Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0709.1191

Thom polynomials of invariant cones, Schur functions, and positivity

Piotr Pragacz·Andrzej Weber
View original source ↗Read source document ↗Export metadata ↓

Abstract

We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the generalized Thom polynomials, expanded in the products of Schur functions of the bundles, have nonnegative coefficients. For classical Thom polynomials associated with maps of complex manifolds, this gives an extension of our former result for stable singularities to nonnecessary stable ones. We also discuss some related aspects of Thom polynomials, which makes the article expository to some extent.

Explore related subjects

math.AGmath.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Piotr Pragacz, Andrzej Weber. 2007-09-08. Thom polynomials of invariant cones, Schur functions, and positivity. https://arxiv.org/abs/0709.1191

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Beyond Harnack Rigidity

The boundary of a plane amoeba is always contained in its contour, and equality is a characteristic feature of simple Harnack curves. We show that the converse fails, even under strong smoothness and nondegeneracy assumptions. For every two-dimensional lattice polygon, except unimodular triangles, we construct a smooth Newton-nondegenerate curve with smooth logarithmic critical locus and smooth embedded contour satisfying $\mathcal C(\mathscr A_f)=\partial\mathscr A_f$, although the curve is not Harnack. We also provide explicit primitive and nonprimitive families that are not torus-equivalent to simple Harnack curves. A concrete primitive example is certified by exact elimination and Sturm root counting. These results disprove contour--boundary rigidity and show that the contour as a set does not detect the real structure or the multiplicity of coincident critical sheets, thereby refining the compensation problem proposed by Lang, Shapiro, and Shustin.

math.AG↗
arXiv2026

Special Cohen--Macaulay sheaves on partial resolutions of rational surfaces singularities

We introduce the categories $\CM(X)$ and $\SCM(X)$ of reflexive sheaves on a minimal partial resolution $f\colon X \to \Spec R$ of a rational surface singularity $\Spec R$. The main result of this paper establishes that $\SCM(X)$ possesses a natural Frobenius structure, serving as a geometric counterpart to the algebraic Frobenius structure on special Cohen--Macaulay $R$-modules, introduced by Iyama--Wemyss and Iyama--Kalck--Wemyss--Yang. Utilizing this geometric framework, we establish an exact equivalence between $\SCM(X)$ and the category of special Cohen--Macaulay $R$-modules equipped with a specific exact structure, which induces a triangle equivalence between their stable categories. Consequently, this provides a direct, geometric proof of the Iyama--Kalck--Wemyss--Yang equivalence and yields a Buchweitz-type equivalence $\underline{\SCM}(X) \simeq D_{\sg}(X)$.

math.AG↗