Chow quotients of Grassmannians. I.

*(English)*Zbl 0811.14043
Gelfand, Sergej (ed.) et al., I. M. Gelfand seminar. Part 2: Papers of the Gelfand seminar in functional analysis held at Moscow University, Russia, September 1993. Providence, RI: American Mathematical Society. Adv. Sov. Math. 16(2), 29-110 (1993).

This paper is an extensive study of the action of the maximal torus \(H \subset GL(n)\) on the Grassmann variety \(G(k,n)\), a problem related to many questions in geometry and analysis. The Chow quotient \(G(k,n) // H\) is first defined in terms of limits in the Chow variety of closures of generic orbits. Then it is shown that it can also be defined:

(1) As the space of limits of closures of generic orbits of the group \(GL(k)\) in the Cartesian power \((P^{k-1})^ n\), and

(2) As the space of limits of special Veronese varieties in the Grassmannian \(G(k-1,n-1) = G(k-1,h)\), where \(h\) is the Lie algebra of the torus \(H\).

Chow quotients of toric varieties by the action of a subtorus of the defining torus give a natural setting for the theory of certain polytopes. The author gives a description for degenerations of orbit closures in \(G(k,n)\) in terms of polyhedra decompositions of a certain polytope. In the case of \(G(2,n)\), he shows that these decompositions are in bijection with trees that describe combinatorics of stable curves. He also shows that \(G(2,n) // H\) is isomorphic to the Grothendieck-Knudsen moduli space \(\overline M_{0,n}\) of stable \(n\)-pointed curves of genus 0.

Finally it is pointed out that the approach in this paper differs from the geometric invariant theory developed by Mumford, more precisely, Mumford’s quotients of \(G(2,n)\) by \(H\) and of \((P^ 1)^ n\) by \(\text{GL}(2)\), although isomorphic to each other, do not coincide with \(\overline M_{0,n}\).

For the entire collection see [Zbl 0777.00036].

(1) As the space of limits of closures of generic orbits of the group \(GL(k)\) in the Cartesian power \((P^{k-1})^ n\), and

(2) As the space of limits of special Veronese varieties in the Grassmannian \(G(k-1,n-1) = G(k-1,h)\), where \(h\) is the Lie algebra of the torus \(H\).

Chow quotients of toric varieties by the action of a subtorus of the defining torus give a natural setting for the theory of certain polytopes. The author gives a description for degenerations of orbit closures in \(G(k,n)\) in terms of polyhedra decompositions of a certain polytope. In the case of \(G(2,n)\), he shows that these decompositions are in bijection with trees that describe combinatorics of stable curves. He also shows that \(G(2,n) // H\) is isomorphic to the Grothendieck-Knudsen moduli space \(\overline M_{0,n}\) of stable \(n\)-pointed curves of genus 0.

Finally it is pointed out that the approach in this paper differs from the geometric invariant theory developed by Mumford, more precisely, Mumford’s quotients of \(G(2,n)\) by \(H\) and of \((P^ 1)^ n\) by \(\text{GL}(2)\), although isomorphic to each other, do not coincide with \(\overline M_{0,n}\).

For the entire collection see [Zbl 0777.00036].

Reviewer: A.Papantonopoulou (Ewing Township)

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14L30 | Group actions on varieties or schemes (quotients) |

14M17 | Homogeneous spaces and generalizations |

14C05 | Parametrization (Chow and Hilbert schemes) |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |