# Tagged Questions

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### stability notion of nets of quadrics

A net of quadrics in $\mathbb{P}^n$ is a plane in $\mathbb{P}^N$, where $N=\frac{n(n+3)}{2}$. So the space of net of quadrics is the Grassmannian $Gr(3,N+1)$. The group $SL_{n+1}(\mathbb{C})$ acts on ...
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### (semi)stability of a net of quadrics and a certain criterion

Let $V:=H^0(\mathbb{P}^4,\mathcal{O}(1))$, $W:=H^0(\mathbb{P}^4,\mathcal{O}(2))$ and $Q_i(i=1,2,3)$ be quadric hypersurfaces in $\mathbb{P}^4$. To a net of quadrics $\Lambda=(Q_1,Q_2,Q_3)$, we ...
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### Research of a reference about $G$-linearizations of line bundles on quasi-projective schemes

I am looking for some references for the following statement: Let $G$ be a linearly reductive algebraic group acting on a quasi-projective scheme $X$, over an algebraically closed field $K$. Let $L$ ...
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### Quotients by the additive group $\mathbb G_a$

Geometric invariant theory doesn't work so well for non-reductive groups, since invariant rings are not generally finitely generated. However, in many cases the action of a non-reductive group has a ...
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### How does one show that slope stability of a vector bundle is an open condition with respect to the polarisation?

I would like a source for the following result, which I expect to be true (probably well known): Let $X$ be a complex projective variety, $L$ an ample line bundle and $E$ a slope stable vector bundle ...
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### Smoothness of fix point components of finite group action on smooth variety

Let $X$ be a smooth complex algebraic variety, and $\varphi: \Gamma\curvearrowright X$ an action (by automorphisms) of a finite group $\Gamma$ on $X$. Can we say that each irreducible component of ...
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### A quotient stack question

Let $X$ be a proper Deligne-Mumford stack, whose normalization, $X'$, is a global quotient stack (that is, a stack of the form [W/GL_n],where W is an algebraic space) with a projective scheme as a ...
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### Why people usually consider reductive groups in GIT?

Where do people essentially use the reductive groups in the theory of GIT? Or how does reductive groups simplify the constructions in GIT? I found that the property of completely reducible of ...
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### Stability conditions for coherent sheaves and GIT

I am learning stability conditions for derived categories of coherent sheaves, following Bridgeland, and coming from a vector bundles background. $\mu$-stability for vector bundles has a clear GIT ...
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### invariants of plane quartics

Does anybody know a good reference where the invariants for plane quartic curves are developed?
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### Action of k* on a variety induces grading?

Let $V$ be a $\Bbbk$-variety such that $\Bbbk^\times$ (as an algebraic group) acts algebraically on $V$. Given any $f\in\Bbbk[V]$, let us call $f$ homogeneous of degree $d$ if for all $v\in V$ and all ...
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### affinization of T^*CP^n

Is there an elementary description of the affinization of the algebraic cotangent bundle of $CP^n$? I know that it can be described as some sort coadjoint orbits, but I am interested in a translation ...
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### When the affine quotient is faithfully flat?

It may be easy for the expert. Consider the map from $n$ by $m$ matrices (over $\mathbb{C}$ )to the $n$ by $n$ symmetric matrices $\phi\colon A\mapsto A A^T$. My question is when this map is ...
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### Quotients of group actions on varieties

Let $Y$ be an affine algebraic variety over $\mathbb{C}$ and let $X$ be its closed subvariety. Let $G$ be a reductive group acting on $Y$ and let $H$ be a reductive subgroup of $G$ preserving $X$ such ...
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### Level n-structure as defined by Mumford in GIT

In Mumford's GIT, the definition of level $n$ structure ($n \geq 2)$ is $2g$ sections $\{\sigma_1, \dots, \sigma_{2g}\} : S \rightarrow A$ such that two conditions hold: (i) For geometric points the ...
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### Hilbert polynomial of an abelian scheme

This is coming out of Mumford's GIT, section 7.2, page 131. $A/S$ an abelian scheme of dimension $g$ with polarization $\bar{\omega}$ of degree $d^2$. Then $\pi_*(L^\Delta(\bar{\omega})^3)$ is ...
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### Verifying claims in the proof of the Rigidity Lemma (Mumford, GIT)

In Chapter 6 of Mumford's Geometric invariant theory, during the proof of the rigidity lemma, there are two statements I'm not sure how to verify. The general setup is: $p : X \rightarrow S$ is ...
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### References to SGA 8 and descent theory

In Geometric Invariant Theory, by Mumford, Fogarty, and Kirwan, if there is a mention of descent theory, it almost always comes along with a reference to SGA 8, Theorem 5.2 (see the end of the proof ...
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### Configuration space of flags

Let $U\subset \mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be the Zariksi open set of ordered quadruple of distinct points in the projective line. The quotient of $U$ by the ...
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### A simple problem on commutative algebra related to G.I.T

Let $G$ be a geometrically reductive algebraic group over an algebraically closed field $k$. Let $X$ be an affine variety over $k$ on which $G$ acts regularly. Then $G$ acts on the coordinate ring $A$ ...
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### Why is the degree:rank ratio of a vector bundle called its “slope”?

Whenever one studies moduli spaces of vector bundles on curves, one of the first things to be introduced is the "slope" of a vector bundle, i.e., its degree:rank ratio. Is there a nice (preferably ...
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### Is an affine “G-variety” with reductive stabilizers a toric variety?

Let $X=Spec(A)$ be a reduced normal affine scheme over an algebraically closed field $k$ of characteristic $0$, with an action of a connected reductive group $G$. Suppose $x\in X$ is a ...
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### If Spec(A) has a G-fixed point and a dense G-orbit, is Spec(A) a cone?

[Edited to include a dense orbit] Let $X=Spec(A)$ be a normal affine scheme over an algebraically closed field $k$, with an action of a linearly reductive group $G$. Suppose $x\in X$ is a ...
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### Are GIT's good categorical quotients just locally ringed space coequalizers?

Introduction: The definition of "good categorical quotient" in geometric invariant theory (given below) seems fairly ad hoc to me, except that it looks very similar to the coequalizer of the action in ...
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### geometric quotient

Let $S$ be a base scheme. Let $X$ be a scheme over $S$ and let $G$ be a group scheme over $S$ acting on $X$ via $\sigma: G \times_S X \to X$. Suppose that we have a scheme $Y$ over $S$ together with ...
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### The canonical divisor of the Hilbert scheme $Hilb^n P^2$?

Hey everyone, I was wondering if anyone knows what the canonical divisor of the Hilbert scheme $Hilb^n P^2$ is --$Hilb^n P^2$ is the Hilbert scheme of degree-n zero dimensional subschemes of the ...
Geometric invariant theory works well when the algebraic group $G$ acting on a variety is reductive. There has been recent work by Doran and Kirwan here and here to find a canonical method of ...
One way to define toric varieties is as quotients of affine $n-$space by the action of some torus. However, this is not strictly true as we need to throw away "bad points" which ruin this ...