141 research outputs found
Thom polynomials and Schur functions: the singularities I_{2,2}(-)
We give the Thom polynomials for the singularities associated with
maps with parameter . Our computations combine the characterization of Thom polynomials via the
``method of restriction equations'' of Rimanyi et al. with the techniques of
Schur functions.Comment: 21 pages; Ann. Inst. Fourier vol.57; this is expanded Sect.4 of
math.AG/0509234; new added results: Theorem 11 (based on P-ideals of
singularities) and explicit expressions for the coefficients of the Thom
polynomials of I_22(-) (Propositions 17, 18, 19); references update
Symmetric polynomials and divided differences in formulas of intersection theory
The goal of the paper is two-fold. At first, we attempt to give a survey of
some recent applications of symmetric polynomials and divided differences to
intersection theory. We discuss: polynomials universally supported on
degeneracy loci; some explicit formulas for the Chern and Segre classes of
Schur bundles with applications to enumerative geometry; flag degeneracy loci;
fundamental classes, diagonals and Gysin maps; intersection rings of G/P and
formulas for isotropic degeneracy loci; numerically positive polynomials for
ample vector bundles.
Apart of surveyed results, the paper contains also some new results as well
as some new proofs of earlier ones: how to compute the fundamental class of a
subvariety from the class of the diagonal of the ambient space; how to compute
the class of the relative diagonal using Gysin maps; a new formula for pushing
forward Schur's Q- polynomials in Grassmannian bundles; a new formula for the
total Chern class of a Schur bundle; another proof of Schubert's and
Giambelli's enumeration of complete quadrics; an operator proof of the
Jacobi-Trudi formula; a Schur complex proof of the Giambelli-Thom-Porteous
formula.Comment: 58 pages; to appear in the volume "Parameter Spaces", Banach Center
Publications vol 36 (1996) AMSTE
Positivity of Thom polynomials II: the Lagrange singularities
We show that Thom polynomials of Lagrangian singularities have nonnegative
coefficients in the basis consisting of Q-functions. The main tool in the proof
is nonnegativity of cone classes for globally generated bundles.Comment: 16 pages, reduced introduction but new chapter about Legendrian
singularities. The title is change. Correction in the formula for A_7 and a
sign correction in formula (29
Positivity of Schur function expansions of Thom polynomials
Combining the Kazarian approach to Thom polynomials via classifying spaces of
singularities with the Fulton-Lazarsfeld theory of numerical positivity for
ample vector bundles, we show that the coefficients of various Schur function
expansions of the Thom polynomials of stable and unstable singularities are
nonnegative.Comment: 12 pages, various modification of the expositio
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