f+ ti gi ⊂ (Σ× Cτ , (z, 0)) −−−→ (Cτ , 0) , i=1 for suitable g1 , . . . , gτ ′ representing a monomial C-basis for the quo- tient  hj(f ), xα y β | xα y β has Newton order ≥ 1i hf, j(f )i . Moreover, in both cases each equisingular deformation of (C, 0) is straight equisingular. Remark 2.12. (1) We can extend the chosen basis of I es (C, z)/hf, j(f )i to a basis g1 , . . . , gτ of OΣ,z /hf, j(f )i, showing that the base space of the semiu- niversal es-deformation is the linear subspace {tτ ′ +1 = P · · · = tτ = 0} of the usual semiuniversal deformation of (C, z) given by f + τi=1 ti gi . (2) Note that in Proposition 2.10(b) and Corollary 2.11 not every (mono- mial) basis of I es (C, z)/hf, j(f )i has the claimed property (this was not clearly formulated in [GLS07]). We illustrate this by an example (due to Marco Mendes), using Singular [DGPS]. 15 Example 2.13. Let f = (y 3 +x7 )(y 3 +x10 ), which is Newton-nondegenerate. The quotient Q := I es (C, z)/hf, j(f )i is the tangent space to the stratum of es-deformations (µ-constant stratum) of f , but in general not the stratum itself. We show that the monomial x10 y 2 is part of a monomial basis of Q. It is, however, below the Newton diagram (the monomials on or above the Newton diagram are those of Newton order1 ≥ 1) and we will see that the deformation f + tx10 y 2 is not equisingular. At the end we construct a basis of monomials of Q of Newton order ≥ 1, explaining some of the peculiarities of Singular. LIB "all.lib"; //loads all libraries ring r = 0,(x,y),ds; //local ring with degree ordering poly f = (y3+x7)*(y3+x10); list L = esIdeal(f,1); //computes all 3 es-ideals: //L[1]: = I^{es} = es-ideal of Wahl //L[2]: = I^{es}_fix //L[3]: = I^s = top. singularity ideal ideal Ifix = std(L[2]); //we continue with I^{es}_{fix} Ifix; //-> Ifix[1]=2xy5+x8y2+x11y2 //-> Ifix[2]=y6 //-> Ifix[3]=x5y4 //-> Ifix[4]=x7y3 //-> Ifix[5]=x10y2 //-> Ifix[6]=x14y //-> Ifix[7]=x17 ideal tj = std(jacob(f)+f); //the Tjurina ideal ideal Ies = std(L[1]); //this is I^{es} NF(Ies,tj)+0; //the non-zero elements not in tj //-> _[1]=x5y4 //-> _[2]=x10y2 //-> _[3]=x14y This shows that the monomial x10 y 2 is part of a monomial basis of Q. The deformation f + tx10 y 2 is not equisingular as we check by computing the Milnor numbers: milnor(f); 1 We say that a monomial has Newton order d ∈ R (w.r.t. f ) iff it corresponds to a point on the hypersurface d · Γ(f ) ⊂ R2 . 16 //-> 71 ring rt = (0,t),(x,y),ds; poly ft = (y3+x7)*(y3+x10) + t*x10y2; milnor(ft); //-> 70 To compute a monomial bases of Q with Newton order ≥ 1, we first construct a set of monomials containing the desired basis: setring r; int d = deg(highcorner(tj)); //determines the monomials inside tj ideal k; int i; for (i = 1; i <= d; i++) { k=k+maxideal(i); } The ideal k contains all monomials of degree ≤ d. The monomials of degree > d belong to the ideal tj (cf. [GP07, Lemma 1.7.17]). To compute the monomials in k on or above the Newton diagram, let n1, n2 be the affine linear polynomials defining the faces of the Newton diagram. We compute the Newton order of the monomials by computing their expo- nents and then take the minimum value of n1 and n2 at these exponents. poly n1 = y + 3/10x - 51/10; poly n2 = y + 3/7x - 6; int e1,e2; poly mi ; ideal J; for (i = 1; i<=size(k); i++) { e1 = leadexp(k[i])[1]; e2 = leadexp(k[i])[2]; mi = min(subst(n1,x,e1,y,e2),subst(n2,x,e1,y,e2)); if (mi >=0) {J = J + k[i];} } The elements of J generate Q, all have Newton order ≥ 1. We can now of course get a basis out of Q by homogenization and applying linear algebra methods. The Singular command reduce(J, tj); creates a basis of Q, however, the reduction process via standard bases produces x10 y 2 , which is a monomial of the generator 2y 5 + x7 y 2 + x10 y 2 of tj, as part of this basis. 17 A good chance to get the right basis with elements of Newton order ≥ 1 is to use the reduction process in a ring with a dp ordering, since this ordering prefers higher degrees (and works in our example). ring R = 0, (x,y),dp; ideal J = imap(r,J); ideal tj = std(imap(r,tj)); ideal N = reduce(J, tj); N = normalize(N+0); N; //just to get a nice-looking basis //-> N[1]=x3y5 //-> N[2]=x4y5 //-> N[3]=x5y4 //-> N[4]=x5y5 //-> N[5]=x6y4 //-> N[6]=x14y //-> N[7]=x15y The 7 elements of N have all Newton order ≥ 1. We check the dimension by computing dimC Q = τ (f ) − τ es (f ). setring r; int tau = tjurina(f); tau; //-> 59 int tes = vdim (Ies); tes; //-> 52 tau - tes; // -> 7 We show now that the functor of straight es-deformations is in general a linear subfunctor of all es-deformations (with and without section). Proposition 2.14. Let (C, z) ⊂ (Σ, z) be a reduced plane curve singularity defined by f ∈ OΣ,z . (1) There exist g1 , . . . , gτ ′ ∈ I s (C, z) inducing a basis of I s (C, z)/hf, mz j(f )i such that  X  ′ pr ′ s ϕ :V f+ ti gi ⊂ (Σ× Cτ , (z, 0)) −−−→ (Cτ , 0) , i is a semiuniversal straight equisingular deformation of (C, z). (2) There exist g1 , . . . , gτ ′′ ∈ hj(f ), I s (C, z)i inducing a basis of of the quotient hj(f ), I s (C, z)i / hf, j(f )i such that  X  ′′ pr ′ ϕs : V f + ti gi ⊂ (Σ× Cτ , (z, 0)) −−−→ (Cτ , 0) , i 18 is a semiuniversal straight equisingular deformation of (C, z) without section. Proof. (1) That the deformation ϕs is straight es follows from Lemma 2.2 and its proof. Extending the chosen basis of I s (C, z)/hf, mz j(f )i to a basis of OΣ,z /hf, mz j(f )i shows that the base space of ϕs is a linear subspace s f ix BC,z of the base space BC,z of the semiuniversal deformation of (C, z) with (trivial) section. If a given deformation over an arbitrary base germ (T, 0) f ix is straight es, it can be induced by a map ϕ : (T, 0) → BC,z . ϕ must factor through BC,zs by definition of I s because otherwise the deformation is not equimultiple along the trivial sections through all infinitely near points q ∈ T ∗ (C, z). This shows that ϕs is complete. Moreover, for extensions of Artinian base spaces (T, 0) ⊂ (T ′ , 0), the equimultiple deformations along the trivial sections over (T, 0) can can be clearly extended to equimulti- ple deformations along the trivial sections over (T ′ , 0). Hence the versality follows. (For “complete” and “versal” see [GLS07, Definition II.1.8]). (2) follows from (1), noting that hj(f ), I s (C, z)i / hf, j(f )i is the tangent space to the functor of isomorphism classes of straight equisingular defor- mation of (C, z) without section. As in Remark 2.12 the base space of the semiuniversal straight equisingular deformation of (C, z) without section can be realized as a linear subspace of the usual semiuniversal deformation of (C, z). 3 Relation to the Rooted Hilbert Scheme In this section we show that straight es-deformations of a plane curve singu- larity (C, z) ⊂ (Σ, z) appear as fibres of a morphism from all es-deformations to the “rooted Hilbert scheme”, a certain punctual Hilbert scheme in Σ parametrizing topological singularity schemes of plane curve singularities with fixed cluster graph. Recall that the Hilbert scheme of points on Σ parametrizes families of zero-dimensional schemes, i.e., complex subspaces of Σ concentrated on finitely many points. Definition 3.1. Let T be an arbitrary complex space. A family of zero- dimensional schemes (of degree n) on Σ over T is a commutative diagram of complex spaces j Z ❂ / Σ×T  ❂❂ ✇✇ ϕ  {✇✇ pr T with j : Z ֒→ Σ × T a closed embedding, pr the projection and ϕ = pr ◦j finite and flat. The fibre Zt := ϕ−1 (t), t ∈ T , is mapped by j to a zero- 19 dimensional subscheme of Σ and if its degree is n we say that j : Z ֒→ Σ × T is a family of zero-dimensional schemes of degree n. We usually identify two families j : Z ֒→ Σ × T and j ′ : Z ′ ֒→ Σ × T if the images j(Z) and j ′ (Z ′ ) coincide, and then we get subspaces Z ⊂ Σ × T (of degree n) which are finite and flat over T . For Σ algebraic and algebraic families this is the Hilbert functor: The Hilbert functor HilbnΣ associates to an algebraic variety T the set   closed algebraic subvarieties Z ⊂ Σ × T HilbnΣ (T ) := . which are finite of degree n and flat over T Hence Z ⊂ Σ × T → T is a family of zero-dimensional schemes of degree n. If Σ is a projective surface, the Hilbert functor is known to be repre- sentable by a smooth projective algebraic variety of dimension 2n, denoted by Hilb nΣ , the Hilbert scheme of points in Σ. Representability means that there exists a closed algebraic subvariety UΣn ⊂ Σ × Hilb nΣ , finite of degree n and flat over Hilb nΣ , such that each element of HilbnΣ (T ) can be induced from the universal family UΣn → Hilb nΣ via base change by a unique algebraic morphism T → Hilb nΣ . There exists a birational morphism, the Hilbert-Chow morphism Φ : Hilb nΣ −→ Symn Σ , Symn Σ denoting the n-th symmetric power of Σ. It assigns to a closed subscheme Z ⊂ Σ of degree n the 0-cycle consisting of the points of Z with multiplicities given by the length of their local rings on Z. If Z consists of n distinct simple points z1 , ..., zn then Φ maps Z to the image of (z1 , ..., zn ) ∈ Σn in Symn Σ = Σn /Sn where Sn is the symmetric group permuting the coordinates. Definition 3.2. (Punctual Hilbert functor and scheme) Fix z ∈ Σ and let x, y be local coordinates at z. Then we have the punctual Hilbert functor HilbnC{x,y} , which associates to a complex space T the set   HilbnC{x,y} (T ) := (Z ⊂ Σ × T ) ∈ HilbnΣ (T )  supp(Z) ⊂ {z} × T . Fogarty [Fog68] and Hartshorne [Har66] showed that the punctual Hilbert functor is representable (in the algebraic category) by a connected projec- tive variety Hilb nC{x,y} , the punctual Hilbert scheme, which can be identified with the closed subvariety Φ−1 (n · z) ⊂ Hilb nΣ . Briançon [Bri77] proved (in the analytic categroy) that Hilb nC{x,y} is irreducible (in general not reduced) of dimension n − 1. 20 Remark 3.3. A functor analogous to the Hilbert functor can be defined for complex spaces T , associating to T the set of closed complex subspaces Z ⊂ Σ × T , finite of degree n and flat over T . This is the Douady functor, denoted by DounΣ , which is representable by a complex space Dou nΣ , the Douady space. It has a universal family VΣn → Dou nΣ with the same universal property as the Hilbert functor but for morphisms of complex spaces, i.e. each element of DounΣ (T ) can be induced from VΣn → Dou nΣ by a unique analytic morphism T → Dou nΣ . The Hilbert scheme and the Douady space exist in much greater gener- ality and their relation is discussed in detail in [GLS18, Section II.2.2.1]. We just mention that for families of zero-dimensional schemes on the smooth projective surface Σ, the analytification of Hilb nΣ and Dou nΣ are isomorphic as complex spaces and that UΣn ⊂ Σ × Hilb nΣ has the universal property for complex spaces T and analytic morphisms T → Hilb nΣ (proved in greater generality in [GLS18, Proposition II.2.28]). We will consider the punctual Hilbert functor as above with z a point in an arbitrary smooth complex analytic surface Σ. Since we consider families (Z ⊂ Σ × T ) ∈ HilbnΣ (T ) with supp(Z) ⊂ {z} × T , we may assume that Σ is projective and the above mentioned results hold in the algebraic as well as in the analytic category and we will always write Hilb, also in the analytic situation. In order to define the “rooted Hilbert functor”, a subfunctor of the punc- tual Hilbert functor, we have to extend the definition of a cluster graph to arbitrary zero-dimensional subschemes of Z ⊂ Σ. We just sketch the defini- tion and refer to [GLS18] for details. We set deg Z = dimC H 0 (Z, OZ ), the degree of Z, and mt(Z, z) the minimum order at z of the elements contained in the ideal IZ defining Z, the multiplicity of Z. Given a zero-dimensional scheme Z0 := Z ⊂ Σ =: Σ0 , let πi : Σi → Σi−1 be the blowing up of supp(Zi−1 ) ⊂ Σi−1 , and let Zi be the strict transform of Zi−1 , i = 1, . . . , r. Note that supp(Zi ) consists of finitely many infinitely near points (of level i) and that deg Zi is strictly decreasing (which can be seen, for instance, by using standard bases). We choose r minimal with the property supp(Zr ) = ∅. Then the (isomorphism class of the) sequence
K := supp(Z0 ), π1 , supp(Z1 ), . . . , πr−1 , supp(Zr−1 ) is called a constellation on Σ (generalizing constellations for plane curve singularities defined in Section 1). Setting mq := mt(Zi , q) for each q ∈ supp(Zi ) , 21 and m = (mq )q then Cℓ(Z) := (K , m) is called the cluster defined by the zero-dimensional scheme Z. Similar as in Definition 1.2 we define the graph ΓK of K and a proximity relation (qj 99K qi ) on the points of K and call the triple G(Z) := G(Cℓ(Z)) := (ΓK , 99K, m) P m (m +1) the cluster graph of the zero-dimensional scheme Z. We set n := q∈K q 2 q . The support of Z is called the root of G. Let G be the cluster graph of a zero-dimensional scheme on Σ. We define the Hilbert functor with fixed cluster graph G on the category of reduced complex spaces as   HilbG n  Σ (T ) = (Z ⊂ Σ × T ) ∈ HilbΣ (T ) G(Zt ) = G for all t ∈ T . It is proved in [GLS18, Proposition I.1.52] that for (Z ⊂ Σ × T ) ∈ HilbG ′ Σ (T ) there exists a complex space T and a finite surjective morphism α : T ′ → T such that the induced family α∗ Z → T ′ is resolvable by a se- (i) quence of blowing ups πi : X (i+1) → X (i) along equimultiple sections σj : T ′ → X (i) (i = 0, · · · , N, j = 0, · · · ki ) with X (0) = Σ × T, Z (0) = Z and Si (i) Z (i+1) the strict transform of Z (i) . Moreover, supp(Z (i) ) = kj=1 σj (T ) and (0) (0) (0) supp(Z (N +1) ) = ∅. The initial sections σj satisfy {σ1 (t), · · · , σk1 (t)} = root(G(Zt )). The rooted Hilbert functor is the subfunctor of HilbG Σ where the initial sections are trivial. Consider now a reduced plane curve singularity (C, z) ⊂ (Σ, z) with essential tree T ∗ and cluster graph G(C, z) = (ΓT ∗ , 99K, m), defined in Section 1. The definition of the cluster graph of a zero-dimensional scheme is modeled in such a way that we have G(C, z) = G(Z s (C, z)), where Z s (C, z) is the topological singularity scheme from Definition 2.3. The root of G(C, z) is {z}. Definition 3.4. (Rooted Hilbert functor) Let G = G(C, z) be the clus- ter graph of a reduced plane curve singularity. The rooted Hilbert functor or the punctual Hilbert functor fixing G is the subfunctor HilbG C{x,y} of HilbΣ G associating to a reduced complex space T the subspaces (Z ⊂ Σ × T ) ∈ (0) HilbGΣ (T ) such that the initial section σ := σ1 passing through z is trivial. 22  The main result about the rooted Hilbert functor is the following theo- rem. It is used in [GLS18, Section IV.6] to prove asymptotically sufficient conditions for the irreducibility of the variety of plane projective curves of given degree with a fixed number of singularities of given topological type. Theorem 3.5. ([GLS18, Theorem I.1.55, I.1.57]) The rooted functor G HilbG C{x,y} is representable by Hilb C{x,y} , the rooted Hilbert scheme, an ir- reducible, quasi-projective subvariety of the projective variety Hilb nC{x,y} , of dimension equal to the number of free vertices in T ∗ (C, z) \ {z}. Let S denote the topological type of (C, z). Since G = G(C, z) is a complete invariant of S, HilbG C{x,y} depends only on S and we can introduce the punctual Hilbert scheme associated to a topological type S, H0s (S) := Hilb G C{x,y} . It is shown in [GLS18, Proposition I.1.17] that each point in H0s (S) cor- responds to a topological singularity scheme Z s (C, w) of a plane curve singu- larity (C, w) of topological type S and with deg Z s (C, w) = dimC OΣ,z /I s (C, w) = P m (m +1) n, with n = q∈T ∗ (C,w) q 2 q , mq the multiplicity of the strict transform of (C, w) at q, and with fixed cluster graph G = G(Z s (C, w)). Before we relate H0s (S) to straight equisingular deformations, we consider several related semiuniversal base spaces. Lemma 3.6. For a reduced plane curve singularity (C, z) the following base spaces of semiuniversal deformations of (C, z) are smooth of the given di- mension. (1) BC,z the base space of the (usual) semiuniversal deformation of dimension τ ea (C, z) = τ (C, z), f ix (2) BC,z the base space of the semiuniversal deformation with section of dimension τfeaix (C, z) = τf ix (C, z), es the base space of the seminuniversal equisingular deformation (3) BC,z of dimension τ ea (C, z) − τ es (C, z), es,f ix (4) BC,z the base space of the seminuniversal es-deformation with sec- tion of dimension τfeaix (C, z) − τfesix (C, z) = τ ea (C, z) − τ es (C, z), 23  s (5) BC,z the base space of the seminuniversal straight es-deformation of dimension τfeaix (C, z) − τ s (C, z). Note that the definitions of τ ea and τ es are consistent in the follow- ing sense. τ es (C, z) is the codimension in BC,z of the base space of the semiuniversal equisingular deformation of (C, z) (which coincides with the µ-constant stratum in BC,z ), while τ ea (C, z) is the codimension in BC,z of the base space of the semiuniversal equianalytic deformation of (C, z) (which is the reduced point). f ix Proof. For BC,z and BC,z the smoothness and the dimension statements are well known. By [GLS07, Theorem II.2.61, Corollary II.2.67 and Proposition es is isomorphic to the µ-constant stratum in B II.2.63] we know that BC,z C,z es,f ix f ix and smooth, while BC,z is isomorphic to the µ-constant stratum in BC,z . s The smoothness of BC,z follows from Proposition 2.14. The dimension state- ments are a consequence of the smoothness and of Corollary 2.8, since then the dimension coincides with the dimension of the Zariski tangent space. es,f ix To see the smoothness of BC,z and the second equality in (4) note that in general deformations with section are isomorphic to deformations with trivial section (cf. [GLS07, Proposition II.2.2]) and we will identify the corresponding semiuniversal base spaces B(C,z) sec = B f ix . Moreover, the for- (C,z) getful morphism of functors from deformations with section to deformations without section is smooth by [GLS07, Corollary II.1.6]. It follows that the es,f ix es is smooth (flat with smooth induced morphism of base spaces BC,z → BC,z es,f ix fibre). Hence BC,z is smooth and of the same dimension as its tangent 1,es,f ix space, namely dimC TC,z = τf ix (C, z) − τfesix (C, z) = τ (C, z) − τ es (C, z) by Corollary 2.8. As this is the dimension of BC,z es by Proposition 2.14, we f ix,es es is an isomorphism (reflecting the fact get that the morphism BC,z → BC,z that the equisingular section is unique). We state now the main result about the relation between equisingular deformations of (C, z) along the trivial section, the induced deformation of the topological singularity scheme Z s (C, z), and straight equisingular defor- mations of (C, z). Theorem 3.7. With the above notations there exists a surjective morphism of germs es,f ix ψ : BC,z → H0s (S), 24 with smooth fibre ψ −1 (Z s (C, z)) ∼ s . Moreover, we have = BC,z dim H0s (S) = #{free vertices q ∈ T ∗ (C, z) \ {z}} = dimC Ifix es (C, z)/I s (C, z) = s deg Z (C, z) − τ es ((C, z) − 2 = τ s (C, z) − τfesix (C, z) P mq (mq+1) = q∈T ∗ (C,z) 2 − τfesix (C, z) with mq the multiplicity of the strict transform of (C, z) at q. Proof. Equisingular deformations C ⊂ Σ × T → T of (C, z) along the trivial section over a reduced germ T are exactly those, which are equimultiple along the trivial section σ(t) = z and equimultiple along the (not necessarily trivial) sections σ(q) , q ∈ T ∗ (C, z), through the non-nodes of the reduced total transform of (C, z) in a resolution of (C, z). Equisingularity implies that the cluster graph G(Ct , z) = G(Z s (Ct , z)) is constant and equal to G = G(Z s (C, z)). Hence deg Z s (Ct , z) is constant and Z = {Z s (Ct , z)}t∈T is a flat family of zero-dimensional schemes in HilbG C{x,y} (T ), a deformation s of Z (C, z). Since the association (C → T ) → (Z → T ) is functorial in T and respects isomorphism classes, we get a morphism of functors Def es,f C,z ix → HilbG C{x,y} , where Def es,f C,z ix is the functor of isomorphism classes of equisingular defor- mations of (C, z) with trivial section. A family in HilbG C{x,y} is trivial if it is given by the subspace Z s (C, z) × T ⊂ Σ × T , i.e., iff the sections through the infinitely near points q ∈ T ∗ (C, z) are trivial. Hence the fibres of the morphism of functors are exactly the straight equisingular deformations of (C, z). es,f ix The morphism of functors induces a morphism of germs ψ : BC,z → s H0 (S), which is surjective (by [GLS18, Proposition I.1.17]) with fibre over Z s (C, z) being isomorphic to BC,z s , since every straight es-deformation of es,f ix (C, z) over T can be induced via a map T → BC,z which must factor −1 s through ψ (Z (C, z)). By Theorem 2.9 we get the first formula for the dimension of H0s (S). For the others we use the theorems of Frisch and Sard to see that there are points Z s (C ′ , z ′ ) in H0s (S) arbitrary close to Z s (C, z), over which ψ is flat and hence satisfies dim(H0s (S), Z s (C ′ , z ′ )) = τ (C ′ , z ′ ) − τ es (C ′ , z ′ ) − dimC I s (C ′ , z ′ )/I ea (C ′ , z ′ ) = deg Z s (C ′ , z ′ ) − τfesix (C ′ , z ′ ). 25 Since deg Z s (C ′ , z ′ ) = deg Z s (C, z) and since H0s (S) is irreducible (by Theo- rem 3.5) its dimension is constant and thus we get τfesix (C ′ , z ′ ) = τfesix (C, z). Using Remark 2.9 we get the last four dimension formulas. Remark 3.8. For any es-deformations there exists (unique) sections σ(q) through q ∈ T ∗ = T ∗ (C, z) along which the deformation of the reduced to- tal transform is equimultiple. For straight es-deformations f + tg over (C, 0) with g ∈ I s (f ) these sections are all trivial. For arbitrary es-deformations es (f ) the sections through satellite points q ∈ T ∗ with tangent directions in Ifix have to stay at satellite points. But the sections through free points in T ∗ \ {z} may move along the exceptional divisor, giving one degree of free- dom for every free point. Since H0s (S) represents es-deformations with trivial initial section, we get a geometric interpretation of the formula dim H0s (S) = #(free vertices q ∈ T ∗ \ {z}). References [Bri77] Briançon, J.: Description de Hilbn C{x, y}. Invent. Math. 41, 45–89 (1977). [CGL07a] Campillo, A.; Greuel, G.-M.; Lossen, C.: Equisingular Deforma- tions of Plane Curves in Arbitrary Characteristic. Compos. Math. 143, 829–882 (2007). [CGL07b] Campillo, A.; Greuel, G.-M.; Lossen, C.: Equisingular calcula- tions for plane curve singularities. J. Symb. Comput. 42, No. 1-2, 89-114 (2007). [Cas00] Casas-Alvero, E.: Singularities of Plane Curves. LMS 276, Cam- bridge Univ. Press (2000). [DGPS] Decker, W.; Greuel, G.-M.; Pfister, G.; Schönemann, H.: Singu- lar 4-1-1 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2018). [Fog68] Fogarty, J.: Algebraic families on an algebraic surface. Amer. J. Math. 90, 511–521 (1968). [GLS07] Greuel, G.-M.; Lossen, C.; Shustin, E.: Introduction to Singulari- ties and Deformations. Springer (2007). [GLS18] Greuel, G.-M.; Lossen, C.; Shustin, E.: Singular Algebraic Curves. Springer (2018). 26 [GP07] Greuel, G.-M.; Pfister, G.: A Singular Introduction to Commuta- tive Algebra. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schnemann. 2nd extended ed., Springer, Berlin, Heidelberg, New York (2007). [Har66] Hartshorne, R.: Connectedness of the Hilbert scheme. Publ. Math. IHES 29, 261–304 (1966). [Wah74] Wahl, J.: Equisingular deformations of plane algebroid curves. Trans. Amer. Math. Soc. 193, 143–170 (1974). Email: [email protected] 27