1316:
703:
550:
777:
413:
994:
1064:
302:
517:
1182:
1149:
822:
545:
915:
331:
80:
1357:
121:
215:
1116:
1090:
1272:
1247:
886:
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is an affine smooth
Deligne–Mumford stack with a non-trivial stabilizer at the origin. If we wish to think about this as a category fibered in groupoids over
336:
920:
996:
Notice that since this quotient is not from a finite group we have to look for points with stabilizers and their respective stabilizer groups. Then
708:
1350:
698:{\displaystyle {\text{Spec}}(\mathbb {C} /(s^{n}-1))\times {\text{Spec}}(\mathbb {C} )(S)\rightrightarrows {\text{Spec}}(\mathbb {C} )(S).}
1381:
1343:
140:
83:
787:
Non-affine examples come up when taking the stack quotient for weighted projective space/varieties. For example, the space
999:
249:
468:
1376:
124:
246:
of some variety where the stabilizers are finite groups. For example, consider the action of the cyclic group
221:
is quasi-compact, has only finitely many automorphisms. A Deligne–Mumford stack admits a presentation by a
1281:
1154:
1121:
790:
522:
891:
307:
53:
1286:
39:
1184:, respectively, showing that the only stabilizers are finite, hence the stack is Deligne–Mumford.
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31:
94:
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148:
100:
1303:
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since this has an infinite stabilizer. Stacks of this form are examples of Artin stacks.
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17:
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136:
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155:
144:
1315:
705:
Note that we could be slightly more general if we consider the group action on
772:{\displaystyle \mathbb {A} ^{2}\in {\text{Sch}}/{\text{Spec}}(\mathbb {Z} )}
222:
1295:
408:{\displaystyle a\cdot \colon (x,y)\mapsto (\zeta _{n}x,\zeta _{n}y).}
989:{\displaystyle \lambda \cdot (x,y)=(\lambda ^{2}x,\lambda ^{3}y).}
242:
Deligne–Mumford stacks are typically constructed by taking the
1268:"The irreducibility of the space of curves of given genus"
1331:
1209:
1203:
One simple non-example of a
Deligne–Mumford stack is
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139:
introduced this notion in 1969 when they proved that
103:
56:
1059:{\displaystyle (x,y)=(\lambda ^{2}x,\lambda ^{3}y)}
297:{\displaystyle C_{n}=\langle a\mid a^{n}=1\rangle }
1241:
1176:
1143:
1110:
1084:
1058:
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512:{\displaystyle ({\text{Sch}}/\mathbb {C} )_{fppf}}
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115:
74:
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8:
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180:A key fact about a Deligne–Mumford stack
102:
55:
824:is constructed by the stack quotient
7:
1312:
1310:
1273:Publications Mathématiques de l'IHÉS
27:Type of object in algebraic geometry
1177:{\displaystyle \lambda =\zeta _{2}}
1144:{\displaystyle \lambda =\zeta _{3}}
169:(also called an Artin stack, after
1330:. You can help Knowledge (XXG) by
165:", then such a stack is called an
25:
817:{\displaystyle \mathbb {P} (2,3)}
540:{\displaystyle S\to \mathbb {C} }
1314:
910:{\displaystyle \mathbb {C} ^{*}}
326:{\displaystyle \mathbb {C} ^{2}}
161:If the "étale" is weakened to "
1236:
1210:
1053:
1021:
1015:
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547:the over category is given by
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107:
86:, quasi-compact and separated.
75:{\displaystyle F\to F\times F}
60:
1:
1398:
1309:
1191:
1382:Algebraic geometry stubs
783:Weighted Projective Line
415:Then the stack quotient
158:Deligne–Mumford stacks.
1326:–related article is a
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116:{\displaystyle U\to F}
76:
50:the diagonal morphism
18:Deligne-Mumford stacks
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883:
819:
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118:
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36:Deligne–Mumford stack
1207:
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1000:
921:
917:-action is given by
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828:
791:
709:
551:
523:
519:then given a scheme
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419:
337:
308:
250:
210:{\displaystyle F(B)}
192:
177:is Deligne–Mumford.
101:
54:
1111:{\displaystyle y=0}
1085:{\displaystyle x=0}
1377:Algebraic geometry
1324:algebraic geometry
1296:10.1007/BF02684599
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89:There is a scheme
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32:algebraic geometry
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16:(Redirected from
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1242:{\displaystyle }
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906:
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881:{\displaystyle }
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458:{\displaystyle }
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149:arithmetic genus
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21:
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1260:Deligne, Pierre
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1066:if and only if
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227:groupoid scheme
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175:algebraic space
167:algebraic stack
130:
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97:surjective map
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1287:10.1.1.589.288
1264:Mumford, David
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1192:Main article:
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244:stack quotient
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133:Pierre Deligne
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14:
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4:
3:
2:
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1342:
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1329:
1325:
1320:
1317:
1313:
1305:
1301:
1297:
1293:
1288:
1283:
1280:(1): 75–109,
1279:
1275:
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1269:
1265:
1261:
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1231:
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1216:
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927:
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733:
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719:
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318:
288:
285:
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272:
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258:
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238:Affine Stacks
237:
232:
230:
228:
224:
220:
201:
195:
187:
183:
178:
176:
172:
171:Michael Artin
168:
164:
159:
157:
154:
150:
146:
145:stable curves
142:
141:moduli spaces
138:
137:David Mumford
134:
126:
110:
104:
96:
92:
88:
85:
84:representable
69:
66:
63:
57:
49:
48:
46:
44:
41:
37:
33:
19:
1332:expanding it
1321:
1277:
1271:
1202:
1194:Stacky curve
1188:Stacky curve
786:
241:
218:
185:
184:is that any
181:
179:
160:
131:
90:
42:
35:
29:
1199:Non-Example
123:(called an
1371:Categories
1253:References
888:where the
45:such that
1282:CiteSeerX
1232:∗
1166:ζ
1159:λ
1133:ζ
1126:λ
1042:λ
1026:λ
969:λ
953:λ
928:⋅
925:λ
903:∗
871:∗
847:−
755:ζ
725:∈
650:⇉
607:×
595:−
530:→
388:ζ
372:ζ
365:↦
347::
344:⋅
333:given by
292:⟩
273:∣
267:⟨
147:of fixed
108:→
67:×
61:→
1266:(1969),
233:Examples
223:groupoid
217:, where
1304:0262240
1302:
1284:
225:; see
173:). An
163:smooth
156:smooth
153:proper
1322:This
125:atlas
95:étale
40:stack
38:is a
1328:stub
1118:and
739:Spec
654:Spec
611:Spec
556:Spec
151:are
135:and
93:and
34:, a
1292:doi
1151:or
1092:or
729:Sch
477:Sch
304:on
188:in
143:of
82:is
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