On unions of scrolls along linear spaces
On vanishing inflection points of plane curves
We study the local behaviour of inflection points of families of plane curves in the projective plane. We develop normal forms and versal deformation concepts for holomorphic function germs which take into account the inflection points of the fibres of . We give a classification of such function- germs which is a projective analog of Arnold’s A,D,E classification. We compute the versal deformation with respect to inflections of Morse function-germs.
On Zariski's theorem in positive characteristic
In the current paper we show that the dimension of a family of irreducible reduced curves in a given ample linear system on a toric surface over an algebraically closed field is bounded from above by , where denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality does not imply the nodality of even if belongs to the...
Osculatory behavior and second dual varieties of Del Pezzo surfaces.
Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants
The set of conjugacy classes appearing in a product of conjugacy classes in a compact, -connected Lie group 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 , where is the complexification of and is a maximal parabolic subgroup. This generalizes the results for of Agnihotri and the second author and Belkale on...
Parameter spaces for quadrics
The parameter spaces for quadrics are reviewed. In addition, an explicit formula for the number of quadrics tangent to given linear subspaces is presented.
Permutations with Kazhdan-Lusztig polynomial . With an appendix by Sara Billey and Jonathan Weed.
Picard Numbers of Surfaces in 3-Dimensional Weighted Projective Spaces.
Picard-Borel-Räume.
Piecewise polynomial functions, convex polytopes and enumerative geometry
Plane curves of genus p and degree 2p as projections of space curves of the same degree.
Plane curves with small linear orbits, I
The “linear orbit” of a plane curve of degree is its orbit in under the natural action of . 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
Let C be a smooth non-degenerate integral curve of degree d and genus g in over an algebraically closed field of characteristic zero. For each point P in let be the linear system on C induced by the hyperplanes through P. By one maps C onto a plane curve , 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 will have only finitely many singular points; or to put it slightly different: The secant scheme parametrizing divisors in...
Plücker conditions on plane rational curves.
Polar classes of singular varieties
Polynomials, Symmetry, and Dynamics: an Undertaking in Aesthetics
Polytopes, quasi-minuscule representations and rational surfaces
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...
Poncelet 5-gons and abelian surfaces.
Positivity of Schur function expansions of Thom polynomials
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
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.