Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1804.08164

Variations on a Theme of Schubert Calculus

Maria Gillespie
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear spaces. It is intended as a guide for readers with a combinatorial bent to understand and appreciate the geometric and topological aspects of Schubert calculus, and conversely for geometric-minded readers to gain familiarity with the relevant combinatorial tools in this area. We lead the reader through a tour of three variations on a theme: Grassmannians, flag varieties, and orthogonal Grassmannians. The orthogonal Grassmannian, unlike the ordinary Grassmannian and the flag variety, has not yet been addressed very often in textbooks, so this presentation may be helpful as an introduction to type B Schubert calculus. This work is adapted from the author's lecture notes for a graduate workshop during the Equivariant Combinatorics Workshop at the Center for Mathematics Research, Montreal, June 12-16, 2017.

Explore related subjects

math.AGmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Maria Gillespie. 2021-01-18. Variations on a Theme of Schubert Calculus. https://arxiv.org/abs/1804.08164

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

KEEP EXPLORING

Related papers

arXiv2026

G3-Criteria and Applications

The G3-property of a subvariety was introduced by Hironaka-Matsumura, and plays an important role for deducing connectedness and extension results. Unfortunately, it's a rather elusive notion, which is not always easy to establish. Most of the existing work is concentrated on subvarieties of homogeneous varieties. The first goal of this article is to show that mobility assumptions on the subvariety, considered in works of Badescu, Chow, Debarre, Voisin, yield a certain partial positivity property, slightly stronger than G3, previously introduced by the author. Second, we apply the result to prove that, in numerous situations, the splitting of the normal bundle of a smooth two-codimensional subvariety implies that it is a complete intersection.

math.AG↗
arXiv2026

Moduli Stacks of $G$-Curves in Homotopy Theory at Height $p-1$

Let $p$ be odd and $G' = \mathbb{Z}/p \rtimes \mathbb{Z}/(p-1)^2$ the maximal finite subgroup of the Morava stabilizer group at height $p-1$. Inverse Galois theory produces from $G'$ alone a curve $X$, the unique curve of minimal genus with $\operatorname{Aut}(X) \simeq G'$; its ramification, its field of definition and its equation are consequences of the group, not choices. We prove a $G'$-equivariant equivalence between the deformations of $X$ and Lubin--Tate space, so that the Lubin--Tate action of $G'$ is the action of $\operatorname{Aut}(X)$ on deformations of the curve. The proof is a coordinate-free Kodaira--Spencer argument reducing to a single character count. The action becomes explicit: $G'$ acts through $\mathbb{F}_p \rtimes \mathbb{F}_p^\times$ shifting and scaling $p+1$ points on $\mathbb{P}^1$. From this we compute $H^*(G', π_* E_{p-1})$ and its Tate cohomology. One identity, $π^{p-1} = -p$, runs through every section.

math.AG↗