Versal Deformation Space Research Articles

Vanishing cycles, plane curve singularities and framed mapping class groups

Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal deformation space, and (b) an easy criterion to decide if a given simple closed curve in the Milnor fiber is a vanishing cycle or not. With the lone exception of singularities of type $A_n$ and $D_n$, we find that both are determined completely by a canonical framing of the Milnor fiber induced by the Hamiltonian vector field associated to f. As a corollary we answer a question of Sullivan concerning the injectivity of monodromy groups for all singularities having Milnor fiber of genus at least 7.

Non-reduced moduli spaces of sheaves on multiple curves

Abstract Some coherent sheaves on projective varieties have a non-reduced versal deformation space; for example, this is the case for most unstable rank 2 vector bundles on ℙ2, see [18]. In particular, some moduli spaces of stable sheaves are non-reduced. We consider some sheaves on ribbons (double structures on smooth projective curves): let E be a quasi locally free sheaf of rigid type and let 𝓔 be a flat family of sheaves containing E. We find that 𝓔 is a reduced deformation of E when some canonical family associated to 𝓔 is also flat. We consider also a deformation of the ribbon to reduced projective curves with two components, and find that E can be deformed in two distinct ways to sheaves on the reduced curves. In particular some components M of the moduli spaces of stable sheaves deform to two components of the moduli spaces of sheaves on the reduced curves, and M appears as the “limit” of varieties with two components, whence the non-reduced structure of M.

Milnor fibers and symplectic fillings of quotient surface singularities

We determine a one-to-one correspondence between Milnor fibers and minimal symplectic fillings of a quotient surface singularity (up to diffeomorphism type) by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3-folds and Pinkham's negative weight smoothing. As by-products, we show that:– Milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions. For this, we classify minimal symplectic fillings of a quotient surface singularity up to diffeomorphism.– Any symplectic filling of a quotient surface singularity is obtained by a sequence of rational blow-downs from a special resolution (so-called the maximal resolution) of the singularity, which is an analogue of the one-to-one correspondence between the irreducible components of the reduced versal deformation space and the so-called P-resolutions of a quotient surface singularity.

Flipping surfaces

We study semistable extremal threefold neighborhoods following earlier work of Mori, Koll\'ar, and Prokhorov. We classify possible flips and extend Mori's algorithm for computing flips of extremal neighborhoods of type k2A to more general neighborhoods of type k1A. In fact we show that they belong to the same deformation family as k2A, and we explicitly construct the universal family of extremal neighborhoods. This construction follows very closely Mori's division algorithm, which we interpret as a sequence of mutations in the cluster algebra of rank 2 with general coefficients. We identify, in the versal deformation space of a cyclic quotient singularity, the locus of deformations such that the total space admits a (terminal) antiflip. We show that these deformations come from at most two irreducible components of the versal deformation space. As an application, we give an algorithm for computing stable one-parameter degenerations of smooth projective surfaces (under some conditions) and describe several components of the Koll\'ar-Shepherd-Barron boundary of the moduli space of smooth canonically polarized surfaces of geometric genus zero.

Irreducibility of versal deformation rings in the [formula omitted]-case for 2-dimensional representations

Let GK be the absolute Galois group of a finite extension K of Qp and ρ¯:GK→GLn(F) be a continuous residual representation for F a finite field of characteristic p. We investigate whether the versal deformation space X(ρ¯) of ρ¯ is irreducible. For n=2 and p>2 we obtain a complete answer in the affirmative based on the results of [2,4]. As a consequence we deduce from recent results of Colmez, Kisin and Nakamura [10,22,28] that for n=2 and p>2 crystalline points are Zariski dense in the versal deformation space X(ρ¯).

The $$F$$ F -pure threshold of a Calabi–Yau hypersurface

We compute the \(F\)-pure threshold of the affine cone over a Calabi–Yau hypersurface, and relate it to the order of vanishing of the Hasse invariant on the versal deformation space of the hypersurface.

Une définition dʼespace de modules locaux de structures CR

In this note, we propose a definition of local moduli space in a general framework including the case of CR structures on a fixed differentiable manifold. We show that it is the same as the notion of versal deformation space for complex structures. Finally, we test this definition on the spaces constructed in [5] and [6].

On the existence of curves with a triple point on a $K$3 surface

Let (S,H) be a general primitively polarized K3 surface of genus p and let p_a(nH) be the arithmetic genus of nH. We prove the existence in |\mathcal O_S(nH)| of curves with a triple point and A_k -singularities. In particular, we show the existence of curves of geometric genus g in |\mathcal O_S(nH)| with a triple point and nodes as singularities and corresponding to regular points of their equisingular deformation locus, for every 1\leq g\leq p_a(nH)-3 and (p,n) \neq (4,1). Our result is obtained by studying the versal deformation space of a non-planar quadruple point.

ANALYTIC CONSTRUCTION OF VERSAL DEFORMATION OF RESOLUTION OF NORMAL ISOLATED SINGULARITIES

We analytically construct the versal deformation space of resolution of normal isolated singularities based on the formalism in (9).

Level Structures on the Weierstrass Family of Cubics

Let W → 𝔸 2 be the universal Weierstrass family of cubic curves over ℂ. For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 𝔸 2. Since W → 𝔸 2 is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0, 0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S 3 with monodromy in SL2 (ℤ/N).

The versal deformation space of a reflexive module on a rational cone

By an approach based on results of A. Ishii, we describe the versal deformation space of any reflexive module on the cone over the rational normal curve of degree m. For each component a resolution is given as the total space of a vector bundle on a Grassmannian. The vector bundle is a sum of copies of the cotangent bundle, the canonical sub-bundle, the dual of the canonical quotient bundle, and the trivial line bundle. Via an embedding in a trivial bundle, we obtain the components by projection. In particular we give equations for the minimal stratum in the Chern class filtration of the versal deformation space. We obtain a combinatorial description of the local deformation relation and a classification of the components. In particular we give a formula for the number of components.

The versal deformation space of a reflexive module on a rational cone

By an approach based on results of A. Ishii, we describe the versal deformation space of any reflexive module on the cone over the rational normal curve of degree m . For each component a resolution is given as the total space of a vector bundle on a Grassmannian. The vector bundle is a sum of copies of the cotangent bundle, the canonical sub-bundle, the dual of the canonical quotient bundle, and the trivial line bundle. Via an embedding in a trivial bundle, we obtain the components by projection. In particular we give equations for the minimal stratum in the Chern class filtration of the versal deformation space. We obtain a combinatorial description of the local deformation relation and a classification of the components. In particular we give a formula for the number of components.

Highly singular quintic curves

The origin of this paper lies in a study of the unfolding space of the stratumN16of singularity theory, and the question, at which points of the stratum the versal deformation space ceases to be topologically trivial over the stratum. This question turns out to be closely related to the study of how a plane section (= binary quintic) of a quintic curve varies as we deform the curve, either rigidly (underGL3) or equisingularly.

Moduli of commutative and non-commutative finite covers

We classify affine, not necessarily commutative,n-coversB of commutativeK-algebrasA using data triples (A, M, α) consisting of the algebraA, a freeA-moduleM of rankn − 1, and an associative, unitary trace-zero structure constant tensorα. We construct a versal deformation space for the deformations of aK-algebraB 0 as a section of the completion at the tensorα 0 ofB 0 of the structure-constant schemeC n . In order to obtain concrete information about the algebraic structure ofC n obtained as above.

Flache T1-Stabilit�t in der Strati-fizierung verseller Entfaltungen

Let 0 be the local ring of a simple singularity defined over the complex numbers and τ the dimension of its versal deformation space. Than it is well known that any nearby singularity in this space is also simple and has smaller unfolding dimension in the hierarchy of simple singularities. In particular this implies that the τ=max-stratum consists just of one point namely the given singularity. We want to generalize this concept as we are interested in families of varieties with formal unchanged singularities. For this we introduce in quite generality the notion of flat T1-stabi1ity which may be checked for any k- algebra 0 where k is for simplicity an algebraically closed field of a priori arbitrary characteristics. We call 0 formal flat T1 stable or for short flat T1-stable if the following is true: if R is any deformation of 0 over an Artin local finite k-algebra A and if T1(R/A,R) is A-flat than R is isomorphic to the trivial deformation\(0\mathop \otimes \limits_K A\). T1(R/A,R) is the first cotangent module of R over A with values in R. Obviously the simple singularities Ak, Dk, E6, E7, E8 fulfill this criterion over C but we look also at fibres of arbitrary stable map germs, generic singularities of algebraic varieties where we have to modify this notion in order to deal with wild ramification and to quasihomo-genous hypersurface singularities where it functorializes because in this case T1 commutes with arbitrary base change. The notion of flat T1-stable singularities is closely related to questions of existence of equisingular families and is used in[12] and [5], [6] to stratify certain Hilbert schemes.

Deformations of Cohen-Macaulay Schemes of Codimension 2 and Non-Singular Deformations of Space Curves

The following work deals with the deformations of embedded affine schemes of codimension 2, which locally have a resolution of length 2. The cases of immediate interest are curves in 3-space and 0-dimensional schemes in the plane. It is first shown that such a scheme X has a global resolution of length 2. Therefore, by a theorem of Burch, the functions defining the ideal of X can be obtained as the maximal minors of a matrix whose columns generate all the relations among these functions. All flat deformations of X can be obtained simply by deforming this matrix, and this permits the construction of the versal deformation space of X. Finally, for X of dimension 3 or less one can construct non-singular deformations of X by taking a parameter space sufficiently large to permit one to change the constant and linear terms of each entry in the matrix. For X of dimension 4, an example is given in XXX [11] in which the scheme not only has no non-singular deformations, but in fact has no non-isomorphic deformations at all. A brief review of the previous literature will help place these results in perspective. It has long been known, by Bertini's theorem, that the generic deformation of a scheme of codimension 1 is non-singular. As a consequence of the work of Fogarty [5], it was also known that every point, or rather, 0-dimensional scheme in the plane has non-singular deformations; Briancon and Galligo [3] give an explicit construction for such a deformation, splitting the scheme into distinct simple points. This led mathematicians interested in algebraic curves to ask if the generic deformation of a space curve is also non-singular, the question which is settled in this paper. Further direct extension of these results in the case of curves is impossible, since the work of Iarrobina [4] permits the construction of a non-reduced curve in affine 4-space which has no non-singular deformations; however, for reduced curves the question is still open as of this writing.

Deformations of generalized complete intersections

The concept of a generalized complete intersection (GCI) of affine schemes is introduced. The proofs of the following theorems are then sketched. Theorem 1. Suppose X X is a G C I GCI of affine Cohen-Macaulay schemes of codimension ⩽ 2 \leqslant 2 . Then if the versal deformation space of X X exists, it is smooth. Theorem 2. Suppose X X is a G C I GCI of determinantal schemes. Then if dim ⁡ X \dim X is sufficiently low, X X is smoothable. In particular, if X X is as in Theorem 1 and dim ⁡ X > 4 \dim X > 4 then X X is smoothable.