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15
Multiplicative rule of Schubert classes
 Invent. Math
"... Let G be a compact connected Lie group and H, the centralizer of a oneparameter subgroup in G. Combining the ideas of BottSamelson resolutions of Schubert varieties and the enumerative formula on a twisted product of 2 spheres obtained in [Du2], we obtain a closed formula for multiplying Schubert ..."
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Cited by 14 (9 self)
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Let G be a compact connected Lie group and H, the centralizer of a oneparameter subgroup in G. Combining the ideas of BottSamelson resolutions of Schubert varieties and the enumerative formula on a twisted product of 2 spheres obtained in [Du2], we obtain a closed formula for multiplying Schubert classes in the flag manifold G/H.
Intersections of Schubert varieties and other permutation array schemes, Algorithms in algebraic geometry
 IMA Vol. Math. Appl
, 2008
"... Abstract. Using a blend of combinatorics and geometry, we give an algorithm for algebraically finding all flags in any zerodimensional intersection of Schubert varieties with respect to three transverse flags, and more generally, any number of flags. In particular, the number of flags in a triple i ..."
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Cited by 13 (1 self)
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Abstract. Using a blend of combinatorics and geometry, we give an algorithm for algebraically finding all flags in any zerodimensional intersection of Schubert varieties with respect to three transverse flags, and more generally, any number of flags. In particular, the number of flags in a triple intersection is also a structure constant for the cohomology ring of the flag manifold. Our algorithm is based on solving a limited number of determinantal equations for each intersection (far fewer than the naive approach). These equations may be used to compute Galois and monodromy groups of intersections of Schubert varieties. We are able to limit the number of equations by using the permutation arrays of Eriksson and Linusson, and their permutation array varieties, introduced as generalizations of Schubert varieties. We show that there exists a unique permutation array corresponding to each realizable Schubert problem and give a simple recurrence to compute the corresponding rank table, giving in particular a simple criterion for a LittlewoodRichardson coefficient to be 0. We describe pathologies of Eriksson and Linusson’s permutation array
Singular Schemes Of Hypersurfaces
 Duke Math. J
, 1996
"... this paper we will use without further mention the following notations. M will denote a smooth ndimensional algebraic variety over an algebraically closed field of characteristic 0; L will be a line bundle on M , and X will be the zeroscheme of a section of L. Typically, X will be a prime divisor ..."
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Cited by 7 (5 self)
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this paper we will use without further mention the following notations. M will denote a smooth ndimensional algebraic variety over an algebraically closed field of characteristic 0; L will be a line bundle on M , and X will be the zeroscheme of a section of L. Typically, X will be a prime divisor of M and L = O(X); we will refer to X as a "hypersurface" on M . The singular scheme of X, SingX, will be the subscheme of M supported on the singular locus SINGULAR SCHEMES OF HYPERSURFACES 3
THE INDEX OF AN ALGEBRAIC VARIETY
"... Abstract. Let K be the field of fractions of a Henselian discrete valuation ring OK. Let XK/K be a smooth proper geometrically connected variety admitting a regular model X/OK. We show that the index δ(XK/K) of the variety XK/K can be explicitly computed using data pertaining only to the special fib ..."
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Cited by 4 (4 self)
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Abstract. Let K be the field of fractions of a Henselian discrete valuation ring OK. Let XK/K be a smooth proper geometrically connected variety admitting a regular model X/OK. We show that the index δ(XK/K) of the variety XK/K can be explicitly computed using data pertaining only to the special fiber Xk/k of the model X. We give two proofs of this theorem, using two moving lemmas. One moving lemma pertains to horizontal 1cycles on a regular projective scheme X over the spectrum of a semilocal Dedekind domain, and the second moving lemma can be applied to 0cycles on an FAscheme X which need not be regular. The study of the local algebra needed to prove these moving lemmas led us to introduce an invariant γ(A) of a singular local ring (A, m): the greatest common divisor of all the HilbertSamuel multiplicities e(Q, A), over all mprimary ideals Q in m. We relate this invariant γ(A) to the index of the exceptional divisor in a resolution of the singularity of Spec A, and we give a new way of computing the index of a smooth subvariety X/K of Pn K over any field K, using the invariant γ of the local ring at the vertex of a cone over X.
The Enumerative Geometry of Projective Algebraic Surfaces and The Complexity of Aspect Graphs
 International Journal of Computer Vision
, 1996
"... The aspect graph is a popular viewercentered representation that enumerates all the topologically distinct views of an object. Building the aspect graph requires partitioning viewpoint space in viewequivalent cells by a certain number of visual event surfaces. If the object is piecewisesmooth alg ..."
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Cited by 4 (1 self)
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The aspect graph is a popular viewercentered representation that enumerates all the topologically distinct views of an object. Building the aspect graph requires partitioning viewpoint space in viewequivalent cells by a certain number of visual event surfaces. If the object is piecewisesmooth algebraic, then all visual event surfaces are either made of lines having specified contacts with the object or made of lines supporting the points of contacts of planes having specified contacts with the object. In this paper, we present a general framework for studying the enumerative properties of line and plane systems. The context is that of enumerative geometry and intersection theory. In particular, we give exact results for the degrees of all visual event surfaces coming up in the construction of aspect graphs of piecewisesmooth algebraic bodies. We conclude by giving a bound on the number of topologically distinct views of such objects.
GALOIS GROUPS OF SCHUBERT PROBLEMS OF LINES ARE AT LEAST ALTERNATING
, 1207
"... Abstract. We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument ..."
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Cited by 2 (1 self)
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Abstract. We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of tworowed tableaux. In most cases, a combinatorial injection proves the inequality. The remaining cases use an integral formula for Kostka numbers of tworowed tableaux which comes from their realization as differences of certain polynomial coefficients that generalize binomial coefficients. This rewrites the inequality as an integral, which we estimate to establish the inequality.
Enumeration of singular algebraic curves
, 2005
"... We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular families in the parameter spaces of plane curves. We suggest an inductive procedure, which is based on t ..."
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Cited by 1 (1 self)
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We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular families in the parameter spaces of plane curves. We suggest an inductive procedure, which is based on the intersection theory combined with liftings and degenerations, and which computes the homology class in question whenever a given singularity type is defined. Our method does not require the knowledge of all possible deformations of a given singularity as it was in
A general Plucker formula
"... We prove a formula which compares intersection numbers of conormal varieties of two projective varieties and their dual varieties. When one of them is linear, we can recover the usual Plücker formula for the degree of the dual variety. The basic strategy of the proof is to study a category of Lagran ..."
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We prove a formula which compares intersection numbers of conormal varieties of two projective varieties and their dual varieties. When one of them is linear, we can recover the usual Plücker formula for the degree of the dual variety. The basic strategy of the proof is to study a category of Lagrangian subvarieties in the cotangent bundle of a projective space under a birational transformation. 1 1 The formula The conormal variety CS of any subvariety S in P n is a Lagrangian subvariety in T ∗ P n with respect to the canonical holomorphic symplectic form. If S is smooth then its Euler characteristic χ (S) equals to the intersection number CS · P n up to a sign (−1) dim S. In general χ (S) is replaced by the Euler characteristic of the Euler obstruction EuS defined by MacPherson [Ma], we denote it as ¯χ (S). More generally CS1 · CS2 is welldefined and equals to the Euler characteristic of the intersection up to a sign provided that S1 and S2 intersect transversely along a smooth subvariety in P n. In this paper we prove the following formula. Theorem 1 Suppose S1 and S2 are two subvarieties in P n which intersect transversely and the same holds true for their dual varieties S ∨ 1 and S ∨ 2 in Pn ∗. Then we have
On the enumeration of complex plane curves with two singular points
, 2008
"... We study equisingular strata of curves with two singular points of prescribed types. The method of the previous work [Kerner04] is generalized to this case. This allows to solve the enumerative problem for plane curves with two singular points of linear singularity types. In the general case this r ..."
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We study equisingular strata of curves with two singular points of prescribed types. The method of the previous work [Kerner04] is generalized to this case. This allows to solve the enumerative problem for plane curves with two singular points of linear singularity types. In the general case this reduces the enumerative questions to the problem of collision of the two singular points. The method is applied to several cases, e.g. enumeration of curves with two ordinary multiple points, with a point of a linear singularity type and a node etc. Explicit numerical results are given. An elementary application of the method is the determination of Thom polynomials for curves with one singular point (for some series of singularity types). Some examples are given.
On the geometry of some strata of unisingular curves
, 2008
"... We study geometric properties of linear strata of unisingular curves. We resolve the singularities of closures of the strata and represent the resolutions as projective bundles. This enables us to study their geometry. In particular we calculate the Picard groups of the strata and the intersection ..."
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We study geometric properties of linear strata of unisingular curves. We resolve the singularities of closures of the strata and represent the resolutions as projective bundles. This enables us to study their geometry. In particular we calculate the Picard groups of the strata and the intersection rings of the closures of the strata. The rational equivalence classes of some geometric cycles on the strata are calculated. As an application we give an example when the proper stratum is not affine. As an auxiliary problem we discuss the collision of two singular points, restrictions on possible resulting singularity types and solve the collision problem in several cases. Then we present some cases of enumeration of