Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies
Abstract In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, al...
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| Autor*in: |
Abe, Hiraku [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2018 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer International Publishing AG, part of Springer Nature 2018 |
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| Übergeordnetes Werk: |
Enthalten in: Selecta mathematica - Springer International Publishing, 1993, 24(2018), 3 vom: 09. März, Seite 2129-2163 |
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volume:24 ; year:2018 ; number:3 ; day:09 ; month:03 ; pages:2129-2163 |
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10.1007/s00029-018-0405-3 |
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| 520 | |a Abstract In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety. | ||
| 650 | 4 | |a Hessenberg varieties | |
| 650 | 4 | |a Peterson varieties | |
| 650 | 4 | |a flag varieties | |
| 650 | 4 | |a local complete intersections | |
| 650 | 4 | |a Flat families | |
| 650 | 4 | |a Schubert varieties | |
| 650 | 4 | |a Newton–Okounkov bodies | |
| 650 | 4 | |a Degree | |
| 700 | 1 | |a DeDieu, Lauren |4 aut | |
| 700 | 1 | |a Galetto, Federico |4 aut | |
| 700 | 1 | |a Harada, Megumi |4 aut | |
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10.1007/s00029-018-0405-3 doi (DE-627)OLC2069542793 (DE-He213)s00029-018-0405-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Abe, Hiraku verfasserin aut Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing AG, part of Springer Nature 2018 Abstract In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety. Hessenberg varieties Peterson varieties flag varieties local complete intersections Flat families Schubert varieties Newton–Okounkov bodies Degree DeDieu, Lauren aut Galetto, Federico aut Harada, Megumi aut Enthalten in Selecta mathematica Springer International Publishing, 1993 24(2018), 3 vom: 09. März, Seite 2129-2163 (DE-627)171156862 (DE-600)1158980-2 (DE-576)038490609 1022-1824 nnns volume:24 year:2018 number:3 day:09 month:03 pages:2129-2163 https://doi.org/10.1007/s00029-018-0405-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 24 2018 3 09 03 2129-2163 |
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10.1007/s00029-018-0405-3 doi (DE-627)OLC2069542793 (DE-He213)s00029-018-0405-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Abe, Hiraku verfasserin aut Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing AG, part of Springer Nature 2018 Abstract In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety. Hessenberg varieties Peterson varieties flag varieties local complete intersections Flat families Schubert varieties Newton–Okounkov bodies Degree DeDieu, Lauren aut Galetto, Federico aut Harada, Megumi aut Enthalten in Selecta mathematica Springer International Publishing, 1993 24(2018), 3 vom: 09. März, Seite 2129-2163 (DE-627)171156862 (DE-600)1158980-2 (DE-576)038490609 1022-1824 nnns volume:24 year:2018 number:3 day:09 month:03 pages:2129-2163 https://doi.org/10.1007/s00029-018-0405-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 24 2018 3 09 03 2129-2163 |
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10.1007/s00029-018-0405-3 doi (DE-627)OLC2069542793 (DE-He213)s00029-018-0405-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Abe, Hiraku verfasserin aut Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing AG, part of Springer Nature 2018 Abstract In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety. Hessenberg varieties Peterson varieties flag varieties local complete intersections Flat families Schubert varieties Newton–Okounkov bodies Degree DeDieu, Lauren aut Galetto, Federico aut Harada, Megumi aut Enthalten in Selecta mathematica Springer International Publishing, 1993 24(2018), 3 vom: 09. März, Seite 2129-2163 (DE-627)171156862 (DE-600)1158980-2 (DE-576)038490609 1022-1824 nnns volume:24 year:2018 number:3 day:09 month:03 pages:2129-2163 https://doi.org/10.1007/s00029-018-0405-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 24 2018 3 09 03 2129-2163 |
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10.1007/s00029-018-0405-3 doi (DE-627)OLC2069542793 (DE-He213)s00029-018-0405-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Abe, Hiraku verfasserin aut Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing AG, part of Springer Nature 2018 Abstract In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety. Hessenberg varieties Peterson varieties flag varieties local complete intersections Flat families Schubert varieties Newton–Okounkov bodies Degree DeDieu, Lauren aut Galetto, Federico aut Harada, Megumi aut Enthalten in Selecta mathematica Springer International Publishing, 1993 24(2018), 3 vom: 09. März, Seite 2129-2163 (DE-627)171156862 (DE-600)1158980-2 (DE-576)038490609 1022-1824 nnns volume:24 year:2018 number:3 day:09 month:03 pages:2129-2163 https://doi.org/10.1007/s00029-018-0405-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 24 2018 3 09 03 2129-2163 |
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Abstract In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety. © Springer International Publishing AG, part of Springer Nature 2018 |
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Abstract In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety. © Springer International Publishing AG, part of Springer Nature 2018 |
| abstract_unstemmed |
Abstract In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety. © Springer International Publishing AG, part of Springer Nature 2018 |
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Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies |
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DeDieu, Lauren Galetto, Federico Harada, Megumi |
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DeDieu, Lauren Galetto, Federico Harada, Megumi |
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| doi_str |
10.1007/s00029-018-0405-3 |
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