## The Space of Triangles, Vanishing Theorems, and Combinatorics (1996)

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Citations: | 5 - 1 self |

### BibTeX

@TECHREPORT{Kallen96thespace,

author = {Wilberd Van Der Kallen and Peter Magyar},

title = {The Space of Triangles, Vanishing Theorems, and Combinatorics},

institution = {},

year = {1996}

}

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### Abstract

We consider compactifications of (P \Delta ij , the space of triples of distinct points in projective space. One such space is a singular variety of configurations of points and lines; another is the smooth compactification of Fulton and MacPherson; and a third is the triangle space of Schubert and Semple.

### Citations

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Citation Context ...onstruct two such resolutions in later sections. The normality, together with the surjectivity of the restriction map above for any multiple of D, is essentially equivalent to projective normality by =-=[15]-=-, Ch II, Ex 5.14(d), given the projective normality of the Plücker embedding. If D does not contain all the columns of D3, then we need a strengthening of Proposition 3. Indeed Ramanathan has proved [... |

473 |
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Citation Context ...show that F3,n has rational singularities. 4.1 Strata Again, we first consider the case n = 3, and we will have a fiber bundle F FM 3,3 → F FM 3,n → Gr(2,Pn−1 ). It is shown in [13] how F FM 3,3 = P2 =-=[3]-=- can be constructed as a union of 8 strata, each consisting of certain configurations of points and tangent vectors in P 2 . For each stratum, there is a natural GL(3) action and an equivariant map to... |

298 |
Representations of algebraic groups
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Citation Context ...oups, needed in the proof of theorem 1. Let G be a connected reductive algebraic group, B a Borel subgroup, P a parabolic subgroup containing B. Let us call a weight λ effective if ind G Bλ ̸= 0. (In =-=[17]-=- an effective weight is called dominant, in [18] it is called anti-dominant.) For the notions of ‘induction’, ‘good filtration’, ‘excellent filtration’, and the basic theorems concerning them we refer... |

256 |
The moment map and equivariant cohomology, Topology 23
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Citation Context ... torus acting on X and L. Throughout this section, we also assume the vanishing of the higher cohomology groups of L: H i (X, L) = 0 for all i > 0 . 24The following formula is due to Atiyah and Bott =-=[2]-=-. Proposition 14 Suppose the torus T acts X with isolated fixed points. Then the character of T acting on the space of global sections of L is given by: tr(x | H 0 (X, L)) = ∑ tr(x | L|p) det(id −x | ... |

179 |
A compactification of configuration spaces
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(Show Context)
Citation Context ...[33], [34], [35], [39].) The geometric theory of F3,n and its desingularizations goes back to Schubert [37] and Semple [38], and has been illuminated recently by Fulton, MacPherson, and others ([10], =-=[13]-=-, [20],[36], [7], [8]). In the current paper, we explore the relations between the algebraic and geometric pictures. We prove a Borel-Weil theorem realizing SD3,n (or the Schur module of any three-row... |

152 | Algebraic geometry - Harris - 1998 |

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Some theorems on actions of algebraic groups
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Citation Context ...xed points of the subtorus are identical to those of the large torus of all diagonal matrices.) Let us also mention that for the smooth spaces F SS 3,n and F FM 3,n , the theorem of Bialynicki-Birula =-=[5]-=- gives cell decompositions of these spaces using the fixed point data. Thus, one can compute their singular cohomology groups and Chow groups as is done in [10] and [13]. 265.2 Character formulas Fir... |

60 |
Frobenius splitting and cohomology vanishing for Schubert varieties
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Citation Context ...e. To prove these facts and the rest of the Theorem, we will need more sophisticated techniques. 2.2 Frobenius splitting The theory of Frobenius splittings invented by Mehta, Ramanan, and Ramanathan (=-=[27]-=-, [30], [31], [18]) is a characteristic-p technique for proving surjectivity and vanishing results about coherent sheaves, even in characteristic 8zero. It is highly practical for dealing with homoge... |

52 | Representation Theory. A First - Fulton, Harris |

37 |
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Citation Context .... . .,xi) → ÔP/(x1, . . .,xi) corresponding with the “residue” of s along x1 = · · · = xi = 0 whose divisor is described by fi. Several other useful properties of Frobenius splittings can be found in =-=[32]-=-. We will apply our theory first in the case n = 3, and then indicate the modifications necessary for general n. Instead of directly splitting the pair F3,n ⊂ Gr(D3), we will find it more convenient t... |

34 |
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Citation Context ...prove these facts and the rest of the Theorem, we will need more sophisticated techniques. 2.2 Frobenius splitting The theory of Frobenius splittings invented by Mehta, Ramanan, and Ramanathan ([27], =-=[30]-=-, [31], [18]) is a characteristic-p technique for proving surjectivity and vanishing results about coherent sheaves, even in characteristic 8zero. It is highly practical for dealing with homogeneous ... |

33 |
Key polynomials and a flagged Littlewood-Richardson rule
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Citation Context ...tation of GL(n).) These objects have been extensively explored. Combinatorists have examined the representations SD,n as part of the theory of generalized Schur modules and Young diagrams. (See [29], =-=[33]-=-, [34], [35], [39].) The geometric theory of F3,n and its desingularizations goes back to Schubert [37] and Semple [38], and has been illuminated recently by Fulton, MacPherson, and others ([10], [13]... |

29 |
A residue formula for holomorphic vector-fields
- Bott
(Show Context)
Citation Context ...ter of S∗ D = H0 (F FM 3,n , π∗LD), this time in terms of the data in Lemma 9. The next result we shall use is based on the theorem of HirzebruchRiemann-Roch [3], combined with Bott’s Residue Formula =-=[6]-=-, [2], according to the method of Ellingsrud and Stromme [11]. Proposition 16 Suppose the torus T = C ∗ is one-dimensional, and acts with isolated fixed points. Let v = 1 in the Lie algebra t = C, and... |

29 | Groupe de Picard et nombres caractéristiques des variétés spheriques - Brion - 1989 |

28 |
Schubert varieties are arithmetically
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(Show Context)
Citation Context ...these facts and the rest of the Theorem, we will need more sophisticated techniques. 2.2 Frobenius splitting The theory of Frobenius splittings invented by Mehta, Ramanan, and Ramanathan ([27], [30], =-=[31]-=-, [18]) is a characteristic-p technique for proving surjectivity and vanishing results about coherent sheaves, even in characteristic 8zero. It is highly practical for dealing with homogeneous variet... |

24 | Specht series for skew representations of symmetric groups - James, Peel - 1979 |

21 | Schur functors and Schur complexes, Adv - Akin, Buchsbaum, et al. - 1982 |

20 |
Normality of Schubert varieties
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Citation Context ...and surjectivity statements also hold for all fields of characteristic zero. Frobenius splitting is also sufficient to establish the normality of our varieties. The main theorem of Mehta and Srinivas =-=[26]-=- states that if Y is a Frobenius-split variety possessing a desingularization with connected fibers, then Y is normal. (Normality in all finite characteristics implies normality in characteristic 0). ... |

18 | The Lefschetz fixed point theorem for elliptic complexes - Atiyah, Bott - 1968 |

13 |
Specht Series for Column-Convex Diagrams
- Reiner, Shimozono
- 1995
(Show Context)
Citation Context ...(n).) These objects have been extensively explored. Combinatorists have examined the representations SD,n as part of the theory of generalized Schur modules and Young diagrams. (See [29], [33], [34], =-=[35]-=-, [39].) The geometric theory of F3,n and its desingularizations goes back to Schubert [37] and Semple [38], and has been illuminated recently by Fulton, MacPherson, and others ([10], [13], [20],[36],... |

8 |
Frobenius splittings and B-modules
- Kallen
- 1993
(Show Context)
Citation Context ...facts and the rest of the Theorem, we will need more sophisticated techniques. 2.2 Frobenius splitting The theory of Frobenius splittings invented by Mehta, Ramanan, and Ramanathan ([27], [30], [31], =-=[18]-=-) is a characteristic-p technique for proving surjectivity and vanishing results about coherent sheaves, even in characteristic 8zero. It is highly practical for dealing with homogeneous varieties be... |

8 |
Frobenius splittings and blowups
- Lakshmibai, Mehta, et al.
- 1998
(Show Context)
Citation Context ...tic 0). These strong properties of split varieties will suffice to prove our Theorem, provided we construct a compatible splitting. This is rendered practical by a criterion that was made explicit in =-=[21]-=- in terms of a notion “residually normal crossing”, which we now recall. 9A divisor D defined by f0 = 0 around a point P on a smooth affine variety X of dimension n has residually normal crossing at ... |

8 |
On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic p
- Mehta, Kallen
- 1992
(Show Context)
Citation Context ... 10 one sees that the Riπ∗KF FM are locally the same as the 3,n Riπ∗OF FM, so they vanish too. Alternatively, one checks that the Grauert3,n Riemenschneider vanishing theorem with Frobenius splitting =-=[28]-=- applies. For this, observe that our splitting of F3,n gives one on the complement of the exceptional locus of π in F FM 3,n . As this exceptional locus has codimension two the splitting extends and i... |

7 |
Borel–Weil theorem for Schur modules and configuration varieties, preprint alg-geom/9411014
- Magyar
- 1994
(Show Context)
Citation Context ...r simple example. Nevertheless, our character formulas for Schur modules are stated in purely elementary terms, and the interested reader can skip directly to sections 5.2 and 5.3. In previous papers =-=[22]-=-, [24], [23] we considered the same problems for diagrams D satisfying the “northwest” or “strongly separated” conditions, for which the geometry of FD is particularly simple. D3 is the smallest diagr... |

7 |
A vanishing theorem for Schur modules
- Woodcock
- 1994
(Show Context)
Citation Context ...These objects have been extensively explored. Combinatorists have examined the representations SD,n as part of the theory of generalized Schur modules and Young diagrams. (See [29], [33], [34], [35], =-=[39]-=-.) The geometric theory of F3,n and its desingularizations goes back to Schubert [37] and Semple [38], and has been illuminated recently by Fulton, MacPherson, and others ([10], [13], [20],[36], [7], ... |

5 | Four new formulas for Schubert polynomials , preprint - Magyar - 1995 |

5 |
Schubert Polynomials and Configuration Varieties, in preparation. 36
- Magyar
- 1990
(Show Context)
Citation Context ...le example. Nevertheless, our character formulas for Schur modules are stated in purely elementary terms, and the interested reader can skip directly to sections 5.2 and 5.3. In previous papers [22], =-=[24]-=-, [23] we considered the same problems for diagrams D satisfying the “northwest” or “strongly separated” conditions, for which the geometry of FD is particularly simple. D3 is the smallest diagram whi... |

5 |
Anzahlgeometrische Behandlung des Dreiecks
- Schubert
- 1954
(Show Context)
Citation Context ...entations SD,n as part of the theory of generalized Schur modules and Young diagrams. (See [29], [33], [34], [35], [39].) The geometric theory of F3,n and its desingularizations goes back to Schubert =-=[37]-=- and Semple [38], and has been illuminated recently by Fulton, MacPherson, and others ([10], [13], [20],[36], [7], [8]). In the current paper, we explore the relations between the algebraic and geomet... |

4 |
Bott’s formula and enumerative geometry, preprint
- Ellingsrud, Stromme
- 1994
(Show Context)
Citation Context ...e data in Lemma 9. The next result we shall use is based on the theorem of HirzebruchRiemann-Roch [3], combined with Bott’s Residue Formula [6], [2], according to the method of Ellingsrud and Stromme =-=[11]-=-. Proposition 16 Suppose the torus T = C ∗ is one-dimensional, and acts with isolated fixed points. Let v = 1 in the Lie algebra t = C, and at each T-fixed point p, let b(p) = tr(v | L|p). Denote the ... |

4 |
Representation of Symmetric Groups by Bad Shapes
- Murphy, Peel
- 1988
(Show Context)
Citation Context ...presentation of GL(n).) These objects have been extensively explored. Combinatorists have examined the representations SD,n as part of the theory of generalized Schur modules and Young diagrams. (See =-=[29]-=-, [33], [34], [35], [39].) The geometric theory of F3,n and its desingularizations goes back to Schubert [37] and Semple [38], and has been illuminated recently by Fulton, MacPherson, and others ([10]... |

4 |
On flagged Schur modules of general shape
- Reiner, Shimozono
- 1993
(Show Context)
Citation Context ... of GL(n).) These objects have been extensively explored. Combinatorists have examined the representations SD,n as part of the theory of generalized Schur modules and Young diagrams. (See [29], [33], =-=[34]-=-, [35], [39].) The geometric theory of F3,n and its desingularizations goes back to Schubert [37] and Semple [38], and has been illuminated recently by Fulton, MacPherson, and others ([10], [13], [20]... |

3 |
Schubert’s enumerative geometry of triangles from a modern viewpoint, Springer LNM 862
- Roberts, Speiser
- 1981
(Show Context)
Citation Context ... [35], [39].) The geometric theory of F3,n and its desingularizations goes back to Schubert [37] and Semple [38], and has been illuminated recently by Fulton, MacPherson, and others ([10], [13], [20],=-=[36]-=-, [7], [8]). In the current paper, we explore the relations between the algebraic and geometric pictures. We prove a Borel-Weil theorem realizing SD3,n (or the Schur module of any three-row diagram) a... |

2 |
Zur Topologie der Fahnenräume: Definition und Fragestellung
- Bozek, Drechsler
- 1980
(Show Context)
Citation Context ... [39].) The geometric theory of F3,n and its desingularizations goes back to Schubert [37] and Semple [38], and has been illuminated recently by Fulton, MacPherson, and others ([10], [13], [20],[36], =-=[7]-=-, [8]). In the current paper, we explore the relations between the algebraic and geometric pictures. We prove a Borel-Weil theorem realizing SD3,n (or the Schur module of any three-row diagram) as the... |

2 |
Faserungen von Fahenenmengen
- Bozek, Drechsler
- 1985
(Show Context)
Citation Context ....) The geometric theory of F3,n and its desingularizations goes back to Schubert [37] and Semple [38], and has been illuminated recently by Fulton, MacPherson, and others ([10], [13], [20],[36], [7], =-=[8]-=-). In the current paper, we explore the relations between the algebraic and geometric pictures. We prove a Borel-Weil theorem realizing SD3,n (or the Schur module of any three-row diagram) as the sect... |

2 |
Variété des triplets complets
- LeBarz, La
- 1988
(Show Context)
Citation Context ...[34], [35], [39].) The geometric theory of F3,n and its desingularizations goes back to Schubert [37] and Semple [38], and has been illuminated recently by Fulton, MacPherson, and others ([10], [13], =-=[20]-=-,[36], [7], [8]). In the current paper, we explore the relations between the algebraic and geometric pictures. We prove a Borel-Weil theorem realizing SD3,n (or the Schur module of any three-row diagr... |