Let G be a compact connected Lie group and H, the centralizer of a one-parameter subgroup in G. Combining the ideas of Bott-Samelson 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.

Abstract. Using a blend of combinatorics and geometry, we give an algorithm for algebraically finding all flags in any zero-dimensional 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 Littlewood-Richardson coefficient to be 0. We describe pathologies of Eriksson and Linusson’s permutation array

this paper we will use without further mention the following notations. M will denote a smooth n-dimensional algebraic variety over an algebraically closed field of characteristic 0; L will be a line bundle on M , and X will be the zero-scheme 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 aspect graph is a popular viewer-centered representation that enumerates all the topologically distinct views of an object. Building the aspect graph requires partitioning viewpoint space in view-equivalent cells by a certain number of visual event surfaces. If the object is piecewise-smooth 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 piecewise-smooth algebraic bodies. We conclude by giving a bound on the number of topologically distinct views of such objects.

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.

We enumerate complex algebraic hypersurfaces in P n, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equisingular strata in the parameter space of hypersurfaces. We suggest an inductive procedure, based on an intersection theory combined with liftings and degenerations. The procedure computes the (co)homology class in question, whenever a given singularity type is properly defined and the stratum possesses good geometric properties. We consider in detail the generalized Newton-non-degenerate singularities. We also give examples of enumeration in some other cases.

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 1-cycles on a regular projective scheme X over the spectrum of a semi-local Dedekind domain, and the second moving lemma can be applied to 0-cycles on an FA-scheme 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 Hilbert-Samuel multiplicities e(Q, A), over all m-primary 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.

We enumerate plane algebraic curves with one singular point of any (prescribed) topological singularity type. We discuss how to generalize the method to the singular hypersurfaces

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

Let G be a compact connected Lie group and H the centralizer of a one-parameter subgroup. We explain a program that expands the product of two arbitrary Schubert classes on the flag manifold G/H in terms of Schubert classes.