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| Preface | vii | |
| Introduction | 1 | |
| 1 | First-Order Deformations | 5 |
| 1. | The Hilbert Scheme | 5 |
| 2. | Structures over the Dual Numbers | 9 |
| 3. | The Tl Functors | 18 |
| 4. | The Infinitesimal Lifting Property | 26 |
| 5. | Deformations of Rings | 35 |
| 2 | Higher-Order Deformations | 45 |
| 6. | Subschemes and Invertible Sheaves | 45 |
| 7. | Vector Bundles and Coherent Sheaves | 53 |
| 8. | Cohen-Macaulay in Codimension Two | 58 |
| 9. | Complete Intersections and Gorenstein in Codimension Three | 73 |
| 10. | Obstructions to Deformations of Schemes | 78 |
| 11. | Obstruction Theory for a Local Ring | 85 |
| 12. | Dimensions of Families of Space Curves | 88 |
| 13. | A Nonreduced Component of the Hilbert Scheme | 91 |
| 3 | Formal Moduli | 99 |
| 14. | Plane Curve Singularities | 100 |
| 15. | Functors of Art in Rings | 106 |
| 16. Schlessinger's Criterion...................................Ill | ||
| 17. | Hilb and Pic are Pro-representable | 118 |
| 18. | Miniversal and Universal Deformations of Schemes | 120 |
| 19. | Versal Families of Sheaves | 128 |
| 20. | Comparison of Embedded and Abstract Deformations | 131 |
| 21. | Algebraization of Formal Moduli | 138 |
| 22. | Lifting from Characteristic p to Characteristic 0 | 144 |
| 4 | Global Questions | 149 |
| 23. | Introduction to Moduli Questions | 150 |
| 24. | Some Representable Functors | 156 |
| 25. | Curves of Genus Zero | 164 |
| 26. | Moduli of Elliptic Curves | 167 |
| 27. | Moduli of Curves | 177 |
| 28. | Moduli of Vector Bundles | 188 |
| 29. | Smoothing Singularities | 199 |
| References | 217 | |
| Index | 225 | |
abelian surface, 84, 123, 126, 141, 143
ACM. See arithmetically Cohen- Macaulay
ACM curves, 66, 68, 70, 72, 73, 90, 96, 97
existence of, 72
specialize to nonACM curves, 70
affine cone, 39, 208, 211
nonsmoothable, 211
rigid, 40
algebraic space, 3, 142, 171, 181, 185, 193
algebraization, 3, 99, 101, 138, 142
analytic isomorphism, 34, 39, 100, 105,
127, 203
André, M., 25
arithmetically Cohen-Macaulay, 63
arithmetically Gorenstein, 75, 76
Artin, Michael, 3, 117, 141-143, 185, 214
Artin's approximation theorem, 142, 143
Artin stack, 186
Auslander, Maurice, 62
automorphisms, 34, 41, 80, 81
affect pro-representability, 121, 122
effect on deformations, 14, 35, 50, 120
infinitesimal, 83, 122, 123, 133
of deformations of node, 124
of invertible sheaf, 50
of coherent sheaf, 53
prevent representability, 155, 181
bare varieties, 141
base change of T% functors, 26
bend and break, 160
Bertini's theorem, 200
biliaison, 69
blowing-up, 48, 83, 94, 126, 188
Bordiga surface, 215
bounded family, 156, 157, 189, 190
Bourbaki seminar, 16, 148, 155
Brauer group, 166
Brauer-Severi variety, 166
Buchsbaum, David, 68, 77
Buchweitz, Ragnar, 77
Burch, L., 58, 68
Calabi-Yau threefold, 148
canonical curve, 77
canonical obstruction theory, 85
Castelnuovo-Mumford regularity, 70, 156
Castelnuovo's criterion, 212
Castelnuovo surface, 215
category, fibered, 186
characteristic p, 144, 161
Chern classes, 90, 150, 188, 198
Chevalley's theorem, 117
Chow variety, 91
classification of algebraic space curves, 87
closed subscheme. See Situation A
coarse moduli, 3, 151, 164, 169, 178, 185
Cohen-Macaulay in codimension 2, 58
is locally unobstructed, 62
module, 57
obstructed in codimension 3, 51, 215
resolution of, 58, 63
ring, 58
subscheme, 2, 47, 49, 58
coherent functor, 201
cohomological functor, 18
cohomology and base change, 17, 166
cohomology class of line bundle, 84, 135
complete family, 100, 156
complete intersection, 7, 73, 74
Hilbert scheme is smooth, 75
resolution of, 74
complex-analytic deformations, 101,
105, 125, 127, 134, 185, 193
complex manifold, 193, 194
cones, 39
deformations of, 208
generic singularity, 211
nonsmoothable, 210
over curves, 214, 215
over d-uple embedding, 40, 44
over Segre embedding, 41
rigid, 210
tangent bundle of, 41, 209
cotangent complex, 2, 19
counting constants, 88, 137, 181
cross-ratio, 100
crude local functor, 120, 123, 128, 154
cubic surface, 92
ACM curves on, 68, 70
deformations of, 83
double line on, 43
lines on, 52, 92
cup product, 84, 135
curves
ACM, 66, 68, 73, 97
canonical, 77
complete intersection, 7, 8
deformations of, 83
d-gonal, 212
dimension of moduli, 39, 42
elliptic, 3, 125, 126, 167, 169, 175
exceptional, 48, 83
extremal, 95
hyperelliptic, 71, 180, 187, 215
in P2, 6, 50
in P3, 66, 70, 88, 90
linear systems on, 180
linked, 68, 69
moduli of, 177, 178
nonreduced, 38, 43, 95
nonsmoothable, 43, 206, 207, 214
nonspecial, 9, 50
not locally complete intersection, 77
of degree 3, 9, 69
of degree 4, 8, 71, 187
of degree 5, 90, 206
of degree 6 and genus 3, 70
of degree 13, 95
of degree 14, 92, 96
of degree 18, 96
of genus 0, 164, 207
of genus 1, 16
of genus 2, 176, 188
of genus 3, 70, 71, 187, 194
of genus 5, 77
of genus g, 39, 56, 123, 163, 177
on cubic surface, 68, 70, 92
on quadric surface, 7, 50, 68
on quartic surface, 73
rational, 3
rational normal, 90, 215
reduced locally complete intersection, 42
resolution of, 90
rigid, 201
semirigid, 90
smoothable, 205, 206
stable, 177, 181, 184, 188
through m general points, 90
trigonal, 77, 180, 215
twisted cubic, 9, 69
union of two, 8, 68, 183
variety of moduli of, 71
cusp, 40, 105
decreasing type, 72
dévissage, 32
deformations
abstract versus embedded, 41, 82, 95, 131, 137
effect of automorphisms, 14, 35, 50
first-order, 2
global versus local, 204
higher-order, 2, 45
nth-order, 45
of Pn, 38, 109
of abstract scheme. See Situation D
of ACM subschemes, 65
of affine scheme, 36
of coherent sheaf. See Situation C
of cones, 39, 208
of double fine, 43
of dual modules, 57
of fine bundles. See Situation B
of finked scheme, 77
of morphism, 159
of nodal elliptic curve, 175
of nonaffine schemes, 41
of nonsingular affine scheme, 30, 50
of scheme with line bundle, 83
of singularity. See Situation D
of subscheme. See Situation A
of surfaces in P3, 133
of vector bundles. See Situation C
trivial, 12, 30
degree of vector bundle, 189
Deligne, Pierre, 3, 137, 148, 177, 178, 187
Deligne-Mumford stack, 186
Del Pezzo surface, 214
depth, 39, 58
derived category, 21
descent theory, 154, 186
determinantal scheme, 58
d-gonal curve, 212
differentials module of, 19
sheaf of, 27
dimension
of fine moduli, 154, 161
of Hilbert scheme, 87
of local ring, 87
divisor
associated to fine bundle, 17, 51
canonical, 161
numerically effective, 161
double line, 43
dual modules, 57
dual numbers, 6, 9
d-uple embedding, 44, 146
effective formal scheme, 140, 142
Ein, Lawrence, 90, 214
Eisenbud, David, 68, 77
Ekedahl, T., 148
elementary biliaison, 69
Ellingsrud, Geir, 62, 65, 68
elliptic curves, 3, 126, 169
affine, 34
Jacobian of, 16
modular family, 171
moduli, 167
pointed, 125
trivial family of, 34
vector bundles on, 17
embedded point, 9
embedded versus abstract deformations, 131
endomorphisms, 15
Enriques surface, 141
equisingular, 100, 106
equivalence relation, 185
equivalent
deformations, 14, 35
extensions, 36, 78
up to power series rings, 127
étale base extension, 178
étale covering, 142, 147
étale topology, 154, 171, 185
Euler sequence, 41, 132
exceptional curve, 48, 83
extension
of deformation, 45, 46, 53, 78
of ring by module, 36
of sheaves, small, 111
trivial, 36
extremal curves, 95
false ruled surface, 148
family
bounded, 156, 162, 189
complete, 100, 156, 162, 188, 193
fiberwise trivial, 153, 163, 165, 169,
172, 176, 187, 198
formal, 107
isotrivial, 167, 176
limit of, 8
locally isotrivial, 167
locally trivial, 166
miniversal, 2, 100, 102, 108, 120
modular, 3, 142, 171, 178
of curves, 7, 143
section of, 169
separated, 156, 162, 163, 176
smooth, 110
tautological, 7, 151, 164, 169
trivial, 153
universal, 7, 100, 120, 152
versal, 2, 100, 108, 128
Fano, G., 136
Fantechi, Barbara, 187
fibered category, 186
fibered product, 115, 117
fiberwise trivial family, 153, 163, 165,
169, 172, 176, 187, 198
fine moduli, 3, 152, 169, 185
first-order deformations, 2, 5, 9
flat
couple, 16
module, 10
morphism, 10
sheaf, 10
flatness, local criterion of, 10
Fogarty, John, 68
forgetful functor, 123
formal deformations
of abstract varieties, 120
of closed subscheme, 118
of vector bundle, 128
formal family, 107, 141
formally smoothable, 3, 200, 202
and analytic isomorphism, 203
is local property, 202
smoothable scheme is, 202
formal moduli, 99
formal scheme, 139, 140
fpqc topology, 154
Frobenius morphism, 161
Fulton, William, 177, 178
functor
coherent, 201
cohomological, 18
crude local, 120, 123
forgetful, 123
formally smooth morphism, 110
Hilbert, 118, 152, 155, 156
moduli, 178
not pro-representable, 109, 121, 130
obstruction theory, 110
of Art in rings, 99, 106
Picard, 119, 155
pro-representable, 106, 107, 109
Quot, 129, 158
representable, 109, 152
semi-exact, 32
sheaf for étale topology, 154
sheaf for Zariski topology, 154, 155
tangent space of, 111, 118
unobstructed, 110
fundamental group, 197 9d, 180, 212
Gaeta, F., 77
Galois cohomology, 166
general curve, 180
general points in P3, 90
generic singularity, 211
geometric fiber, 31
geometric genus, 141
geometric invariant theory, 3, 178
Gieseker, David, 197
global moduli, 3
Gorenstein ring, 76
Gorenstein subscheme, 2
of codimension 3, 47, 49, 73, 76
resolution of, 76
Grassmann variety, 18, 88, 94, 207
Grauert, Hans, 125, 127
Grothendieck, Alexandre, 5, 6, 16, 30,
34, 68, 92, 110, 119, 140, 154
Grothendieck topology, 171
group action, 178
Gruson, Laurent, 88, 95
h-vector, 72
Halphen, G., 88, 91, 137
Harris, Joe, 90
Hartshorne, Robin, 68, 212
Hartshorne's conjecture, 160
Hilbert, David, 58, 67
Hilbert-Burch theorem, 58
Hilbert-flag scheme, 52, 91, 95, 119, 135, 158
obstruction theory, 52
of lines on cubic surface, 52
of lines on quartic surface, 53
Zariski tangent space, 52
Hilbert scheme, 2, 5, 99, 119, 149, 152
dimension of, 6, 7, 87, 88, 90
existence of, 6, 162
infinitesimal study of, 88
irreducible, 66
is connected, 68
nonreduced, 91, 95
nonsingular, 49, 50
not irreducible, 42, 72, 73, 96
of ACM curves, 70, 73
of ACM subscheme, 65
of complete intersection curves, 8
of curves in P2, 6
of curves in P3, 92, 96
of double lines, 43
of points in P2 irreducible, 67
of twisted cubic curves, 9
of zero-dimensional subschemes, 42,
51, 66, 72
singular points of, 9, 51, 66, 71, 72, 96
smooth, 8, 50, 65, 75
universal property of, 5, 13, 49
Zariski tangent space of, 13
Hirokado, M., 148
Hirschowitz, André, 214
Hodge index theorem, 135
homological dimension, 54, 55
Huneke, Craig, 77
Hurwitz scheme, 177
hyperelliptic curves, 71, 180, 212, 215
hypersurface singularity, 105
Iarrobino, A., 72
Illusie, Luc, 25, 87, 148
indecomposable vector bundle, 189
infinitesimal automorphisms, 83, 122,
123, 133
infinitesimal deformations, 99
infinitesimal lifting property, 2, 26, 27, 49
and smoothness, 27, 30, 34
of local ring, 30
relative, 34
infinitesimal thickening, 27
International Congress of Mathemati- cians, 188
intersection theory, 135
irreducible moduli problem, 187
isolated singularity, 103, 105, 120, 127, 202
Isom scheme, 160, 162, 191, 208
isotrivial, 167, 176
j-invariant, 100, 106, 125, 151, 167, 169
Jacobian criterion, 26
matrix, 26
of elliptic curves, 16
variety, 16, 88
jump phenomena, 100, 152, 155, 163
curves of genus 1, 126, 152, 175
plane curve singularities, 100, 105, 126, 152
prevents coarse moduli, 100, 152
vector bundles, 16, 130, 152, 188, 198
K3 surface, 133, 141, 148
Kaplansky, Irving, 68
Kerner, H., 127
Kleiman, S. L., 158
Kleppe, J. O., 77, 95
Knutson, Donald, 3, 185
Kollar, Janos, 87, 119
Koszul complex, 24, 33, 74, 77
/-adic cohomology, 137
lambda line, 176
Lang, W. E., 148
Langton, S. G., 193
Laudal, O. A., 6, 25, 87, 95
Lefschetz pencil, 137
Lefschetz, S., 137
liaison theory. See linkage
licci scheme, 78, 216
Lichtenbaum, Stephen, 2, 18, 25, 34, 82
lifting
from char p, 144, 145
of morphism, 27, 28
limit
of affine elliptic curves, 39
of flat family, 8
stable curve as, 188
linear resolution, 91
line bundles
associated divisor of, 17, 51
deformations of. See Situation B
obstructed, 51, 134
lines
double, 43
in cubic surface, 52
in quartic surface, 51, 53
linkage, 69, 73, 93, 95-97, 216
of deformations, 77, 78, 127
local analytic space, 127
local complete intersection, 2, 32, 47, 49
and T* functors, 32
relative, 33
locally formally smoothable, 203
local rings
analytic isomorphism of, 34
regular, 27
Martin-Deschamps, Mireille, 95
Maruyama, M., 197
Migliore, Juan, 77
miniversal family, 2, 100, 108, 109, 110, 120
Mirö-Roig, R. M., 76
Mittag-Leffier condition, 204
modular family, 3, 142, 149, 171
curves of genus 0, 176
curves of genus > 2, 178, 208
elliptic curves, 171
no smallest, 173
vector bundles, 191, 199
module
Cohen-Macaulay, 57
dual, 57
of differentials, 19
reflexive, 58
moduli
abelian varieties, 141
coarse, 3, 151, 164, 178
completion of, 174, 181
curves of genus 0, 151
curves of genus >2, 177
curves of genus 3, 187, 194
elliptic curves, 151, 167
fine, 3, 152, 185
formal, 99
functor, 178
global, 3, 149
irreducible, 187
nonreduced, 153
nonseparated, 177, 193, 194ff n points on line, 176
of curves, 2, 120, 177, 178
pointed curves, 166, 168, 169
polarized varieties, 141
vector bundles, 188
Moishezon space, 185
monodromy, 137
Mori, Shigefumi, 3, 6, 87, 138, 160, 162
Mori's theorem, 138, 161
Mumford, David, 2, 3, 92, 95, 119, 148, 156, 162, 176, 177, 178, 187, 188, 197, 207
Nagata, M., 196
Narasimhan, M. S., 197
natural, use of word, 15, 46
negative grading, 209
Nitsure, N., 156
nodal cubic curve, 9, 174
node, 25, 33, 40, 101, 123, 182
Noether, Max, 88, 91, 99, 131, 135
Noether's theorem, 135, 138
nonlicci scheme, 77, 78, 216
nonliftable scheme, 146
nonreduced Hilbert scheme, 91, 95
nonseparated moduli space, 177, 193,
194fT
nonsingular variety, 26, 132
and 7n functors, 30
first-order deformations, 38
formal deformations, 120
higher-order deformations, 82
plane curves, 137
surface in P3, 132
nonsmoothable
curve, 43, 206, 207, 214
scheme, 200, 211
nonspecial curve, 9, 50
normal bundle, 7, 12
computation of sections, 93
normal module, 23
normal sheaf, 12, 89
normalized vector bundle, 17
null-correlation bundle, 18, 56
numerically effective, 161
obstructed deformation
of curves in P3, 92
of curves in P4, 71
of curve on surface, 51
of line bundle, 51, 53, 134
of nonsingular 3-fold, 94
of zero-dimensional subscheme, 41, 51, 71
obstruction theory, 2, 6, 45, 47
and T% functors, 78, 81
of abstract schemes, 78, 81
of Cohen-Macaulay subscheme, 62
of coherent sheaf, 53, 55, 57
of complete intersection, 75
of functor, 110, 118, 154
of Hilbert-flag scheme, 52
of invertible sheaf, 49, 51
of local ring, 85-87
of nonsingular affine scheme, 50
of Quot functor, 55
of subscheme, 47, 51
Oh, K., 90
Oort, Frans, 148
open problems, 90, 148, 201
orbit space, 178
ordinary double point, 40
Perrin, Daniel, 91, 95
Peskine, Christian, 62, 68, 88, 95
Pfaffian, 76
Picard group, 92, 135, 138, 162
Picard scheme, 3, 149, 158, 162
Piene, Ragni, 68
pinching a curve, 207
Pinkham, Henry, 208, 211, 212, 214, 215
plane curves, 50, 70
plane curve singularities, 99, 100
cusp, 41, 105, 126
minimal family, 103
node, 25, 33, 40, 101, 123
not pro-representable, 109, 121
ordinary fourfold, 100, 106
ordinary fivefold, 106
tacnode, 105, 106
pointed curve, 166, 168
pointed elliptic curves, 125
polarized varieties, 141
postulation function, 69
principal homogeneous space. See torsor principal parts, sheaf of, 84
projective general linear group, 179
projectively normal variety, 40
pro-representable functor, 2, 99, 107, 112, 118
and automorphisms, 122, 123
Hilb is, 118
of stable curves, 184
Pic is, 119
projective scheme, 122
Schlessinger's criterion, 99
pro-representable hull, 108
pseudotorsor, 46, 47
punctual scheme, 29, 34, 51. See also
zero-dimensional schemes Purdue conference, 176
quadric surface, 7, 50, 68
quartic surface, 51, 96, 133-136, 138, 140
lines on, 53, 96, 138
obstructed line bundle on, 51, 53, 134
obstructed line on, 51, 53
Quillen, Daniel, 25
Quot functor, 54, 56, 129, 158
quotient by equivalence relation, 180, 185
Quot scheme, 191
rank of vector bundle, 189
Rao, A. P., 90
Rao module, 69, 95, 96
rational curve, 3, 215
Raynaud, Michel, 148
reflexive module, 58
regularity, 70
regular local ring, 27, 30, 31
and Ti functors, 31
regular sequence, 73
relative local complete intersection, 33
represent able functor, 152, 156
resolution linear, 91
of Cohen-Macaulay subscheme, 58, 63
of complete intersection, 74
of curves, 90
of Gorenstein subscheme, 76
of singularities, 163
rigid scheme, 38, 83, 127, 163, 199, 201
affine cone is, 40
affine nonsingular variety is, 38
cones, 210
double line is, 44
is not smoothable, 204
nonreduced, 44
Pn is, 38
union of planes is, 41
Rim, D. S., 87
ring
Cohen-Macaulay, 58
complete intersection, 33
Gorenstein, 75
of dual numbers, 6
regular local, 26, 30, 31
Röhn, K., 137
ruled surface, 165
Schaps, Mary, 60, 62, 68, 206
scheme
punctual, 29
rigid, 38, 201
with H\Ox) = 0, 16
Schlessinger, Michael, 2, 18, 25, 34, 40,
68, 82, 105, 111, 117, 214
Schlessinger's criterion, 2, 111, 113, 118
Schroer, S., 148
scroll, 215
Segre embedding, 41
semicontinuity, 106, 208
semi-exact functor, 32
semirigid curve, 90
semistable vector bundle, 189, 198
separated family, 156
Sernesi, Edoardo, 73, 96, 117, 119
Serre, Jean-Pierre, 2, 16, 77, 146
Seshadri, C. S., 197
sheaf
automorphisms of, 14
deformation of, 14
for Zariski topology, 154, 171, 176
of principal parts, 84
sheafification, 167
simple vector bundle, 56, 129, 162
singularities
analytically isomorphic, 34
equisingular, 100
generic, 211
isolated, 103
nonsmoothable, 210
not local complete intersection, 77
of plane curves, 100, 103
resolution of, 163
rigid, 210
smoothable, 199
Situation A, 1. See also Hilbert scheme affine case, 11, 12
Cohen-Macaulay, 60, 65
first-order, 11, 13
formal, 99, 118, 140
global case, 12, 149, 156
Gorenstein, 76
higher-order, 46, 47
local complete intersection, 74, 75
obstructed, 92, 96
Situation B, 1
first-order, 13
formal, 99, 119, 141
global, 149, 158
higher-order, 49, 51
obstructed, 134
Picard scheme, 158
Situation C, 1. See also vector bundles first-order, 13, 14
for coherent sheaves, 57
for locally free sheaves, 53
for sheaves of hd < 1, 55
formal, 99, 128, 129, 141
global moduli, 191
higher-order, 53, 55
Situation D, 1, 19, 35
first-order, 38, 82
for nonsingular variety, 38, 82
formal, 99, 120, 122, 123, 125, 141
global, 164, 167, 177
higher-order, 78, 81
obstructed, 94, 127
obstructions, 78, 80
small extension, 111
smooth morphisms, 26, 30, 31
and Tl functors, 31
infinitesimal lifting property, 34
of functors, 108
smoothable curve, 8
curves in Pn, 205, 206
formally, 3, 200
hypersurface is, 200
rigid scheme is not, 204
scheme, 39, 200
singularities, 199
space curves, 88
speciality function, 69
spectral sequence, 42, 82
stable curves, 177, 181, 184, 188
limit of nonsingular, 188, 206
stable vector bundles, 3, 162, 188, 193
on P2, 199
stack, 3, 142, 149, 171, 176, 177, 184fT
Stevens, J., 214
strongly surjective, 108, 110, 111, 204, 208
subscheme. See also Hilbert scheme Cohen-Macaulay, 2
deformation of, 11
Gorenstein, 2
locally complete intersection, 2
universal, 5
surface, 83
abelian, 84, 123, 141, 143
Bordiga, 215
Castelnuovo, 215
cubic, 43, 52, 68, 70, 83, 92
Del Pezzo, 214
Enriques, 141
in P3, 132, 137, 138, 143
K3, 133, 141, 148
of general type, 141
quadric in P3, 123, 131, 132, 143
quartic, 133, 138
rational, 121, 123, 130, 212, 215
ruled, 141, 148, 165
Szpiro, Lucien, 62, 68
T functors, 2, 18
and local complete intersection, 32, 33
and obstructions, 78
and smoothing, 201
and smooth morphisms, 31
as Ext, 25
base change of, 26
construction of, 19
exact sequence of, 21, 34
examples of, 25
global construction of, 26
localization of, 26
notation of, 25
obstructed, 51
of a polynomial ring, 23
of affine cone, 39
of analytically isomorphic local rings, 34
of nonsingular affine scheme, 30
of quotient ring, 24, 34
sheaves, 25, 34, 80, 82
tacnode, 105
tangent bundle
of abelian surface, 84
of cone, 41
tangent module, 23
tangent sheaf, 13, 38, 183
of blowing-up, 83, 94
tangent space of functor, 111, 117
tangent theory of functor, 118
tautological family, 7, 151, 164
not universal, 174
thick point, 25, 33
torsor, 15, 29, 46, 47, 49, 51, 54, 78
tricanonical embedding, 178
trigonal curve, 77, 180, 212, 215
trivial
extension, 36
deformation, 12, 15
family, 153
twisted cubic curves, 9, 69
Ulrich, Bernd, 77
union
of curves, 8, 42, 68
of two planes in A4, 41
unitary representations, 197
universal
family, 7, 152
subscheme, 5
unobstructed deformation, 45
functor, 110
unstable vector bundle, 189
Vakil, Ravi, 96, 148
valuative criteria, 162
variety of moduli, 1, 88
vector bundles
deformations of, 15
degree, 189
indecomposable, 189
moduli of, 188, 197
normalized, 17
null-correlation, 18, 56
of rank 2, 17
on curve of genus 3, 194
on curves, 16, 17, 56, 129, 189
on elliptic curve, 17
on P1, 16, 129
on P2, 198
on P3, 18, 56
rank, 189
semistable, 189
simple, 56, 129, 162, 189
simple unstable, 195
stable, 3, 152, 188, 189
unstable, 189
Veronese surface, 40, 215
versal family, 2, 100, 101, 108, 128, 131
very general, 135
Vistoli, A., 184
Wahl, Jonathan, 100
Watanabe, J., 76, 77
Weyr, E., 91
Witt vectors, 110, 144
Yoneda's interpretation of Ext, 14
Zariski, Oscar, 105
Zariski tangent space, 6, 13, 92, 102, 153
Zariski topology, 154
zero-dimensional schemes
Hilbert scheme not irreducible, 42, 72
in P2, 65, 67
in P3, 51, 72
in P4, 41
