Picard Groups of Singular Affine Curves over a Perfect Fields.
Picard groups of Zariski surfaces
Picard numbers of surfaces in 3-dimensional weighted projective spaces.
Picard Numbers of Surfaces in 3-Dimensional Weighted Projective Spaces.
Picard schemes of quotients by finite commutative group schemes.
Picard-Borel-Eigenschaft und Anwendungen.
Picard-Borel-Räume.
Picardindex und ganzalgebraische Punkte.
Picard-Lefschetz theory and characters of a semisimple Lie group.
Picard-Lefschetz theory for the coadjoint quotient of a semisimple Lie algebra.
Picard-Zahlen komplexer Tori.
Piecewise polynomial functions, convex polytopes and enumerative geometry
Pieri Type formula for isotropic Grassmannians; the operator approach.
Pieri's formula for flag manifolds and Schubert polynomials
We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary or a complete symmetric polynomial. Thus, we generalize the classical Pieri’s formula for Schur polynomials (associated to Grassmann varieties) to Schubert polynomials (associated to flag manifolds). Our primary technique is an explicit geometric description of certain...
Pieri-type formulas for maximal isotropic Grassmannians via triple intersections
We give an elementary proof of the Pieri-type formula in the cohomology ring of a Grassmannian of maximal isotropic subspaces of an orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The multiplicities (which are powers of 2) in the Pieri-type formula are seen to arise from the intersection of a collection of quadrics with a linear space.
Pieri-type intersection formulas and primary obstructions for decomposing 2-forms
We study the homological intersection behaviour for the Chern cells of the universal bundle of G(d,Qₙ), the space of [d]-planes in the smooth quadric Qₙ in over the field of complex numbers. For this purpose we define some auxiliary cells in terms of which the intersection properties of the Chern cells can be described. This is then applied to obtain some new necessary conditions for the global decomposability of a 2-form of constant rank.
Pinceaux de courbes planes et invariants polaires
We study pencils of plane curves , t ∈ ℂ, using the notion of polar invariant of the plane curve f = 0 with respect to a smooth curve l = 0. More precisely we compute the jacobian Newton polygon of the generic fiber , t ∈ ℂ. The main result gives the description of pencils which have an irreducible fiber. Furthermore we prove some applications of the local properties of pencils to singularities at infinity of polynomials in two complex variables.
Pinceaux de droites et automorphismes des surfaces affines.
Plane curve singularities and carousels
In this paper we give a direct and explicit description of the local topological embedding of a plane curve singularity using the Puiseux expansions of its branches in a given set of coordinates.
Plane curves and their fundamental groups: Generalizations of Uludağ's construction.