arXiv subjects

Rachel Webb

Publications and source records attributed to Rachel Webb.

15 recordsLinked to original sources

arXiv2025

Weyl-invariant subspaces are (usually) not generic

Let $V$ be a linear representation of a connected complex reductive group $G$. Given a choice of character $\theta$ of $G$, Geometric Invariant Theory defines a locus $V^{ss}_\theta(G) \subseteq V$ of semistable points. We give necessary, sufficient, and in some cases equivalent conditions for the existence of $\theta$ such that a maximal torus $T$ of $G$ acts on $V^{ss}_\theta(T)$ with finite stabilizers. In such cases, the stack quotient $[V^{ss}_\theta(G)/G]$ is is known to be Deligne-Mumford. Our proof uses the combinatorial structure of the weights of irreducible representations of semisimple groups. As an application we generalize the Grassmannian flop example of Donovan-Segal.

math.RT↗

arXiv2025

Twisted stable maps with colliding points

We study moduli spaces of stable maps from pointed curves, where the points are allowed to coincide, with target a tame Deligne-Mumford stack. This generalizes the Abramovich-Vistoli theory of twisted stable maps as well as work of Hassett, Alexeev and Guy, and Bayer and Manin, who studied stable maps to projective varieties from curves with weighted marked points.

math.AG↗

arXiv2025

Finding the walls for quiver moduli

We give an effective characterisation of the walls in the variation of geometric invariant theory problem associated to a quiver and a dimension vector.

math.AG↗

arXiv2025

Stringy Chow rings and weighted blow ups

We compute the stringy chow ring of a general Deligne-Mumford stack of the form [X/G] for a smooth variety X and diagonalizable group scheme G, working over a base field that is not necessarily algebraically closed. We then specialize to the stringy chow ring of the weighted blow up of a smooth variety along a smooth center. We explore finite generation properties of this ring.

math.AG↗

arXiv2024

Curves with colliding points: logarithmic and stacky

We introduce a new notion of generalized log twisted curves, which are marked nodal curves with additional data at the marked points. In the case when the markings are distinct this notion agrees with the notion of twisted curve introduced by Abramovich and Vistoli. In addition to developing the basic notions and results, we study in this article the moduli of such curves as well as contraction maps between them. This is motivated, in part, by applications to twisted stable maps which will be studied in a subsequent article.

math.AG↗

arXiv2024

Ample vector bundles and moduli of tame stacks

We explain how to define an embedding of a tame stack over a noetherian ring into a certain generalization of a weighted projective stack using a notion of ample vector bundle on the stack. As applications we construct algebraic moduli stacks of tame stacks equipped with an ample vector bundle and algebraic stacks of tame orbicurves.

math.AG↗

arXiv2024

Crepant Transformation Conjecture For the Grassmannian Flop

We prove an explicit form of the Crepant Transformation Conjecture for Grassmannian flops. Our approach uses abelianization to first relate the restrictions of the Lagrangian cones to degree-2 classes, and then deduces the general result using ``explicit reconstruction'' (also known as the method of big I-functions).

math.AG↗

arXiv2022

Subtleties of Quantum Lefshetz without Convexity

Let $Y\mathord{/\mkern-6mu/}_\theta G$ be a complete intersection in a GIT quotient $X\mathord{/\mkern-6mu/}_\theta G$ cut out by a $G$-representation $E$. We show that the $E$-twisted quasimap $I$-function recovers invariants of $(Y, G, \theta)$ if its nonequivariant limit exists before restriction to $Y\mathord{/\mkern-6mu/}_\theta G$. This corrects a conjecture credited to Coates-Corti-Iritani-Tseng. We explain how to correct computations in the literature based on the faulty conjecture.

math.AG↗

arXiv2022

Some Applications of Abelianization in Gromov-Witten Theory

Let $G$ be a complex reductive group and let $X$ and $E$ be two linear representations of $G$. Let $Y$ be a complete intersection in $X$ equal to the zero locus of a $G$-equivariant section of the trivial bundle $E \times X \to X$. We explain some general techniques for using quasimap formulas to compute useful $I$-functions of $Y\mathord{/\mkern-6mu/} G$. We work several explicit examples, including a rigorous derivation of a quantum period computed conjecturally by Oneto-Petracci.

math.AG↗

arXiv2021

Abelianization and Quantum Lefschetz for Orbifold Quasimap $I$-Functions

Let $Y$ be a complete intersection in an affine variety $X$, with action by a complex reductive group $G$. Let $T \subset G$ be a maximal torus. A character $\theta$ of $G$ defines GIT quotients $Y//_\theta G$ and $X//_\theta T$. We prove formulas relating the small quasimap I-function of $Y//_\theta G$ to that of $X//_\theta T$. When $X$ is a vector space, this provides a completely explicit formula for the small $I$-function of $Y//_\theta G$.

math.AG↗

arXiv2021

The Moduli of Sections Has a Canonical Obstruction Theory

We give a detailed proof that locally Noetherian moduli stacks of sections carry canonical obstruction theories. As part of the argument we construct a dualizing sheaf and trace map, in the lisse-etale topology, for families of tame twisted curves, when the base stack is locally Noetherian.

math.AG↗

arXiv2019

Virtual cycles of stable (quasi)-maps with fields

We generalize the results of Chang-Li, Kim-Oh and Chang-Li on the moduli of $p$-fields to the setting of (quasi-)maps to complete intersections in arbitrary smooth Deligne-Mumford stacks with projective coarse moduli. In particular, we show that the virtual cycle of stable (quasi-)maps to a complete intersection can be recovered by the cosection localized virtual cycle of the moduli of $p$-fields of the ambient space.

math.AG↗

arXiv2019

Quasimaps and some examples of stacks for everybody

This note introduces the theory of quasimaps to GIT quotients with intuition and concrete examples, with the goal of explaining a closed formula for the quasimap $I$-function. Along the way, it emphasizes aspects of this story that illustrate general stacky concepts.

math.AG↗

arXiv2018

The Abelian-Nonabelian Correspondence for $I$-functions

We prove the abelian-nonabelian correspondence for quasimap $I$-functions. That is, if $Z$ is an affine l.c.i. variety with an action by a complex reductive group $G$, we prove an explicit formula relating the quasimap $I$-functions of the GIT quotients $Z//_{\theta} G$ and $Z//_{\theta} T$ where $T$ is a maximal torus of $G$. We apply the formula to compute the $J$-functions of some Grassmannian bundles on Grassmannian varieties and Calabi-Yau hypersurfaces in them.

math.AG↗

arXiv2015

Landau-Ginzburg Mirror Symmetry Conjecture

We prove the Landau-Ginzburg mirror symmetry conjecture between invertible quasi-homogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito-Givental theory (LG B-model) of the mirror polynomial.

math.AG↗