BUILDING INFINITESIMAL NEIGHBOURHOODS OF VARIETIES

Description

BUILDING INFINITESIMAL NEIGHBOURHOODS OF VARIETIES. by Catriona Maclean ABSTRACT. We develop a deformation-type tool for the study of embeddings of a singular variety X . Given a variety X ? Y there is a natural series of schemes X = X0 ? X1 ? X2 . . . ? Y of infinitesimal neighbourhoods of X defined as follows Xn = zero(InX/Y ). Under certain assumptions, we calculate the obstructions to the existence of infinitesimal neighbourhoods and classify them when they exist. 1. INTRODUCTION. 1.1 A MOTIVATING EXAMPLE. Consider the following question. Let X be a complex normal crossing variety : for simplicity's sake we consider the case where X is the union of two smooth irreducible varieties X1 and X2 glued together along divisors D1 ? X1 and D2 ? X2 via an isomorphism ? : D1 ? D2 . The common image of D1 and D2 in X is denoted D . It is natural to ask whether X can be embedded in a smooth variety Y as a normal crossing divisor. The following example (based on Friedman's paper [1]) shows that the answer is “no” in general. Suppose that X ? Y is an inclusion of X as a normal crossing divisor in a smooth variety. Then there is an exact sequence 0 ? N?X|Y ? ?1Y ? OX ? ?1X ? 0 where N?X/Y is the conormal bundle of X in Y , N?X/Y = IX/Y/I2X/Y .

• let xn


• xn ?


• xn can


• smooth then


• x1 ?


• order infinitesimal


• surjective map


• ?0 any isomorphism

Sujets

• Surjective function

Informations

• Publié par: pefav
• Nombre de lectures: 7
• Langue: English

Extrait

BUILDINGINFINITESIMALNEIGHBOURHOODSOFVARIETIES.
byCatrionaMaclean

A
BSTRACT
.Wedevelopadeformation-typetoolforthestudyofembeddingsofasingularvariety
X
.Givenavariety
X
⊂
Y
thereisanaturalseriesofschemes
X
=
X
0
⊂
X
1
⊂
X
2
...
⊂
Y
of
innitesimalneighbourhoodsofX
denedas
follows
X
n
=
zero(
I
Xn
/
Y
)
.
Undercertainassumptions,wecalculatetheobstructionstotheexistenceofinnitesimalneighbourhoodsandclassify
themwhentheyexist.

1.I
NTRODUCTION
.

1.1A
MOTIVATINGEXAMPLE
.
Considerthefollowingquestion.Let
X
beacomplexnormalcrossingvariety:forsimplicity'ssakewe
considerthecasewhere
X
istheunionoftwosmoothirreduciblevarieties
X
1
and
X
2
gluedtogetheralong
divisors
D
1
⊂
X
1
and
D
2
⊂
X
2
viaanisomorphism
φ
:
D
1
→
D
2
.Thecommonimageof
D
1
and
D
2
in
X
isdenoted
D
.
Itisnaturaltoaskwhether
X
canbeembeddedinasmoothvariety
Y
asanormalcrossingdivisor.
Thefollowingexample(basedonFriedman'spaper[1])showsthattheanswerisnoingeneral.
Supposethat
X
⊂
Y
isaninclusionof
X
asanormalcrossingdivisorinasmoothvariety.Thenthereisan
exactsequence
0
→
N
X
∗|
Y
→
Ω
Y
1
⊗
O
X
→
Ω
X
1
→
0
where
N
X
∗
/
Y
istheconormalbundleof
X
in
Y
,
N
X
∗
/
Y
=
I
X
/
Y
/
I
X
2
/
Y
.Wenotethat
I
X
1
|
Y
⊗
O
X
1
=
N
X
∗
1
/
Y
and
ecnehN
X
∗
1
|
Y
⊗
O
D
=
I
X
1
|
Y
⊗
O
X
1
⊗
O
D
=
I
X
1
|
Y
⊗
O
D
=
I
D
|
X
2
⊗
O
D
=
N
D
∗|
X
2
.
Likewise,
N
X
∗
2
|
Y
⊗
O
D
=
N
D
∗|
X
1
,butsince
X
isanormalcrossingdivisorin
Y
,
N
X
∗|
Y
|
D
=
N
X
∗
1
|
Y
|
D
⊗
N
X
∗
2
|
Y
|
D
so
N
X
∗|
Y
|
D
=
N
D
∗|
X
1
⊗
N
D
∗|
X
2
.
Thepointisthattheright-handsideofthisequationistherestrictionto
D
ofa
linebundledenedon
X
,whereastheleft-handsidedoesnotdependon
Y
.Wethereforehavethefollowing
result.
P
ROPOSITION
1.1.
LetX
=
X
1
∪
D
X
2
beanormalcrossingvarietyasabove.IfXcanbeembeddedin
asmoothvarietyYasanormalcrossingdivisorthenthelinebundleN
D
∗|
X
1
⊗
N
D
∗|
X
2
canbeextendedtoa
linebundleonX.
Considerapair(
X
,
D
),where
D
isasmoothdivisorin
X
suchthattherestrictionmapPic(
X
)
→
Pic(
D
)
isnotsurjective.(Thistypicallyholdsif
X
isasurfaceand
D
isasmoothcurveofgenus
>
0.)Choosea
linebundle
L
on
D
whichisnottherestrictionofalinebundleon
X
andset
X
1
=
Proj(
L
⊕
O
D
),
X
2
=
X
,
D
1
=
Proj(
L
)and
D
2
=
D
.
Wethenhavethat
N
D
∗|
X
1
=
L
,
N
D
∗|
X
2
=
O
(
−
D
)andbydenition,
N
D
∗|
X
1
⊗
N
D
∗|
X
2
doesnotextendtoa

linebundleon
X
2
.Itfollowsthat
X
cannotbeembeddedinanysmoothvarietyasanormalcrossingdivisor.
Theobstructiongivenaboveto
X
beinganormalcrossingdivisorisinfact
innitesimal
:inotherwords,it
isanobstructionnotonlytotheexistenceof
Y
butalsototheexistenceofthescheme
X
ǫ
=
zero(
I
X
2
|
Y
).
Indeed,supposegivenaschemesupportedon
X
,
X
ǫ
,suchthat
21.
I
X
|
X
ǫ
isalinebundle,
L
on
X
.(Inparticular,
I
X
|
X
ǫ
=
0.)
2.Thesheaf
Ω
X
1
ǫ
⊗
O
X
isalocallyfreesheafon
X
.(Thisconditionmeansthat
X
ǫ
ispotentiallytherst
innitesimalneighbourhoodof
X
ina
smooth
variety
Y
).
Itthenturnsoutthat,asabove,wecanbuildanexactsequence
0
→
I
X
|
X
ǫ
→
Ω
1
X
ǫ
⊗
O
X
→
Ω
X
1
→
0
.
111∗Since
Ω
X
ǫ
⊗
O
X
isassumedtobelocallyfree,thereisasurjectivemap
I
X
|
X
ǫ
→
E
xt
(
Ω
X
,
O
X
)
.
Thesheaf
E
xt
1
(
Ω
X
1
,
O
X
)isalinebundleon
D
whichisprovedin[1](page85)tobegivenby
N
D
|
X
1
⊗
N
D
|
X
2
.The
∗∗∗∗∗left-handsideisalinebundleon
X
:itfollowsthat
I
X
|
X
ǫ
|
D
∼
=
N
D
|
X
1
⊗
N
D
|
X
2
andhence
N
D
|
X
1
⊗
N
D
|
X
2
extends
toalinebundleon
X
.
1.2D
EFINITIONSANDSTATEMENTOFTHEOREMS
.
Hopefully,theaboveexamplehasconvincedthereaderthatinnitesimalconsiderationscanproduceinter-
estinginformationaboutembeddingsof
X
andasystematicstudyofinnitesimalobstructionstotheexistence
ofembeddingscanbeuseful.
Throughoutwhatfollowsweworkover
k
,analgebraicallyclosedeldofcharacteristiczero.Givenan
inclusionof
k
-schemes
X
⊂
Y
thereisanassociatedsequence
X
=
X
0
⊂
X
1
⊂
X
2
⊂
X
3
⊂
...
ofnilpotent
schemessupportedon
X
givenby
X
i
=
zero(
I
iX
|
+
Y
1
).Ouraimistoclassifytheseinnitesimalmodelsunder
certainassumptions.Westartbydeninginnitesimalneighbourhoods,whichshouldbethoughtofasnilpotent
1+nschemeswhoseunderlyingbaseschemeis
X
andwhicharepotentiallyschemesoftheformzero(
I
X
|
Y
)for
good
X
and
Y
.
D
EFINITION
1.2.Let
X
beareducedlocallycompleteintersection
k
-variety,where
k
isanalgebraically
closedeld.Let
V
beavectorbundleon
X
.An
n
-thorderinnitesimalneighbourhoodof
X
withnormal
bundle
V
,
X
n
,isthedataofatriple
X
n
=
(
X
n
,
i
X
n
,α
X
n
)suchthat
1.
X
n
isa
k
-schemeofnitetype,
2.Themap
i
X
n
:
X
→
X
n
isaninclusion,
1+n3.Theidealsheaf
I
X
|
X
n
=
0,
2∗4.Themap
α
X
n
:
V
→
I
X
|
X
n
/
I
X
|
X
n
isanisomorphismof
O
X
-modules,
nn5.ThemultiplicationmapSym(
α
X
n
):Sym(
V
∗
)
→
I
Xn
|
X
n
isanisomorphism.
Thebundle
V
∗
iscalledthe
conormal
bundleoftheinnitesimalneighbourhood
X
n
.
R
EMARK
1.3.If
X
n
=
(
X
n
,
i
X
n
,α
X
n
)isa
n
-thorderinnitesimalneighbourhoodof
X
withnormalbundle
V
thenforany1
≤
i
≤
n
thereisan
i
-thorderinnitesimalneighbourhoodof
X
withnormalbundle
V
,
X
i
=
(
X
i
,
i
X
i
,α
X
i
)whichisdenedasfollows:
X
i
=
zero(
I
iX
|
+
X
1
n
);
i
X
i
=
i
X
n
|
X
i
;
α
X
i
=
α
X
n
.
Theinnitesimalneighbourhood
X
i
iscalledthe
i
-thordertruncationof
X
n
.
Wexanopenset
U
⊂
X
.Wewillneedtoknowwhatwemeanbytherestictionsofann-thorder
innitesimalneighbourhoodof
X
to
U