Fragmented deformations

# Fragmented deformations of primitive multiple curves

Jean–Marc Drézet Institut de Mathématiques de Jussieu, Case 247, 4 place Jussieu, F-75252 Paris, France
###### Resume.

A primitive multiple curve is a Cohen-Macaulay irreducible projective curve that can be locally embedded in a smooth surface, and such that is smooth.

This paper studies the deformations of to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity of ).

We are particularly interested in deformations to disjoint smooth irreducible components, which are called fragmented deformations. We describe them completely. We give also a characterization of primitive multiple curves having a fragmented deformation.

###### Summary

Mathematics Subject Classification: 14M05, 14B20

## 1. Introduction

A primitive multiple curve is an algebraic variety over which is Cohen-Macaulay, such that the induced reduced variety is a smooth projective irreducible curve, and that every closed point of has a neighborhood that can be embedded in a smooth surface. These curves have been defined and studied by C. Bănică and O. Forster in [ba_fo]. The simplest examples are infinitesimal neighborhoods of projective smooth curves embedded in a smooth surface (but most primitive multiple curves cannot be globally embedded in smooth surfaces, cf. [ba_ei], theorem 7.1).

Let be a primitive multiple curve with associated reduced curve , and suppose that . Let be the ideal sheaf of in . The multiplicity of is the smallest integer such that . We have then a filtration

 C=C1⊂C2⊂⋯⊂Cn=Y

where is the subscheme corresponding to the ideal sheaf and is a primitive multiple curve of multiplicity . The sheaf    is a line bundle on , called the line bundle on associated to .

The deformations of double (i.e. of multiplicity 2) primitive multiple curves (also called ribbons) to smooth projective curves have been studied in [gon]. In this paper we are interested in deformations of primitive multiple curves of any multiplicity to reduced curves having exactly components which are smooth ( is the maximal number of components of deformations of ). In this case the number of intersection points of two components is exactly . We give some results in the general case (no assumption on ) and treat more precisely the case , i.e. deformations of to curves having exactly disjoint irreducible components.

###### 1.1.

Motivation – Let be a flat projective morphism of algebraic varieties, a closed point of such that , a very ample line bundle on and a polynomial in one variable with rational coefficients. Let

 τ:MOC(1)(P)⟶S

be the corresponding relative moduli space of semi-stable sheaves (parametrizing the semi-stables sheaves on the fibers of with Hilbert polynomial with respect to the restriction of , cf. [si]).

We suppose first that there exists a closed point such that is a smooth projective irreducible curve. Then in general is not flat (some other examples on non flat relative moduli spaces are given in [in2]). The reason is that the generic structure of torsion free sheaves on is more complicated than on smooth curves, and some of these sheaves cannot be deformed to sheaves on the smooth fibers of .

A coherent sheaf on a smooth algebraic variety is locally free on some nonempty open subset of . This is not true on . But a coherent sheaf on is quasi locally free on some nonempty open subset of , i.e. on this open subset, is locally isomorphic to a sheaf of the form  , the sequence of non negative integers being uniquely determined (cf. [dr2], [dr4]). It is not hard to see that if can be extended to a coherent sheaf on , flat on , then    must be a multiple of . For example, it is impossible to deform the stable sheaf on in sheaves on the smooth fibers, if .

Now suppose that all the fibers , , are reduced with exactly smooth components. I conjecture that (with suitable hypotheses) a torsion free coherent sheaf on can be extended to a coherent sheaf on , flat on , using the fact that we allow coherent sheaves of the reducible fibers that have not the same rank on all the components. This would be a step in the study of the flatness of . For example (for suitable ), there exists a coherent sheaf on , flat on , such that , and that for , is the structural sheaf of an irreducible component of .

Moduli spaces of sheaves on reducible curves have been studied in [tei1], [tei2], [tei3].

###### 1.2.

Maximal reducible deformations – Let be the germ of a smooth curve. Let be a primitive multiple curve of multiplicity and an integer. Let   be a flat morphism, where is a reduced algebraic variety, such that

1. For every closed point such that , the fiber has irreducible components, which are smooth and transverse, and any three of these components have no common point.

2. The fiber is isomorphic to .

We show that by making a change of variable, i.e. by considering a suitable germ and a non constant morphism , and replacing with , we can suppose that has exactly irreducible components, inducing on every fiber , the irreducible components of . In this case is called a reducible deformation of of length .

We show that . We say that (or ) is a maximal reducible deformation of if .

Suppose that is a maximal reducible deformation of . We show that if is the union of irreducible components of , and is the restriction of , then , and is a maximal reducible deformation of . Let . We prove that the irreducible components of have the same genus as . Moreover, if are distinct irreducible components of , then consists of points.

###### 1.3.

Fragmented deformations (definition) – Let be a primitive multiple curve of multiplicity and a maximal reducible deformation of . We call it a fragmented deformation of if , i.e. if for every , is the disjoint union of smooth curves. In this case has irreducible components which are smooth surfaces.

The variety appears as a particular case of a gluing of along (cf. LABEL:ecl2c). We prove (proposition LABEL:ecl2b) that such a gluing is a fragmented deformation of a primitive multiple curve if and only if every closed point in has a neighborhood in that can be embedded in a smooth variety of dimension 3. The simplest gluing is the trivial or initial gluing . An open subset of (and ) is given by open subsets of respectively, having the same intersection with , and

 OA(U) = {(α1,…,αn)∈OC1(U∩C1)×⋯×OCn(U∩Cn);α1∣C=⋯=αn∣C},

and appears as a subalgebra of , hence we have a canonical morphism .

We can view elements of as -tuples , with . In particular we can write  .

###### 1.4.

A simple analogy – Consider copies of glued at 0. Two extreme examples appear: the trivial gluing (the set of coordinate lines in ), and a set of lines in . We can easily construct a bijective morphism sending each coordinate line to a line in the plane

But the two schemes are of course not isomorphic: the maximal ideal of 0 in needs generators, but 2 are enough for the maximal ideal of 0 in .

Let be a morphism sending each component linearly onto , and . The difference of and can be also seen by using the fibers of 0: we have

 π−1C0(0)≃spec(C[t]/(tn))andπ−1A0(0)≃spec(C[t1,…,tn−1]/(t1,…,tn−1)2) .

Let be a general gluing of copies of at 0, such that there exists a morphism inducing the identity on each copy of . It is easy to see that we have    if and only if some neighborhood of 0 in can be embedded in a smooth surface.

###### 1.5.

Fragmented deformations (main properties) – Let be a fragmented deformation of . Let be a proper subset, its complement, and the subscheme union of the . We prove (theorem LABEL:ecl19) that the ideal sheaf of is isomorphic to

In particular, the ideal sheaf of is generated by a single regular function on . We show that we can find such a generator such that for , , its -th coordinate can be written as , with and such that . If and , we can then obtain a generator that can be written as

 uij = (u1,…,um),

with

 ui=0,um = α(m)ijπpimm  for m≠i,α(j)ij=1.

The constants    have interesting properties (propositions LABEL:ecl17, LABEL:ecl11). Let for . The symmetric matrix is called the spectrum of (or ).

It follows also from the fact that that is a simple primitive multiple curve, i.e. the ideal sheaf of in is isomorphic to . Conversely, we show in theorem LABEL:theo_sim that if is a simple primitive multiple curve, then there exists a fragmented deformation of .

We give in LABEL:const_def and LABEL:const_def2 a way to construct fragmented deformations by induction on . This is used later to prove statements on fragmented deformations by induction on .

###### 1.6.

-stars and structure of fragmented deformations – An -star of is a gluing of copies of at , together with a morphism which is an identity on each . All the -stars have the same underlying Zariski topological space .

An -star is called oblate if some neighborhood of can be embedded in a smooth surface. This is the case if and only  .

Oblate -stars are analogous to fragmented deformations but simpler. We provide a way to build oblate -stars by induction on .

Let be a fragmented deformation of . We associate to it an oblate -star of . Let be the Zariski topological space of . We have an obvious continuous map  . For every open subset of , is the set of such that for . We obtain also a canonical morphism   . We prove (theorem LABEL:st_fr_2) that is flat. Hence it is a flat family of smooth curves, with  . The converse is also true, i.e. starting from an oblate -star of and a flat family of smooth curves parametrized by it, we obtain a fragmented deformation of a multiple primitive curve of multiplicity .

###### 1.7.

Fragmented deformations of double curves – Let be a primitive double curve, its associated smooth curve, a fragmented deformation of , of spectrum , and , the irreducible components of . For , , let be the infinitesimal neighborhood of order of in (defined by the ideal sheaf ). It is a primitive multiple curve of multiplicity .

It follows from LABEL:ecl12 that and are isomorphic, and , are two extensions of in primitive multiple curves of multiplicity . According to [dr1] these extensions are parametrized by an affine space with associated vector space (where is the tangent bundle of ). Let be the vector from to .

Similarly, the primitive double curves with associated smooth curve such that are parametrized by (cf. [ba_ei], [dr1]).

We prove in theorem LABEL:theo_cf that the point of corresponding to is .

###### 1.8.

Notation: Let be an algebraic variety and a closed subvariety. We will denote by (or if there is no risk of confusion) the ideal sheaf of in .

## 2. Preliminaries

###### 2.1.

Local embeddings in smooth varieties

###### 2.1.1.

Proposition: Let be an algebraic variety, a closed point of and a positive integer. Then the three following properties are equivalent:

1. There exist a neighborhood of and an embedding in a smooth variety of dimension .

2. The -module (maximal ideal of ) can be generated by elements.

3. We have  .

###### Proof.

It is obvious that (i) implies (ii), and (ii),(iii) are equivalent according to Nakayama’s lemma. It remains to prove that (iii) implies (i).

Suppose that (iii) is true. There exist an integer and an embedding  . Let be the ideal sheaf of in . Let be the biggest integer such that there exists whose images in the -vector space    are linearly independent. Then we have

 IX,x ⊂ (f1,⋯,fp)+m2PN,x.

In fact, let  . Since is maximal, the image of in    is a linear combination of those of . Hence we can write

 f=p∑i=1λifi+g,withλi∈C, g∈m2PN,x ,

and our assertion is proved. It follows that we have a surjective morphism

 α:OX,x/m2X,x⟶OPN,x/((f1,⋯,fp)+m2PN,x) .

We have

 dimC(OX,x/m2X,x) ≤ n+1,dimC(OPN,x/((f1,⋯,fp)+m2PN,x)) =N−p+1 .

Hence  , i.e.  . We can take for a neighborhood of in the subvariety of defined by , which is smooth at . ∎

###### 2.2.

Flat families of coherent sheaves

Let be the germ of a smooth curve and a generator of the maximal ideal. Let    be a flat morphism. If is a coherent sheaf on , is flat on at if and only if the multiplication by is injective. In particular the multiplication by is injective.

###### 2.2.1.

Lemma: Let be a coherent sheaf on flat on . Then, for every open subset of , the restriction    is injective.

###### Proof.

Let whose restriction to vanishes. We must show that . By covering with smaller open subsets we can suppose that is affine: . Hence  . Let , it is an -module. We have   and . Hence if the restriction of to vanishes, there exists an integer such that . Since the multiplication by is injective (because is flat on ), we have . ∎

Let be a coherent sheaf on flat on . Let    be a subsheaf. For every open subset of we denote by the subset of of elements that can be extended to sections of on . If is an open subset, the restriction    induces a morphism   .

###### 2.2.2.

Proposition: is a subsheaf of , and is flat on .

###### Proof.

To prove the first assertion, we must show that if is an open subset of and is an open cover of , then

1. If is such that for every we have , then .

2. For every let . Then if for all , we have   , then there exists such that for every we have  .

This follows easily from lemma 2.2.1.

Now we prove that is flat on . Let and such that . We must show that . Let over . Then we have . Let be a neighborhood of such that comes from . This means that . Since is invertible on we can write , with . We have then on . Hence and . ∎

###### 2.3.

Primitive multiple curves

(cf. [ba_fo], [ba_ei], [dr2], [dr1], [dr4], [dr5], [dr6], [ei_gr]).

Let be a smooth connected projective curve. A multiple curve with support is a Cohen-Macaulay scheme such that .

Let be the smallest integer such that , being the -th infinitesimal neighborhood of , i.e.   . We have a filtration    where is the biggest Cohen-Macaulay subscheme contained in . We call the multiplicity of .

We say that is primitive if, for every closed point of , there exists a smooth surface , containing a neighborhood of in as a locally closed subvariety. In this case, is a line bundle on and we have  ,  for . We call the line bundle on associated to . Let . Then there exist elements , of (the maximal ideal of ) whose images in form a basis, and such that for we have   .

The simplest case is when is contained in a smooth surface . Suppose that has multiplicity . Let and a local equation of . Then we have    for , in particular , and   .

We will write and we will see as a coherent sheaf on with schematic support if .

If is a coherent sheaf on one defines its generalized rank and generalized degree (cf. [dr4], 3-): take any filtration of

 0=E0⊂E1⊂⋯⊂En=E

by subsheaves such that is concentrated on for , then

 R(E)=n∑i=1rk(Ei/Ei−1)andDeg(E)=n∑i=1deg(Ei/Ei−1).

Let be a very ample line bundle on . Then the Hilbert polynomial of is

 PE(m) = R(E)deg(OC(1))m+Deg(E)+R(E)(1−g)

(where is the genus of ).

We deduce from proposition 2.1.1:

###### 2.3.1.

Proposition: Let be a multiple curve with support . Then is a primitive multiple curve if and only if is zero, or a line bundle on .

###### 2.3.2.

Parametrization of double curves - In the case of double curves, D. Bayer and D. Eisenbud have obtained in [ba_ei] the following classification: if is of multiplicity 2, we have an exact sequence of vector bundles on

 0⟶L⟶ΩY∣C⟶ωC⟶0

which is split if and only if is the trivial curve, i.e. the second infinitesimal neighborhood of , embedded by the zero section in the dual bundle , seen as a surface. If is not trivial, it is completely determined by the line of induced by the preceding exact sequence. The non trivial primitive curves of multiplicity 2 and of associated line bundle are therefore parametrized by the projective space .

###### 2.4.

Simple primitive multiple curves

Let be a smooth projective irreducible curve, an integer and a primitive multiple curve of multiplicity and associated reduced curve . Then the ideal sheaf of in is a line bundle on .

We say that is simple if  .

In this case the line bundle on associated to is . The following result is proved in [dr6] (théorème 1.2.1):

###### 2.4.1.

Theorem: Suppose that is simple. Then there exists a flat family of smooth projective curves    such that and that is isomorphic to the -th infinitesimal neighborhood of in .

## 3. Reducible reduced deformations of primitive multiples curves

###### 3.1.

Connected Components

Let be the germ of a smooth curve and a generator of the maximal ideal. Let be an integer and a projective primitive multiple curve of multiplicity .

Let be an integer. Let   be a flat morphism, where is a reduced algebraic variety, such that

1. For every closed point such that , the fiber has irreducible components, which are smooth and transverse, and any three of these components have no common point.

2. The fiber is isomorphic to .

It is easy to see that the irreducible components of are reduced surfaces.

Let be the open subset of of points belonging to only one irreducible component of . Then the restriction of is a smooth morphism. For every , let  . It is the open subset of smooth points of .

Let and . There exist a neighborhood (for the Euclidean topology) of , isomorphic to , and a neighborhood of such that , , the restriction of being the projection on the first factor. We deduce easily from that the following facts:

1. let and an irreducible component of . Let . Then there exist neighborhoods (in , for the Euclidean topology) , of , respectively, such that if , are such that , then and belong to the same irreducible component of .

2. for every continuous map    and every such that there exists a lifting of ,  such that . Moreover, if    is another lifting of such that , then and are in the same irreducible component of . More generally, if we only impose that is in the same irreducible component of as , then and are in the same irreducible component of .

###### 3.1.1.

Lemma: Let    be two continuous maps such that , . Suppose that they are homotopic. Let , be liftings of , respectively, such that . Then and belong to the same irreducible component of .

###### Proof.

Let

 Ψ:[0,1]×[0,1]⟶S∖{P}

be an homotopy:

 Ψ(0,t)=σ0(t),Ψ(1,t)=σ1(t),Ψ(t,0)=σ0(0),Ψ(t,1)=σ0(1)

for . For every and let . By using the local structure of for the Euclidean topology it is easy to see that for every , there exists an such that the restriction of

 Iu,ϵ×[0,1]⟶S∖{P}

can be lifted to a morphism

 Ψ′:Iu,ϵ×[0,1]⟶Z

such that    for every . It follows that if , then and are in the same irreducible component of . We have just to cover with a finite number of intervals to obtain the result. ∎

Let , be the irreducible components of and for . Let be a loop of with origin , defining a generator of . Let be an integer such that . The liftings of such that end up at a component which does not depend on . Hence we can write

 j =αC(i).

###### 3.1.2.

Lemma: is a permutation of .

###### Proof.

Suppose that and . By inverting the paths we find liftings of paths from to and . This contradicts lemma 3.1.1. ∎

Let be an integer such that . Let be a generator of the maximal ideal of , the field of rational functions on and . Let be the germ of the curve corresponding to , canonical the morphism and the unique point of . Let . We have therefore a cartesian diagram

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