Topology of Blow-ups and Enumerative Geometry

Duan, Haibao; Li, Banghe

doi:10.48550/arxiv.0906.4152

Cited by 3 publications

(6 citation statements)

References 25 publications

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“…IV) fail to be flag manifolds, but can be constructed by performing a finite number of steps of blow-ups on flag manifolds (see examples in Fulton [38], § 10.4, Eisenbud and Harris [34], Chap. 13, or in [23] for the constructions of the parameter spaces of complete conics and quadrics in CP 3 ). As a result, the relevant characteristics can be computed from those of flag manifolds via strict transformations (see, for instance, [23], Examples 5.11 and 5.12).…”

Section: Schubert Problem Of Characteristicsmentioning

confidence: 99%

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Schubert calculus and intersection theory of flag manifolds

Duan

Zhao

2022

Russian Math. Surveys

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Hilbert's 15th problem called for a rigorous foundation of Schubert calculus, of which a long-standing and challenging part is the Schubert problem of characteristics. In the course of securing a foundation for algebraic geometry, Van der Waerden and Weil attributed this problem to the intersection theory of flag manifolds. This article surveys the background, content, and solution of the problem of characteristics. Our main results are a unified formula for the characteristics and a systematic description of the intersection rings of flag manifolds. We illustrate the effectiveness of the formula and the algorithm by explicit examples. Bibliography: 71 titles.

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Section: Schubert Problem Of Characteristicsmentioning

confidence: 99%

“…13, or in [23] for the constructions of the parameter spaces of complete conics and quadrics in CP 3 ). As a result, the relevant characteristics can be computed from those of flag manifolds via strict transformations (see, for instance, [23], Examples 5.11 and 5.12).…”

Section: Schubert Problem Of Characteristicsmentioning

confidence: 99%

Schubert calculus and intersection theory of flag manifolds

Duan

Zhao

2022

Russian Math. Surveys

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“…6 Further remarks on the characteristics 6.1. Certain parameter spaces of the geometric figures concerned by Schubert [51,Chapter 4] may fail to be flag manifolds, but can be constructed by performing finite number of blow-ups on flag manifolds along the centers again in flag manifolds, see the examples in Fulton [27,Example 14.7.12], or in [18] for the construction of the parameter spaces of the complete conics and quadrics on the 3-space P 3 . As results the relevant characteristics can be computed from those of flag manifolds via strict transformations (e.g.…”

Section: The Ring H * (G/t ) For a Classical Gmentioning

confidence: 99%

“…As results the relevant characteristics can be computed from those of flag manifolds via strict transformations (e.g. [18,Examples 5.11;5.12]).…”

Section: The Ring H * (G/t ) For a Classical Gmentioning

confidence: 99%

On Schubert's Problem of Characteristics

Duan,

Zhao

2019

Preprint

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The Schubert varieties on a flag manifold G/P give rise to a cell decomposition on G/P whose Kronecker duals, known as the Schubert classes on G/P , form an additive base of the integral cohomology H * (G/P ). The Schubert's problem of characteristics asks to express a monomial in the Schubert classes as a linear combination in the Schubert basis.We present a unified formula expressing the characteristics of a flag manifold G/P as polynomials in the Cartan numbers of the group G. As application we develop a direct approach to our recent works on the Schubert presentation of the cohomology rings of flag manifolds G/P .

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“…Theorem 6.12 (Differential slice theorem, [2, pp. [14][15]). There exists an equivariant diffeomorphism from an equivariant open neighborhood of the zero section in G× G x V x to an open neighborhood of G• x in X, which sends the zero section G/G x onto the orbit G • x by the orbit map.…”

Section: Proof Due To Bijectivity Of ∇ E and (38)mentioning

confidence: 99%

Cohomology rings and algebraic torus actions on hypersurfaces in the product of projective spaces and bounded flag varieties

Solomadin¹

2019

Preprint

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In this paper, we present a necessary condition for a non-singular projective variety with an effective algebraic torus action of positive complexity with isolated fixed points to be a toric variety. The method is based on the monodromy map on the weight hypergraph associated with the torus action. The introduced notion of a weight hypergraph generalises the GKM-graph in the case of a torus action satisfying all axioms from the definition of a GKM-manifold, except 2-linear independence of weights at fixed points. We apply this method to the generalised Buchstaber-Ray varieties BR i , j ⊂ BF i × P j , i , j 0, as well as the Ray's varieties R i , j ⊂ BF i × BF j and give a full list of toric varieties among them. The importance of the varieties R i , j is due to the problem of representatives in the unitary bordism ring in the family of quasitoric TTS-and TNS-manifolds. Also in this paper the automorphism group Aut H i , j is computed for the Milnor hypersurface H i , j ⊂ P i × P j . As a corollary, any effective action of (S 1 ) k on the Milnor hypersurface H i , j preserving the natural complex structure has k max {i , j }. * (0 i j ). However, BR i,j are not TNS-manifolds, generally speaking. In [7], the diamond sum operation was introduced, in order to replace a disjoint union of two quasitoric manifolds (in sense of [13]) by a complex bordant quasitoric manifold. As a corollary ([7]), in any class x ∈ Ω U 2n a quasitoric representative was constructed from the Buchstaber-Ray varieties, n > 1. It remained unknown whether R i,j are toric varieties. The importance of this question is due to the problem of representatives of the complex bordism ring Ω U * in the family of quasitoric TTS-and TNS-manifolds. This problem was completely solved in [25], by constructing a family of toric TTS-and TNS-manifolds whose bordism classes multiplicatively generate the ring Ω U * . However, this construction is rather involved. On the other hand, if R i,j would be toric varieties, then they form a family of toric TTS-and TNS-generators of Ω U * , which is simpler to describe. This provides a motivation to study torus actions on the aforementioned varieties.The Picard group Pic(BF i × BF j ) is isomorhic to H 2 (BF i × BF j ; Z) (see [21, p.127, §15.9]). It implies that the first Chern class c 1 (β i ⊗ β ′ j ) does not belong to the cone of effective classes (spanned by maximal torus-invariant effective divisors of the toric variety), hence it is not represented by any non-singular closed subvariety of codimension 1 in X (see [9]). This justifies our choice of the linear bundle β i ⊗ β ′ j on BF i × BF j . The choice of the particular variety R i,j among all possible dualisations of β i ⊗ β ′ j on BF i × BF j is due to its relation to the well-known MilnorThe publication has been prepared with the support of the "RUDN University Program 5-100" and by the grant of "Young mathematics of Russia" foundation.

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Topology of Blow-ups and Enumerative Geometry

Cited by 3 publications

References 25 publications

Schubert calculus and intersection theory of flag manifolds

Schubert calculus and intersection theory of flag manifolds

On Schubert's Problem of Characteristics

Cohomology rings and algebraic torus actions on hypersurfaces in the product of projective spaces and bounded flag varieties

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