Canonical Artin stacks over log smooth schemes
We develop a theory of toric Artin stacks extending the theories of toric Deligne–Mumford stacks developed by Borisov–Chen–Smith, Fantechi–Mann–Nironi, and Iwanari. We also generalize the Chevalley–Shephard–Todd theorem to the case of diagonalizable group schemes. These are both applications of our...
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| Published in | Mathematische Zeitschrift Vol. 274; no. 3-4; pp. 779 - 804 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.08.2013
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0025-5874 1432-1823 |
| DOI | 10.1007/s00209-012-1096-7 |
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| Abstract | We develop a theory of toric Artin stacks extending the theories of toric Deligne–Mumford stacks developed by Borisov–Chen–Smith, Fantechi–Mann–Nironi, and Iwanari. We also generalize the Chevalley–Shephard–Todd theorem to the case of diagonalizable group schemes. These are both applications of our main theorem which shows that a toroidal embedding
is canonically the good moduli space (in the sense of Alper) of a smooth log smooth Artin stack whose stacky structure is supported on the singular locus of
. |
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| AbstractList | We develop a theory of toric Artin stacks extending the theories of toric Deligne–Mumford stacks developed by Borisov–Chen–Smith, Fantechi–Mann–Nironi, and Iwanari. We also generalize the Chevalley–Shephard–Todd theorem to the case of diagonalizable group schemes. These are both applications of our main theorem which shows that a toroidal embedding
is canonically the good moduli space (in the sense of Alper) of a smooth log smooth Artin stack whose stacky structure is supported on the singular locus of
. |
| Author | Satriano, Matthew |
| Author_xml | – sequence: 1 givenname: Matthew surname: Satriano fullname: Satriano, Matthew email: satriano@umich.edu organization: Department of Mathematics, University of Michigan |
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| Keywords | Stacky fan 14D23 Log structure 14M25 Chevalley–Shephard–Todd Toric stack |
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| References | VistoliAIntersection theory on algebraic stacks and on their moduli spacesInvent. Math.198997361367010050080694.1400110.1007/BF01388892 Satriano, M.: de Rham Theory for tame stacks and schemes with linearly reductive singularities. Ann. l’Institut Fourier. http://arxiv.org/abs/0911.2056 (2012, to appear) SatrianoMThe Chevalley–Shephard–Todd Theorem for finite linearly reductive group schemes Algebra Number Theory20126–1126295015910.2140/ant.2012.6.1 JiangYThe orbifold cohomology ring of simplicial toric stack bundlesIll. J. Math.2008522493514 WehlauDWhen is a ring of torus invariants a polynomial ring?Manuscr. Math.19948216117012561570809.1400810.1007/BF02567695 OlssonM.Logarithmic geometry and algebraic stacksAnn. Sci. École Norm. Sup. (4)2003365747791 IwanariIThe category of toric stacksCompos. Math.2009145371874625077461177.1402410.1112/S0010437X09003911 IwanariILogarithmic geometry, minimal free resolutions and toric algebraic stacksPubl. Res. Inst. Math. Sci.20094541095114025971301203.1405810.2977/prims/1260476654 FantechiBMannENironiFSmooth toric Deligne–Mumford stacksJ. Reine Angew. Math.201064820124427743101211.14009 WehlauD.A proof of the Popov conjecture for toriProc. Am. Math. Soc.19921143839845 FultonWIntroduction to Toric Varieties1993PrincetonPrinceton University Press0813.14039 AbramovichDOlssonMVistoliATame stacks in positive characteristicAnn. Inst. Fourier2008581057109124279541222.1400410.5802/aif.2378 Ogus, A.: Lectures on Logarithmic Algebraic Geometry (unpublished notes). http://math.berkeley.edu/~ogus/preprints/log-book/logbook.pdf Alper, J.: Good Moduli Spaces for Artin Stacks. http://arxiv.org/abs/0804.2242 (2009) Kato, K.: Logarithmic structures of Fontaine-Illusie. In: Algebraic analysis, geometry, and number theory (Baltimore, 1988), pp. 191–224. Johns Hopkins University Press, Baltimore (1989) LafforgueLChtoucas de Drinfeld et correspondance de LanglandsInvent. Math.2002147124118751841038.1107510.1007/s002220100174 BourbakiNGroupes et algèbres de Lie1968ParisCh. V. Hermann0186.33001 BorisovLChenLSmithGThe orbifold Chow ring of toric Deligne–Mumford stacksJ. Am. Math. Soc.200518119321521148201178.1405710.1090/S0894-0347-04-00471-0 KatoF.Log smooth deformation theoryTohoku Math. J. (2)1996483317354 B Fantechi (1096_CR5) 2010; 648 1096_CR13 1096_CR14 1096_CR16 I Iwanari (1096_CR7) 2009; 145 N Bourbaki (1096_CR4) 1968 1096_CR10 1096_CR11 W Fulton (1096_CR6) 1993 L Borisov (1096_CR3) 2005; 18 1096_CR18 A Vistoli (1096_CR19) 1989; 97 D Wehlau (1096_CR17) 1994; 82 L Lafforgue (1096_CR12) 2002; 147 1096_CR1 D Abramovich (1096_CR2) 2008; 58 M Satriano (1096_CR15) 2012; 6–1 I Iwanari (1096_CR8) 2009; 45 Y Jiang (1096_CR9) 2008; 52 |
| References_xml | – reference: FantechiBMannENironiFSmooth toric Deligne–Mumford stacksJ. Reine Angew. Math.201064820124427743101211.14009 – reference: OlssonM.Logarithmic geometry and algebraic stacksAnn. Sci. École Norm. Sup. (4)2003365747791 – reference: Kato, K.: Logarithmic structures of Fontaine-Illusie. In: Algebraic analysis, geometry, and number theory (Baltimore, 1988), pp. 191–224. Johns Hopkins University Press, Baltimore (1989) – reference: JiangYThe orbifold cohomology ring of simplicial toric stack bundlesIll. J. Math.2008522493514 – reference: WehlauD.A proof of the Popov conjecture for toriProc. Am. Math. Soc.19921143839845 – reference: WehlauDWhen is a ring of torus invariants a polynomial ring?Manuscr. Math.19948216117012561570809.1400810.1007/BF02567695 – reference: Satriano, M.: de Rham Theory for tame stacks and schemes with linearly reductive singularities. Ann. l’Institut Fourier. http://arxiv.org/abs/0911.2056 (2012, to appear) – reference: SatrianoMThe Chevalley–Shephard–Todd Theorem for finite linearly reductive group schemes Algebra Number Theory20126–1126295015910.2140/ant.2012.6.1 – reference: FultonWIntroduction to Toric Varieties1993PrincetonPrinceton University Press0813.14039 – reference: AbramovichDOlssonMVistoliATame stacks in positive characteristicAnn. Inst. Fourier2008581057109124279541222.1400410.5802/aif.2378 – reference: IwanariILogarithmic geometry, minimal free resolutions and toric algebraic stacksPubl. Res. Inst. Math. Sci.20094541095114025971301203.1405810.2977/prims/1260476654 – reference: Alper, J.: Good Moduli Spaces for Artin Stacks. http://arxiv.org/abs/0804.2242 (2009) – reference: IwanariIThe category of toric stacksCompos. Math.2009145371874625077461177.1402410.1112/S0010437X09003911 – reference: Ogus, A.: Lectures on Logarithmic Algebraic Geometry (unpublished notes). http://math.berkeley.edu/~ogus/preprints/log-book/logbook.pdf – reference: KatoF.Log smooth deformation theoryTohoku Math. J. (2)1996483317354 – reference: VistoliAIntersection theory on algebraic stacks and on their moduli spacesInvent. Math.198997361367010050080694.1400110.1007/BF01388892 – reference: BourbakiNGroupes et algèbres de Lie1968ParisCh. V. Hermann0186.33001 – reference: LafforgueLChtoucas de Drinfeld et correspondance de LanglandsInvent. Math.2002147124118751841038.1107510.1007/s002220100174 – reference: BorisovLChenLSmithGThe orbifold Chow ring of toric Deligne–Mumford stacksJ. Am. Math. Soc.200518119321521148201178.1405710.1090/S0894-0347-04-00471-0 – volume: 82 start-page: 161 year: 1994 ident: 1096_CR17 publication-title: Manuscr. Math. doi: 10.1007/BF02567695 – volume: 18 start-page: 193 issue: 1 year: 2005 ident: 1096_CR3 publication-title: J. Am. Math. Soc. doi: 10.1090/S0894-0347-04-00471-0 – volume: 147 start-page: 1 year: 2002 ident: 1096_CR12 publication-title: Invent. Math. doi: 10.1007/s002220100174 – volume-title: Groupes et algèbres de Lie year: 1968 ident: 1096_CR4 – volume-title: Introduction to Toric Varieties year: 1993 ident: 1096_CR6 doi: 10.1515/9781400882526 – ident: 1096_CR18 – volume: 145 start-page: 718 issue: 3 year: 2009 ident: 1096_CR7 publication-title: Compos. Math. doi: 10.1112/S0010437X09003911 – ident: 1096_CR16 doi: 10.5802/aif.2741 – volume: 52 start-page: 493 issue: 2 year: 2008 ident: 1096_CR9 publication-title: Ill. J. Math. doi: 10.1215/ijm/1248355346 – volume: 97 start-page: 613 issue: 3 year: 1989 ident: 1096_CR19 publication-title: Invent. Math. doi: 10.1007/BF01388892 – volume: 45 start-page: 1095 issue: 4 year: 2009 ident: 1096_CR8 publication-title: Publ. Res. Inst. Math. Sci. doi: 10.2977/prims/1260476654 – volume: 648 start-page: 201 year: 2010 ident: 1096_CR5 publication-title: J. Reine Angew. Math. – ident: 1096_CR13 – volume: 58 start-page: 1057 year: 2008 ident: 1096_CR2 publication-title: Ann. Inst. Fourier doi: 10.5802/aif.2378 – ident: 1096_CR14 – ident: 1096_CR11 – ident: 1096_CR10 – volume: 6–1 start-page: 1 year: 2012 ident: 1096_CR15 publication-title: Algebra Number Theory doi: 10.2140/ant.2012.6.1 – ident: 1096_CR1 |
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| Title | Canonical Artin stacks over log smooth schemes |
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