# Questions tagged [geometric-invariant-theory]

for questions on geometric invariant theory (or GIT), including stability criteria and symplectic quotients.

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### Moduli of smooth curves in $|\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(2,2)|$ and their invariants

It is well known that any smooth curve $C\in |\mathcal{O}_{\mathbf{P}^1\times\mathbf{P}^1}(2,2)|$ has geometric genus equal to 1, so its isomorphism class is determined by its $j$-invariant. ...
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### How big is the complement of stable locus $\operatorname{Bun}G$

Let $\Sigma$ be a smooth projective curve, and $G$ a reductive group. Let $\operatorname{Bun}G$ be the stack of principal $G$ bundles on $\Sigma$ (with a fixed topological type). What is the ...
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### Descent of projective bundles

A problem studied in GIT is the descending of vector bundles (or more in general coherent sheaves) to quotients. It is a result of Kempf that whenever we have a vector bundle over a quasiprojective ...
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### How exactly do we calculate an affine covering of a scheme?

Is there a general algorithm that takes in some scheme $X$ and outputs some affine covering of $X$ ? If not, how are affine coverings of schemes usually calculated ? I'm particularly interested in ...
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### Quotients of toric varieties

This is a follow up of this question. Given a toric variety $X$ with a fan $\Sigma$ and a finite group $G$ acting on $X$, we know that the GIT quotient $X/G$ exists. However, as stated in the answer ...
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### Fiber product arising from reductive group action on varieties

Let $G$ be a reductive group acting on a smooth projective $X$. Let $P$ be a parabolic subgroup of $G$, and $Y$ a locally closed subvariety invariant under $P$. Assume in addition $Y$ is smooth. ( ...
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### Motivating the Quotient of an Algebraic Variety

Let $X$ be a variety with a $G$-action by an algebraic group. My question refers to a motivating example from: https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf Here is the relevant ...
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### Secondary fan and KN strata

Let $\mathbb{G}_m^r$ act on the affine space $\mathbb{A}^n$ through an embedding into the open dense torus. Is there a way to calculate the 1-parameter subgroups that determine the KN strata from the ...
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### Preparation for GIT (Geometric Invariant Theory)

I am trying to read Mumford's Geometric Invariant Theory, however, I find my knowledge in algebraic geometry is inadequate. My knowledge is at the level of Hartshorne's Algebraic Geometry. Mumford ...
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### Quotient of a Fano variety by a torus

We work over an algebraically closed field of characteristic zero. Let $X$ be a Fano variety, and $T\cong \mathbb{G}_m^r$ a torus acting faithfully on $X$. I think we can canonically linearize the ...
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### Invariants of linear endomorphisms of tensor products

Let $V$ and $W$ be two finite dimensional vector spaces over an algebraically closed field $K$ of characteristic zero. Consider the coordinate ring $K[\mathrm{End}(V\otimes W)]$ of the affine space ...
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### Moduli space of nilpotent Lie algebras

Fix a nilpotent Lie algebra $L$ over some char 0 field $k$ which is naturally graded, i. e. isomorphic to graded algebra $\bar L$ associated to lower central filtration. I'm interested in some ...
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### Complexifed Gauge action on determinant line bundle and change of metric

Within the GIT setup for Hermitian-Einstein connection, Donaldson exhibited a holomorphic line bundle over $\mathcal{A}$ the space of unitary connections such that its cutvature equals $−2πi\Omega$ ...
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### Alternatives to the ring of invariants depicting the orbit closures?

Let $G$ be an affine algebraic group over $\mathbb{C}$ and $V=\textrm{Spec}(A)$ an affine $G$-variety. Assume that $G$ is reductive. My understanding is that a central object in studying the action of ...
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### Reference on reductive group acting on quotient algebra

In unpublished notes by Yi Hu (which appear to be no longer online), I found the following: Corollary 2.4.5. Let the characteristic of $k$ is zero. Assume that a reductive group $G$ acts rationally ...
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### A duality result for Coxeter groups

Short version: if $G$ is a Coxeter group and $H \subset G$ is a parabolic subgroup, both acting on a space $V$, is it true that the invariant-coinvariant algebra $(S(V)_G)^H$ has a natural bilinear ...
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### un-ordered distinct $n$-tuples of points on $\mathbb P^1$

I want to study the space $Y$ of all un-ordered $d$-tuples of points on $\mathbb P^1$. By considering the space $V_d$ of homogeneous polynomials of degree $d$ in two variables, one may identify $Y$ ...
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### Orbit of a transverse manifold under the action of an algebraic group

Consider an algebraic group acting on a affine manifold. Suppose that S is an affine submanifold transverse to the action. Are there some conditions on the group, the action or S such that the orbit ...
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### Injectivity of a standard map in quiver representation

Let $X$ be a smooth projective variety, and assume its divisor class group is finite and free. Let $E_1,E_2,\ldots,E_n$ be line bundles on $X$. Define $L_k=E_1+\ldots E_k$, and let $Q$ be the ...
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### Standard Monomial basis for other types

For the algebraic group $SL_n$ (type $A_{n-1}$) and for a dominant weight $\lambda$ the standard monomials are indexed by the semi-standard young tableaux of shape $\lambda$ and they form a basis for ...
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### Is the conjugation action linearizable?

Let $G$ be a reductive algebraic group over some algebraically closed field $k$. Recall that, given an algebraic variety $X$ with an action of $G$, it is said that this action is linearizable if there ...
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### Interactions (functors) between equivariant sheaves for different groups?

Let $G$ be a finite group and $k$ a field (alg. closed char 0 for simplicity). To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. It is ...
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### Is $\widetilde M_{0, n}$ a Mori Dream space?

I'm reading on $\overline M_{0, n}$ and $\widetilde M_{0, n}$. I know that $\overline M_{0, n}$ is a Mori Dream space for $n \leq 6$ and not a Mori Dream space for $n \geq 13$. Is there a similar ...
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### Cohen-Macaulay rings in GIT

I'm starting to learn about Cohen-Macaulay rings in my commutative Algebra course, and my teacher mentioned its use in Geometric Invariant Theory. Having some basic knowledge on the subject and ...
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### Projective and Quasiprojective quotients

Let $G$ be a finite group acting on a projective variety $X$. Then $G$ also acts on $X-X^G$, where $X^G$ is the fixed locus. The GIT quotient varieties $X/G$ and $(X-X^G)/G$ are projective and quasi-...
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### Categorical quotients for quasi-affine varieties

Let $X$ be an affine variety and let $G$ be a reductive algebraic group acting on $X$. Let $U \subset X$ be a $G$-invariant open set. Under what hypothesis there exists a categorical quotient of $U$ ...