Deformations of pairs (X,L) when X is singular

We give an elementary construction of the tangent-obstruction theory of the deformations of the pair $(X,L)$ with $X$ a reduced local complete intersection scheme and $L$ a line bundle on $X$. This generalizes the classical deformation theory of pairs in case $X$ is smooth. A criteria for sections of $L$ to extend is also given.

map H 1 (T X ) ∪c(L) / / H 2 (O X ) ∼ = C is surjective for every nontrival line bundle L. This means that L deforms along a 19-dimensional subspace of H 1 (T X ), because h 1 (X, T X ) = 20.
In this paper, we give an elementary approach to the deformation theory of the pair (X, L) for X a separated reduced local complete intersection scheme (l.c.i) of finite type over C. We prove that even though X could be singular, the functor of Artin rings Def (X,L) (A) = {Flat deformations of (X, L) over A}/isomorphisms still behaves well in the sense that there is a tangent-obstruction theory for this deformation functor, with tangent space Ext 1 O X (P 1 X (L), L) and obstruction space Ext 2 O X (P 1 X (L), L), where P 1 X (L) is the sheaf of one jets or sheaf of principle parts of L on X. Moreover, there is a natural map analogous to M characterizing obstructions for sections of L to extend. Therefore, all the nice consequences mentioned above generalize to reduced l.c.i schemes. If X is smooth, P 1 X (L) = D 1 (L) * ⊗ L, where D 1 (L) is the sheaf of first-order differential operators on L, and Ext i O X (P 1 X (L), L) = H i (X, D 1 (L)). We go back to the classical case. The tangent and obstruction spaces for deformations of (X, L) was known to experts and was stated implicitly in [7], [8]. Our approach is more elementary and does not use the more abstract machinery of cotangent complexes. It seems to the author that the criteria for sections of L to extend is new.
Acknowledgements. The author would like to thank his advisor Herb Clemens for valuable suggestions and constant support, E. Sernesi and M. Manetti for pointing out a gap in the previous version of the paper.

the sheaf of one jets
In this section, we briefly review some basic facts and definitions about the sheaf of one jets.
Let g : X → Y be a morphism between two algebraic schemes (separated schemes of finite type over C), L be a line bundle on X, and let ∆ ⊂ X × Y X be the diagonal defined by ideal sheaf I ∆ . Consider the first order neighborhood Spec of ∆ with two projections π 1 , π 2 to X. The sheaf of one jets P 1 X/Y (L) of X over Y is defined to be P 1 X/Y (L) := π 1 * π * 2 (L). P 1 X/Y (L) has a natural left O X -module structure induced by π 1 and a right O X -module structure induced by π 2 which, in general, is not equivalent to the left one. Throughout this paper, we will only use the left O X -module structure of P 1 X/Y (L). Consider the short exact sequence Tensoring the above sequence with π * 2 L then applying the functor π 1 * , we get a short exact sequence of left O X -modules on X where Ω 1 X/Y is the sheaf of relative Kähler differentials. The sequence is exact on the right because there is no higher derived image for π 1 * (π 1 has relative dimension 0). When Y = Spec(C), we will write P 1 X (L) for P 1 X/Y (L). The "fibre"of the sheaf P 1 X/Y (L) at a closed point x ∈ X is the stalk of L| g −1 (g(x)) at x mod the maximal ideal squared, i.e.
This is the reason P 1 X/Y (L) is called the sheaf of (relative) one jets. There is a O Y -linear splitting p 1 : L → P 1 X/Y (L), which sends a section s of L to its one jet π 1 * π * 2 s. p 1 satisfies the property that is the sheaf of first-order differential operators on L.

computation of the tangent space
In this section, let X be a reduced algebraic scheme. Applying the functor Hom O X (−, L) to (2.1), we get a long exact sequence is the tangent space of the deformations of X, and Ext 1 O X (L, L) = H 1 (O X ) is the tangent space of deformations of L with the base X fixed. This suggests that Ext 1 O X (P 1 X (L), L) is the tangent space of deformations of the pair (X, L) and Ext 2 O X (P 1 X (L), L) is an obstruction space. If X is smooth, Hom O X (P 1 X (L), L) is the sheaf of firstorder differential operators D 1 (L), and Ext 1 O X (P 1 X (L), L) = H 1 (X, D 1 (L)) is the correct tangent space. In this section and the next, we will prove this is indeed the correct generalization of the tangent-obstruction theory for deformations of the pair (X, L).
Let's first recall that for any reduced algebraic scheme over C, we have an one-to-one correspondence between isomorphism classes of extensions of X by a coherent locally free O X -module I and Ext 1 O X (Ω 1 X , I) in the following way: Given an isomorphism class of extension of O X by I, when X ⊂ X is a closed immersion defined by ideal sheaf I, and I 2 = 0 in O X , we associate to it (the isomorphism class of) the conormal sequence (which is also exact on the left.) This conormal sequence corresponds to an element c E in Ext 1 O X (Ω 1 X , I).
Conversely, for any O X -module extension We get a commutative diagram: For the deformations of the pair (X, L), we have the following result: Theorem 3.1. Let X be a reduced scheme of finite type over C, L be a line bundle on X.
(1) The tangent space of the functor of Artin rings Def (X,L) is canonically identified with (2) There exists a natural pairing such that for any first-order deformation of the pair (X, L) corresponding to ξ ∈ Ext 1 O X (P 1 X (L), L), a section s ∈ H 0 (L) extends to first order along ξ if and only if ξ and s pair to zero under p.

Proof.
(1) Given a first-order deformation of the pair (X, L), i.e. the following fibered diagram with O X flat over Spec(C[ǫ]) and L line bundle on X : We have a diagram of (left) O X -modules: The two right columns are exact by (2.1), and the fact that restriction to X is (left) exact The first row is the conormal sequence of X ⊂ X twisted by L, which is exact. Thus by Snake Lemma, ker(r) = L and the second row is exact . Therefore, we can associate any first-order deformation of the pair (X, L) the second row exact sequence, which corresponds to an element of Ext 1 O X (P 1 X (L), L). Now consider the commutative diagram where p ′ 1 is the composition of p 1 : L → P 1 X (L) and the restriction map to X. Thus p ′ 1 factors through L × P 1 X (L) P 1 X (L)| X and therefore L ∼ = L × P 1 X (L) P 1 X (L)| X . This fact suggests that we can recover L from P 1 X (L)| X and L.
Conversely, for any element ξ ∈ Ext 1 O X (P 1 X (L), L) corresponding to an O X -module extension: sits naturally in the diagram The first row exact sequence corresponds to an element in Ext 1 , which corresponds to a first-order infinitesimal deformation X of X as described in the beginning of this section.
To recover the deformation of L, let E ′′ = E ′ ⊗ L −1 and let In order to see L is a locally free O X -module of rank one, it suffices to prove the case L is the trivial bundle since the question is local. In this case, (2.1) splits (as left O X -module) and P 1 The statement follows immediately from this.

obstructions
In this section, let X be as in section 3 and we assume furthermore that X is a local complete intersection scheme. We will show that Ext 2 O X (P 1 X (L), L) is an obstruction space for deformations of the pair (X, L).
The general idea is to apply Vistoli's construction of obstruction spaces for deformations of l.c.i schemes (cf. sections 3, 4 of [10]) to the total space of L ∨ and keep track of the bundle structure using a C * -action.
For any z ∈ C * , denote φ z : L ∨ → L ∨ be the multiplication map by z in the fiber direction. Define a C * -action on O L ∨ and Ω 1 L ∨ by for local sections f ∈ O L ∨ , ω ∈ Ω 1 L ∨ .
Let O C * L ∨ and Ω C * L ∨ be the sheaf of sections which are invariant under the C * -action. Both O C * L ∨ and Ω C * L ∨ have natural O X -module structures. Under some trivialization of L ∨ over U ⊂ X: and Ω C * L ∨ consists of 1-forms f (x)d L ∨ t+ω(x)t where f is the pull back of a function on U and ω ∈ Ω 1 U .
We have natural isomorphisms of O X -modules O C * L ∨ ∼ = L and P 1 X (L) ∼ = Ω C * L ∨ . The isomorphisms can be described as follows: for any section s ∈ L, we can naturally view it as a function on the total space of L ∨ which restricts to a linear function on the fiber. Such functions are invariant under the C * -action and vice versa. This gives the first isomorphism. The second isomorphism is the natural one which identifies p 1 (s) with d L ∨ (f s ), where s is any section of L, f s is the function on L ∨ corresponding to s and d L ∨ is the exterior derivative on L ∨ . Under some local trivialization of L ∨ , it sends (f, ω) ∈ P 1 Let ( X α , L α ) and ( X β , L β ) be two liftings of (X , L) to Spec( A). We would like to measure the difference of two such liftings.
Let's restrict ourselves to the local situation first. Suppose that X is affine, embedded in S = Spec(A[x 1 , ..., x n ]) and the total space of L ∨ i are both embedded into Spec( A[x 1 , ..., Taking the invariant part under the C * -action we get an exact sequence of O X -modules The difference of L ∨ α and L ∨ β as embedded deformations corresponds to an O L ∨ -module homomorphism v αβ : I 0 Now, take the push-out of (4.3) under v ′ αβ , we obtain an O X -module extension E αβ of P 1 X (L) by J ⊗ C L: Lemma 4.1. The extension E αβ does not depend on the choice of S.
Proof. Suppose there are two embeddings L ∨ i → S j × A 1 , where i = α, β, j = 1, 2; reducing to embeddings L ∨ → S 1 × A 1 and L ∨ → S 2 × A 1 . These induce embeddings Let C 1 , C 2 , C 12 be the conormal bundles of L ∨ in S 1 × A 1 , S 2 × A 1 , S 1 × S 2 × A 1 respectively. Denote by v ′ j : C C * j → J ⊗ C L the invariant part of the corresponding sections of the normal bundles, E j = v ′ j * Ω C * S j ×A 1 | L ∨ , E 12 = v ′ 12 * Ω C * S 1 ×S 2 ×A 1 | L ∨ and p j : C C * j → C C * 12 be the natural map between conormal bundles. Then We have the following diagram By the universal property of push out, this diagram induces isomorphism of extensions ψ j : E j ∼ = E 12 . We define the canonical isomorphism between E 2 and E 1 to be ψ 1 • ψ −1 2 .
Proposition 4.2. For any two liftings of line bundles L ∨ α , L ∨ β inside S × A 1 as above, there is an O X -module extension E αβ of P 1 X (L) by J ⊗ C L, well defined up to canonical isomorphism, with the following properties.
(a) For any three liftings L ∨ α , L ∨ β , and L ∨ γ , there is a canonical isomorphism of extensions such that for any four liftings, as homomorphism of extensions from E αβ + E βγ + E γδ to E αδ . (b) Given an O X -module extension E of P 1 X (L) by J ⊗ C L, and an lifting L ∨ α of L ∨ , there is an abstract lifting L ∨ β such that E αβ is isomorphic to E. (c) There is a natural bijection between bundle isomorphisms Φ : Proof.
(a) As embedded deformations we certainly have v ′ αβ + v ′ βγ = v ′ αγ as homomorphisms from ( I 0 By the universal property of push-out, there is a unique isomorphism F −1 αβγ : E αγ → E αβ +E βγ such that (ψ αβ , ψ βγ ) factors through F −1 αβγ . The compatibility condition (4.5) follows from the universal property of push-out as well.
(b) Applying the derived functor Hom O X (−, J ⊗ C L) to (4.3), we obtain exact sequence where the last term is zero because X is affine and Ω C * S×A 1 | L ∨ is locally free. Thus for any such that the difference of L ∨ β and L ∨ α as embedded deformations corresponds to v, then by construction E αβ ∼ = E.
(c) First notice that by the construction of push-out, to give a splitting s : 1 The sum of two extensions of OX -module 0 The oposite extension −E is defined to be 0 It is easy to check that This gives a spliting of E αβ . Conversely, any O X -module homomorphism One checks easily that π β − D vanishes on the ideal sheaf of L ∨ α thus factors through π α , and therefore we recover the bundle isomorphism φ from such D.
Remark. Proposition 4.2 still holds in the global case. Since the local extension does not depending on the choice of embeddings, one can construct a global extension for any two abstract liftings L ∨ 2 and L ∨ 1 by glueing together the local extensions using the canonical isomorphisms in lemma 4.1 on the overlap of two open affine subsets. One checks easily that the glued extension satisfies the properties in the proposition. We will not need the global case in the construction of the obstruction space.
The rest of the proof is entirely based on the construction in [10]. The idea is to use extension cocycles to measure the obstructions to patching together local liftings (which always exist since X is l.c.i) coherently.
Here we collect some useful results about extension cocycles and refer to [10] for details. of F by G on {U α } is a collection of extensions {E αβ } of F| U αβ by G| U αβ , and isomorphisms F αβγ : E αβ + E βγ ∼ = E αγ on U αβγ satisfying the compatibility condition as in (4.5).

Two extension cocycles ({E
Definition 4.4. We say an extension cocycle is a boundary if it is isomorphic to for a collection of extensions {E α } of F| Uα by G| Uα , where is the obvious isomorphism. The set of isomorphism classes of extension cocycles form an abelian group, and the boundaries form a subgroup. The quotient group is called the group of extension classes, and is denoted by Ξ O X (U α ; F, G). We refer to section 3 in [10] for the proofs of the above facts.
To finish the proof, we cover X by open affine subscheme {U α } such that L ∨ α = L ∨ | Uα has a lifting L ∨ α over U α . The difference of L ∨ α and L ∨ β on the overlap corresponds to an extension E αβ of P 1 U αβ (L αβ ) by J ⊗ C L αβ . For each triple α, β, γ, consider the isomorphism F αβγ : E αβ + E βγ ∼ = E αγ in proposition 4.2 (a).
Then (E αβ , F αβγ ) is an extension cocycle, which we will denote simply by (E αβ ). If L ∨ α is another collections of liftings, coresponding to another extension cocycle (E ′ αβ ), we get isomorphisms by proposition 4.2 (a). One checks that this is an isomorphism of extension cocycles. Thus the class of [E αβ ] ∈ Ξ O X (U α ; P 1 X (L), J ⊗ C L) is independent of the choice of local liftings.
A global lifting exists if and only if we can choose local liftings L ∨ α and isomorphisms of line bundles φ αβ : L ∨ α → L ∨ β satisfying the cocycle condition φ αβ • φ βγ = φ αγ .
By proposition 4.2 (c), to give φ αβ is equivalent to assigning splittings for E αβ . It is easy to check that φ αβ satisfies cocycle condition if and only if (E αβ ) is isomorphic to the trivial extension cocycle.
Conversely, if the class [E αβ ] ∈ Ξ O X (U α ; P 1 X (L), J ⊗ C L) is zero, (E αβ ) is isomorphic to a boundary (E α − E β ). By proposition 4.2 (b), we can choose local lifting L ∨ α such that E( L ∨ α , L ∨ α ) ∼ = E α . Then L ∨ α will patch together to give a global lifting.
Combine the above discussion with theorem 4.5 and the fact that Ext 2 O X (P 1 X (L), L) = 0 (since (4.3) is a locally free resolution of P 1 X (L)), we get