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These de Doctorat de l’Universite· Paris 6
Specialit· e· : Mathematiques·
present· ee· par :
Mlle Catriona Maclean
pour obtenir le grade de
Docteur de l’Universite· Paris 6
sur le sujet :
Quelques resultats·
en theorie· des deformations·
en geom· etrie· algebrique·
Soutenue le 25 avril 2003 devant le jury compose· de:
M. Arnaud Beauville Nice
M. Olivier Debarre Strasbourg Rapporteur
M. Geir Ellingsrud Oslo Rapporteur
M. Daniel Huybrechts Paris
M. Joseph Le Potier? Paris
Mme. Claire Voisin Paris Directrice2Remerciements
Tout d’abord, je voudrais remercier de tout mon coeur ma directrice de these,
Claire Voisin, qui a fait preuve d’une gen· erosit· e· exceptionelle dans le partage de
ses connaissances et de sa creati· vite· mathematique.· Cette these n’aurait jamais vu
le jour sans son aide continuelle, et je voudrais exprimer ma profonde reconnais-
sance pour tout le temps et l’en· er· gie qu’elle y a consacre.·
Olivier Debarre et Geir Ellingsrud ont accepte· d’etre? rapporteurs de cette these,
laquelle a et· e· enormement· amelioree· par leurs nombreux commentaires detaill· es.·
Je les en remercie tres chaleureusement, ainsi de s’etre? deplaces· pour la soute-
nance.
Je remercie aussi Arnaud Beauville, Daniel Huybrechts et Joseph Le Potier d’avoir
accepte· de faire partie du jury.
Je tiens a remercier vivement tous ceux a l’Ecole Normale Superieure· qui m’ont
accueillie quand je suis arrivee· en France il y a cinq ans, et qui m’ont soutenue
par la suite. De meme,? je remercie l’Universite· Paris 6 et plus particulierement
l’equipe· de Geom· etrie· et Topologie Algebrique· pour leur accueil et leur soutien.
Je remercie chaleureusement Ania Otwinowska pour les echanges· tres interessants
que nous avons eus.
C’est pendant mes etudes· a l’Universite· de Cambridge que je me suis passionnee·
pour la premiere fois pour la geom· etrie· algebrique.· Je tiens a remercier tous ceux
qui m’y ont enseigne· les mathematiques,· avec une mention toute particuliere pour
Alessio Corti, directeur de mon memoire· de ma? trise,qui m’a appris tant de choses
sur la geom· etrie· et pourquoi elle merite· d’etre? aimee.·
Et pour terminer avec les remerciements professionels : je remercie Phillip Heyes,
mon professeur de mathematiques· de college et de lycee.· C’est un homme qui
34
savait etre? a la fois gen· ereux· et rigoureux, et qui pendant des annees· d’enseignement
a lance· la carriere de maints jeunes scienti ques dont la mienne. Je l’en remer-
cie: je sais que je ne suis pas la seule.
Je remercie tous ceux qui ont et· e· mes camarades ou voisins de bureau pendant
ces quatres annees· de m’avoir empech? ee· de travailler si souvent, c’etait· mieux
ainsi. Donc merci a Aurelie,· Gwendal, Layla, Kristian, Madga, Marcos, Masha,
Olivier, Pietro et Samy. Je suis en particuliere tres reconaissante a mes freres
mathematiques,· Lorenz et Gianluca, pour quatre ans de soutien professionel et
pas-professionel-du-tout, et a Ingo de m’avoir pret? e· un coin de son bureau, parmi
d’autres choses. Et bien sur? , merci a Tanguy.
Les amis qui m’ont accompagnee· et qui m’ont supportee· sont nombreux et chers:
je les remercie tous. Dave en particulier s’est montre· plus patient que je n’aurais
cru possible.
Ma famille a toujours et· e· la pour moi, et je ne serais jamais parvenue au bout
sans eux. Merci donc a mon pere, ma mere et ma soeur: ca me fait tres plaisir
qu’ils puissent etre? la aujourd’hui. C’est a eux que je dedie· ce travail.Contents
56 CONTENTSChapter 1
Introduction
This thesis contains the work on various problems that I have considered over
the last three years. It is divided into four independent sections dealing with four
independent problems.
1.0.1 Section 1 Deformation theory
This section corresponds to chapter 2 of my thesis. In it, we recall the basic results
of deformation theory and then extend these results to a more general problem,
namely, the construction of formal neighbourhoods of a given scheme X with
speci ed normal bundle. All our schemes will be of nite type over a base eld.
More precisely, we de ne more general deformations in the following way.
De nition 1 Let X be an l.c.i. scheme and V a vector bundle. An n-th order
formal deformation ofX with normal bundleV is a schemeX together with ann
embeddingi :X !X and an isomorphismn
2 ⁄j :I =I ’VX X
(I is here the ideal sheaf ofX inX ) such thatX n
n+11. I = 0 inO ,XnX
n›n ⁄ n2. The multiplication mapj : Sym V !I is an isomorphism.X
Although the results of abstract deformation theory do not translate directly into
this context, the results for embedded deformations carry over. We de ne an
embedded generalised deformation in the following way.
De nition 2 An embedded generalisedn-th order deformation ofX with normal
bundleV is given by the following data.
78 CHAPTER 1. INTRODUCTION
1. A smooth schemeP and an embeddingX !P ,
~ ~2. A vector bundleV !P and an isomorphismVj =V ,X
n+1~3. A subschemeX ofP (the subscheme ofV whose ideal sheaf isI ) suchn n ~P=V
thatX \P =X and the restriction and multiplication mapsn
⁄ 2r :V !I =IX=Xn X=Xn
›n n ⁄ n n+1r : Sym V !I =IX=Xn X=Xn
are isomorphisms.
ByI we mean the ideal sheaf ofX inX .X=X nn
We then prove the following results for these embedded deformations.
Theorem 1 LetX be ann-th order embedded deformation ofX. We can assignn
1 2to any pair (X ;X ) of extensions ofX to (n + 1)-st order embedded deforma-n
n+11 2 ⁄ ⁄tions ofX an elementd(X ;X )2 Hom(N ; Sym V ) such thatX
3 1 2 2 3 1 31. IfX is another extension ofX , thend(X ;X )+d(X ;X ) =d(X ;X ) andn
1 2 2 1d(X ;X ) =¡d(X ;X ),
2. IfX is generically smooth and the push-forward of
⁄ 1 10!N ! › ›O ! › ! 0XX P X
1 2 1 2alongd(X ;X ) is a trivial extension thenX andX are isomorphic as
abstract in nitesimal neighbourhoods ofX ,n
3. If any extensions ofX exist they form a principal homogeneous space un-n
n+1⁄ ⁄der Hom(N ; Sym V ).X
Theorem 2 We can associate to anyX an elementn
1 ⁄ n+1 ⁄! 2H (X; Hom(N ; Sym V ))Xn X
(or alternatively
1 n+1 ⁄! 2H (X;N › Sym V ))X Xn
in such a way thatX can be extended to an (n + 1)-st order embedded deforma-n
tion ofX if and only if! = 0.Xn9
⁄ 2In all the above,N denotes the conormal bundleI =I , andN denotes itsX=P XX X=P
dual.
These results are fairly straightforward generalisations of the equivalent results
for ordinary deformations, but I have not found them in the literature. The re-
sults for abstract higher-order deformations cannot be directly translated into this
context, since they depend on the existence of a canonical isomorphism between
1 2I 1 andI 2, (X andX being two separate (n + 1)-st order formal neigh-X=X X=X
bourhoods ofX which have the samen-th part) which does not now exist.
In certain circumstances, all obstruction groups vanish for appropriate choices
of normal bundlesV . We can then apply these theorems to prove the following
results.
Theorem 3 LetX be a projective local complete intersection scheme. Then there
exists a smooth formal neighbourhoodX ofX, a vector bundleV onX and a1 1
section? :X !V such that1
† V is a direct sum of line bundles,
† The rank of the vector bundleV is equal to the codimension ofX inX ,1
† X is schematically the zero locus of? .
This last result was the subject of a short article in the C.R.A.S [?].
1.0.2 Section 2 The Noether-Lefschetz locus
This section corresponds to chapters 3-5 of my thesis. The material in sections 3-4
form an article submitted for publication, [?], and the material in Chapter 5 form
another. Whilst writing this thesis I learnt that similar results have been indepen-
dently obtained by Ania Otwinowska in her articles [?] and [?]. I am grateful to
her for communicating her results to me and for the very interesting discussions
we have had.
3If X is a generic degree d surface in P and d > 3, the Noether-Lefschetz the-
orem says that
1;1
H (X;Z) = 0:prim
1;1
Here, H (X;Z) means the group of all primitive Hodge classes onX. A co-prim
homology class onX is said to be primitive if it is orthogonal for the cup product
to all cohomology classes inherited from projective space. The space of surfaces
which do not satisfy this condition is known as the Noether-Lefschetz locus; it is10 CHAPTER 1. INTRODUCTION
a union of countably many algebraic subvarieties of U , the parameter space ofd
3smooth degreed surfaces in P .
LetW be a component of the Noether-Lefschetz locus. It can be shown thatW
0;2
is locally the zero locus of a certain section of the bundle H . The latter is theUd
0;2bundle on the parameter spaceU whose bre over the point [X] isH (X;C); itd ¡ ¢
d¡10;2is therefore a bundle of rankh (X;C) = . This gives us a prediction of the
3¡ ¢
d¡1codimension ofW ? namely ? and we say that a component is exceptional
3
if its is strictly smaller than this predicted dimension. In this section
we will study the in nitesimal geometry of these exc