Selecta Mathematica

, Volume 24, Issue 3, pp 2129–2163 | Cite as

Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

  • Hiraku Abe
  • Lauren DeDieu
  • Federico Galetto
  • Megumi Harada


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 

Mathematics Subject Classification

Primary 14M17 14M25 Secondary 14M10 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Hiraku Abe
    • 1
    • 2
  • Lauren DeDieu
    • 3
  • Federico Galetto
    • 1
  • Megumi Harada
    • 1
  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  2. 2.Osaka City University Advanced Mathematical InstituteOsakaJapan
  3. 3.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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