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Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants

Constantin Teleman, Christopher Woodward (2003)

Annales de l’institut Fourier

The set of conjugacy classes appearing in a product of conjugacy classes in a compact, 1 -connected Lie group K can be identified with a convex polytope in the Weyl alcove. In this paper we identify linear inequalities defining this polytope. Each inequality corresponds to a non-vanishing Gromov-Witten invariant for a generalized flag variety G / P , where G is the complexification of K and P is a maximal parabolic subgroup. This generalizes the results for S U ( n ) of Agnihotri and the second author and Belkale on...

Parameter spaces for quadrics

Anders Thorup (1996)

Banach Center Publications

The parameter spaces for quadrics are reviewed. In addition, an explicit formula for the number of quadrics tangent to given linear subspaces is presented.

Plane curves with small linear orbits, I

Paoli Aluffi, Carel Faber (2000)

Annales de l'institut Fourier

The “linear orbit” of a plane curve of degree d is its orbit in ℙ d ( d + 3 ) / 2 under the natural action of PGL ( 3 ) . In this paper we compute the degree of the closure of the linear orbits of most curves with positive dimensional stabilizers. Our tool is a nonsingular variety dominating the orbit closure, which we construct by a blow-up sequence mirroring the sequence yielding an embedded resolution of the curve. The results given here will serve as an ingredient in the computation of the analogous information for...

Plane projections of a smooth space curve

Trygve Johnsen (1996)

Banach Center Publications

Let C be a smooth non-degenerate integral curve of degree d and genus g in ℙ 3 over an algebraically closed field of characteristic zero. For each point P in ℙ 3 let V P be the linear system on C induced by the hyperplanes through P. By V P one maps C onto a plane curve C P , such a map can be seen as a projection of C from P. If P is not the vertex of a cone of bisecant lines, then C P will have only finitely many singular points; or to put it slightly different: The secant scheme S P = ( V P ) 2 1 parametrizing divisors in...

Polytopes, quasi-minuscule representations and rational surfaces

Jae-Hyouk Lee, Mang Xu, Jiajin Zhang (2017)

Czechoslovak Mathematical Journal

We describe the relation between quasi-minuscule representations, polytopes and Weyl group orbits in Picard lattices of rational surfaces. As an application, to each quasi-minuscule representation we attach a class of rational surfaces, and realize such a representation as an associated vector bundle of a principal bundle over these surfaces. Moreover, any quasi-minuscule representation can be defined by rational curves, or their disjoint unions in a rational surface, satisfying certain natural...

Positivity of Schur function expansions of Thom polynomials

Piotr Pragacz, Andrzej Weber (2007)

Fundamenta Mathematicae

Combining the approach to Thom polynomials via classifying spaces of singularities with the Fulton-Lazarsfeld theory of cone classes and positive polynomials for ample vector bundles, we show that the coefficients of the Schur function expansions of the Thom polynomials of stable singularities are nonnegative with positive sum.

Positivity of Thom polynomials II: the Lagrange singularities

Małgorzata Mikosz, Piotr Pragacz, Andrzej Weber (2009)

Fundamenta Mathematicae

We study Thom polynomials associated with Lagrange singularities. We expand them in the basis of Q̃-functions. This basis plays a key role in the Schubert calculus of isotropic Grassmannians. We prove that the Q̃-function expansions of the Thom polynomials of Lagrange singularities always have nonnegative coefficients. This is an analog of a result on the Thom polynomials of mapping singularities and Schur S-functions, established formerly by the last two authors.

Predegree Polynomials of Plane Configurations in Projective Space

Tzigantchev, Dimitre (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 14N10, 14C17.We work over an algebraically closed field of characteristic zero. The group PGL(4) acts naturally on PN which parameterizes surfaces of a given degree in P3. The orbit of a surface under this action is the image of a rational map PGL(4) ⊂ P15→PN. The closure of the orbit is a natural and interesting object to study. Its predegree is defined as the degree of the orbit closure multiplied by the degree of the above map restricted to a general Pj,...

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