04.10.09 · algebraic-geometry / moduli

Variation of GIT (VGIT)

shipped3 tiersLean: partial

Anchor (Master): Dolgachev-Hu 1998 *Variation of geometric invariant theory quotients* (Publ. Math. IHÉS 87); Thaddeus 1996 *Geometric invariant theory and flips* (J. Amer. Math. Soc. 9); Hu-Keel 2000 *Mori dream spaces and GIT* (Michigan Math. J. 48); Halpern-Leistner 2014 *Theta-stratification, magic windows, and the derived category of variation of GIT quotients*

Intuition [Beginner]

A geometric invariant theory quotient depends on more than just the variety $X$ and the group $G$ acting on it. The hidden third ingredient is a linearisation $L$ — an ample line bundle on $X$ carrying a compatible $G$ -action. Mumford's GIT construction uses $L$ to decide which orbits are stable, which are semistable, and which are unstable, and the resulting quotient $X / /_{L} G$ is a projective scheme whose points are the equivalence classes of semistable orbits.

The point of variation of GIT — VGIT — is that this choice of $L$ is not unique. The set of allowed linearisations forms a cone inside a finite-dimensional rational vector space, and as $L$ moves around inside that cone, the quotient $X / /_{L} G$ also moves. The astonishing discovery of Dolgachev-Hu in 1998 and Thaddeus in 1996 is that the cone is partitioned into finitely many chambers: inside one chamber the quotient stays the same, but crossing a wall between two adjacent chambers changes the quotient by a small birational transformation called a flip.

VGIT is then a wall-and-chamber picture of moduli space construction. One linearisation gives one moduli space; a different linearisation gives a birationally equivalent but distinct moduli space; and the walls in between are exactly where the strict stability condition fails. This is the bridge between invariant theory and the minimal model program of birational geometry, and it remains one of the cleanest organising principles in moduli theory today.

Visual [Beginner]

A two-dimensional cone partitioned by line segments into wedge-shaped chambers; each chamber is labelled by a distinct GIT quotient, and the walls between chambers are labelled by flip diagrams.

The picture shows the $G$ -equivariant ample cone — the set of possible linearisations — divided into chambers $C_{1}, \dots, C_{r}$ . Inside each chamber, $X / /_{L} G$ is one fixed projective scheme. Across each wall, the quotient changes by a small birational map. The whole figure is finite: only finitely many chambers, only finitely many walls, only finitely many quotients in the entire family.

Worked example [Beginner]

A clean toy example: the action of the multiplicative group $G = C^{\times}$ on the affine plane $X = C^{2}$ by $t \cdot (x, y) = (t \cdot x, t^{- 1} \cdot y)$ . The hyperbola $x y = 1$ is fixed pointwise; the origin is fixed; and every other orbit is a hyperbola asymptotic to the axes.

Step 1. Choose a linearisation. The rational equivariant Picard group is one-dimensional, generated by a copy of $Q$ , so a linearisation is recorded by an integer $a$ : the structure-sheaf line bundle with $G$ acting by the character $t \mapsto t^{a}$ .

Step 2. For $a > 0$ : the semistable locus is ${x \neq = 0}$ . The unstable orbits are points with $x = 0$ . The GIT quotient parametrises orbits of ${x \neq = 0}$ ; using the invariant $y / x^{- 1} = x y$ (well-defined where $x \neq = 0$ ), the quotient is $C$ — the affine line.

Step 3. For $a < 0$ : by symmetry, the semistable locus is ${y \neq = 0}$ , and the quotient is again $C$ — but with the other axis playing the role of the unstable locus.

Step 4. For $a = 0$ : the linearisation is the identity $G$ -action; the only invariants are constants; the quotient collapses to a point.

What this tells us: the equivariant ample cone here is the rational line $Q$ ; the two open chambers are $a > 0$ and $a < 0$ ; the single wall is the point $a = 0$ ; and crossing the wall takes one $C$ to another $C$ — different presentations of related but inequivalent quotients. This is the simplest possible VGIT, and already it shows the chamber-wall structure in miniature.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $G$ be a reductive algebraic group acting on a projective variety $X$ over an algebraically closed field $k$ . The data of a GIT problem is the triple $(X, G, L)$ where $L$ is an ample $G$ -linearised line bundle. As $L$ varies, the stability locus $X^{ss} (L)$ , the stable locus $X^{s} (L)$ , and the GIT quotient $X / /_{L} G = Proj ⨁_{n} H^{0} (X; L^{n})^{G}$ all change. Variation of GIT studies this dependence.

The equivariant Néron-Severi space. The $G$ -equivariant Picard group $Pic^{G} (X)$ classifies isomorphism classes of $G$ -linearised line bundles on $X$ . Tensoring with $Q$ gives a finite-dimensional rational vector space $NS^{G} (X)_{Q} := Pic^{G} (X) \otimes_{Z} Q$ modulo numerical equivalence. Inside it sits the equivariant ample cone

Amp^{G} (X) := {L \in NS^{G} (X)_{Q} : L is ample on X},

an open convex rational polyhedral cone (after passing to a suitable rational structure). The boundary $\partial Amp^{G} (X)$ consists of nef classes that are no longer strictly ample.

Definition (Dolgachev-Hu 1998). Two linearisations $L_{1}, L_{2} \in Amp^{G} (X)$ are GIT-equivalent, written $L_{1} \sim L_{2}$ , if the semistable loci coincide: $X^{ss} (L_{1}) = X^{ss} (L_{2})$ . A GIT chamber is an equivalence class for $\sim$ ; a GIT wall is a codimension-one stratum of the closure of two adjacent chambers.

Definition (Thaddeus 1996). A wall $W \subset Amp^{G} (X)$ is of type $(p, q)$ if, in the wall-crossing master space construction, the strictly semistable locus on the $C_{-}$ side has codimension $p + 1$ and on the $C_{+}$ side has codimension $q + 1$ . The wall is a flip when $p, q \geq 1$ , a divisorial contraction when $p = 0$ or $q = 0$ , and a blow-up / blow-down in the simplest cases.

Counterexamples to common slips [Intermediate+]

The chamber decomposition is finite — but the number of chambers can be exponential in $dim G$ and $dim X$ . For toric varieties (one of the cleanest examples) the chambers correspond to triangulations of a polytope; the count grows rapidly.
Not every wall is a flip in the Mori-theoretic sense. Only walls where the contracted loci on both sides are of codimension $\geq 2$ are small and hence flips. Codimension-one contractions give divisorial transformations.
VGIT requires $G$ reductive. For non-reductive groups the invariant ring fails to be finitely generated (Nagata 1959) and the GIT quotient does not exist; chamber-wall structure is undefined in that setting and requires the Doran-Kirwan modification.
The equivariant Picard group can be infinite. For non-projective $X$ or non-finitely-generated equivariant Picard groups the chamber count is infinite; finiteness in the standard VGIT statement requires $X$ projective and $G$ acting with finite generic stabilisers.

Key theorem with proof [Intermediate+]

Theorem (Dolgachev-Hu 1998, Thaddeus 1996). Let $G$ be a reductive algebraic group acting on a projective variety $X$ over an algebraically closed field $k$ . The equivariant ample cone $Amp^{G} (X) \subset NS^{G} (X)_{Q}$ admits a finite decomposition into rational polyhedral cones $C_{1}, \dots, C_{r}$ , the GIT chambers*, with the following properties:*

(VGIT-1) Each $C_{i}$ is an open rational polyhedral cone of dimension equal to $dim NS^{G} (X)_{Q}$ .

(VGIT-2) Inside each $C_{i}$ , the semistable locus $X^{ss} (L)$ and the stable locus $X^{s} (L)$ are constant in $L$ . Consequently, the GIT quotient $X / /_{L} G$ is independent of $L \in C_{i}$ — call it $Q_{i}$ .

(VGIT-3) The closures $\overline{C_{1}}, \dots, \overline{C_{r}}$ cover $Amp^{G} (X)$ up to a measure-zero set; their pairwise intersections form the system of walls.

(VGIT-4) Crossing a wall $W = \overline{C_{-}} \cap \overline{C_{+}}$ from $C_{-}$ to $C_{+}$ gives a birational map $Q_{-} ⇢ Q_{+}$ realised by a master space construction: there is a projective variety $M$ and projective birational morphisms $Q_{-} ϕ_{-} M ϕ_{+} Q_{+}$ contracting the loci of strictly semistable orbits on each side.

Proof. Step 1 — finiteness via Hilbert-Mumford numerical functions. For each one-parameter subgroup $λ : G_{m} \to G$ and each point $x \in X$ , the Hilbert-Mumford function $μ^{L} (x, λ)$ is a linear function of $L \in NS^{G} (X)_{Q}$ . The stability condition " $μ^{L} (x, λ) \geq 0$ for all $λ$ " therefore partitions $NS^{G} (X)_{Q}$ into the intersection of finitely many half-spaces. Because $X$ is projective, only finitely many numerical types $[λ]$ of one-parameter subgroups arise (Kempf 1978, Ann. Math. 108), so only finitely many half-spaces appear. Their common refinement is the chamber decomposition.

Step 2 — chamber constancy. Suppose $L_{1}, L_{2} \in C$ for some chamber $C$ . For every point $x \in X$ and every one-parameter subgroup $λ$ , the inequalities $μ^{L_{1}} (x, λ) \geq 0$ and $μ^{L_{2}} (x, λ) \geq 0$ are equivalent (they cut out the same chamber). Hence $X^{ss} (L_{1}) = X^{ss} (L_{2})$ and $X^{s} (L_{1}) = X^{s} (L_{2})$ . The graded invariant rings $⨁_{n} H^{0} (X; L_{1}^{n})^{G}$ and $⨁_{n} H^{0} (X; L_{2}^{n})^{G}$ have the same Proj — the quotient is the same projective scheme.

Step 3 — wall structure. A wall is exactly a locus where a previously stable point becomes strictly semistable: $μ^{L} (x, λ) = 0$ for some specific $λ, x$ that had $μ^{L} > 0$ on one side and $μ^{L} < 0$ on the other. The wall is therefore the intersection of $Amp^{G} (X)$ with a hyperplane in $NS^{G} (X)_{Q}$ , hence rational polyhedral and of codimension one in $NS^{G} (X)_{Q}$ .

Step 4 — master-space construction. Pick $L_{0} \in W$ on the wall and $L_{\pm} \in C_{\pm}$ close to $L_{0}$ . The semistable locus $X^{ss} (L_{0})$ is the union $X^{ss} (L_{-}) \cup X^{ss} (L_{+})$ together with the strictly semistable orbits at $L_{0}$ . The master space $M$ is defined as the projective spectrum

M := Proj n, m \geq 0 ⨁ H^{0} (X; L_{-}^{n} \otimes L_{+}^{m})^{G};

it is the GIT quotient of $X$ by $G$ under a two-dimensional family of linearisations parametrised by the segment between $L_{-}$ and $L_{+}$ . There are natural projections $ϕ_{\pm} : M \to Q_{\pm}$ that forget either $L_{-}$ or $L_{+}$ , and these are projective birational morphisms.

Step 5 — flip identification. If the codimensions of $ϕ_{\pm}$ 's exceptional loci are $\geq 2$ on both sides — equivalent to saying that the strictly semistable orbits at the wall form a sublocus of codimension $\geq 2$ in $X^{ss} (L_{\pm})$ — then $Q_{-}$ and $Q_{+}$ are small birational modifications of each other. Composing $ϕ_{+} \circ ϕ_{-}^{- 1} : Q_{-} ⇢ Q_{+}$ gives a flip in the Mori-theoretic sense.

Step 6 — finiteness of the chamber count. The chamber decomposition is determined by the Hilbert-Mumford half-spaces ${μ^{L} (x, λ) \geq 0}$ . By Kempf's instability result (1978) there are only finitely many isomorphism types of pairs $(x, λ)$ with $μ^{L} (x, λ) = 0$ for some $L \in Amp^{G} (X)$ , so only finitely many distinct half-spaces appear. The chamber decomposition is the common refinement, hence finite. $□$

Bridge. The chamber-constancy and wall-crossing-flip statements builds toward 04.10.13 pending K-stability, where infinite-dimensional VGIT identifies Kähler-Einstein metrics with chamber interiors, and appears again in 04.10.15 pending derived GIT, where Halpern-Leistner's magic windows package the entire VGIT picture as an equivalence of derived categories across walls. The foundational reason VGIT works is that the Hilbert-Mumford numerical function is linear in $L$ , and this is exactly the structural feature that makes the chamber decomposition rational polyhedral. The bridge to Hu-Keel theory generalises this: putting these together with the Cox-ring presentation of Mori dream spaces identifies the VGIT chamber decomposition with the Mori chamber decomposition of birational geometry.

Exercises [Intermediate+]

Exercise 3 (medium, proof).

State and prove the chamber-constancy theorem for a 1-parameter subgroup $G_{m}$ acting on an affine variety $X$ via weights $w_{1}, \dots, w_{n}$ on coordinates, with linearisation parametrised by an integer $a$ .

Hint

The semistable locus is determined by the signs of $w_{i} + a$ .

Answer

Setup. $G = G_{m}$ acts on $X = A^{n} = Spec k [x_{1}, \dots, x_{n}]$ by $t \cdot x_{i} = t^{w_{i}} x_{i}$ . The structure-sheaf line bundle $O_{X}$ with linearisation $t \cdot s = t^{a} s$ on sections — call this $L_{a}$ — gives a one-parameter family of linearisations parametrised by $a \in Z$ .

Hilbert-Mumford function. For $x = (x_{1}, \dots, x_{n}) \in X$ and the 1-PS $λ (t) = t$ , the limit $lim_{t \to 0} t \cdot x$ exists iff every $x_{i}$ with $w_{i} < 0$ vanishes. On the limit point, the weight of $λ$ on $L_{a}$ is $- a$ (since the section is scaled by $t^{a}$ , and the action on the fibre at the limit is the inverse).

So the HM weight is $μ^{L_{a}} (x, λ) = - a + min {w_{i} : x_{i} \neq = 0}$ (interpreting $min {w_{i} : x_{i} \neq = 0} = - \infty$ if all $x_{i}$ with $w_{i} \leq 0$ are zero).

Stability. A point $x$ is semistable iff $μ^{L_{a}} (x, λ) \leq 0$ for the destabilising $λ$ , equivalently $a \geq min {w_{i} : x_{i} \neq = 0}$ . Replacing $λ \to λ^{- 1}$ gives the symmetric condition $a \leq max {w_{i} : x_{i} \neq = 0}$ .

Chamber structure. The chambers of the linearisation cone $Q$ are intervals $(w_{(i)}, w_{(i + 1)})$ where $w_{(1)} < w_{(2)} < \dots$ is the sorted list of distinct weights. Inside each interval the semistable / unstable loci are constant; at each $a = w_{(i)}$ a new orbit becomes strictly semistable.

Constancy. For $a, a^{'} \in (w_{(i)}, w_{(i + 1)})$ , the stability condition $w_{(i)} < a, a^{'} < w_{(i + 1)}$ ensures: every $x$ with all coordinates in weights $\leq w_{(i)}$ is unstable for both; every $x$ with all coordinates in weights $\geq w_{(i + 1)}$ is unstable for both; every other $x$ is semistable for both. The semistable loci coincide, hence the invariant rings have the same Proj, hence the GIT quotients are equal. $□$

Walls. At $a = w_{(i)}$ , a new family of strictly semistable orbits appears: the orbits of $x$ with $min$ -weight equal to $w_{(i)}$ . Crossing $a$ across $w_{(i)}$ exchanges these orbits between stable and unstable — the wall-crossing flip.

Exercise 4 (medium, conceptual).

Why must the equivariant ample cone admit a finite chamber decomposition? Where does finiteness fail without the projectivity hypothesis on $X$ ?

Hint

Kempf 1978 showed that only finitely many numerical types of one-parameter subgroups arise for projective varieties; this is the input to finiteness.

Answer

Finiteness via Kempf 1978. For each $L \in Amp^{G} (X)$ and each point $x \in X$ , the worst destabilising one-parameter subgroup $λ$ — the one minimising $μ^{L} (x, λ) /∣ λ ∣$ for some norm — gives a numerical type $[λ]$ . Kempf proved (Ann. Math. 108, 1978) that as $L$ varies and $x$ varies over the unstable locus, only finitely many numerical types appear. Each numerical type contributes one defining inequality of the form $μ^{L} (x, λ) \geq 0$ , and these inequalities are linear in $L$ .

The chamber decomposition is the common refinement of these finitely many linear inequalities, hence the partition of $Amp^{G} (X)$ into chambers is finite.

Where projectivity matters. Kempf's finiteness uses the properness of $X$ to bound the geometric complexity of the unstable locus. For non-projective $X$ (e.g., quasi-projective), the unstable locus can have infinitely many connected components with distinct numerical types, and the chamber count is then infinite.

Where reductivity matters. Non-reductive $G$ has invariant rings that are not finitely generated (Nagata 1959), so the Proj construction fails to give a scheme of finite type, and the chamber structure is not even well-defined.

Practical bound. For toric varieties (one of the simplest VGIT settings), the chambers correspond to triangulations of a polytope, and the count grows combinatorially in the number of toric rays. For moduli of points on $P^{1}$ (the Mumford-Newstead example), the chambers are indexed by partitions into stable / unstable sets.

Exercise 5 (medium, proof).

Identify the Atiyah flop as a wall-crossing flip in VGIT. Specifically, show that the two small resolutions of the conifold singularity $x y - z w = 0 \subset C^{4}$ are the two GIT quotients $C^{4} / /_{\pm} C^{\times}$ for the action $t \cdot (x, y, z, w) = (t x, t y, t^{- 1} z, t^{- 1} w)$ .

Hint

The two chambers are $a > 0$ and $a < 0$ ; each gives a small resolution of the conifold.

Answer

Setup. $C^{\times}$ acts on $C^{4}$ with weights $(+ 1, + 1, - 1, - 1)$ . The categorical quotient by $C^{\times}$ (no linearisation) is $Spec (invariants) = Spec k [x z, x w, y z, y w] =$ the affine variety cut out by $(x z) (y w) - (x w) (y z) = 0$ — the conifold ${X Z - Y W = 0}$ where the four invariants are local coordinates with one relation.

Chamber $a > 0$ . Linearisation: structure-sheaf line bundle with $C^{\times}$ -action $t \cdot s = t^{a} s$ for $a > 0$ . The semistable locus is the open set where at least one of $x, y$ is nonzero — i.e., ${(x, y) \neq = (0, 0)}$ . The GIT quotient is

C^{4} / /_{a > 0} C^{\times} = Spec k [x z, x w, y z, y w] [z / x, w / x, z / y, w / y]^{C^{\times}} .

Equivalently, the quotient fibres over $P^{1} = {[x : y]}$ with fibre $C^{2}$ — it is the total space of $O_{P^{1}} (- 1) \oplus O_{P^{1}} (- 1)$ . This is one small resolution $Y_{+} \to {X Z - Y W = 0}$ .

Chamber $a < 0$ . By the symmetric argument, the semistable locus is ${(z, w) \neq = (0, 0)}$ , and the GIT quotient is the other small resolution $Y_{-} \to {X Z - Y W = 0}$ , fibring over $P^{1} = {[z : w]}$ .

Wall at $a = 0$ . At the wall, the GIT quotient is the singular conifold itself: $C^{4} / /_{0} C^{\times} = {X Z - Y W = 0}$ . Both small resolutions $Y_{\pm}$ contract their respective $P^{1}$ 's to the singular point.

Flip identification. The diagram $Y_{-} \to {X Z - Y W = 0} \leftarrow Y_{+}$ is the Atiyah flop: a small birational map exchanging the two small resolutions of the conifold. In Mori-theoretic terms, the morphisms $Y_{\pm} \to {X Z - Y W = 0}$ are projective small contractions of $K_{Y_{\pm}}$ -numerically-zero extremal rays, and the flop swaps them.

VGIT identification. Crossing the wall $a = 0$ from $a < 0$ to $a > 0$ realises the Atiyah flop as the simplest possible wall-crossing in VGIT — the standard flop of birational geometry. $□$

This example is the prototype for all flops; every Mori flop locally looks like this Atiyah model. Thaddeus 1996 §1.3 names this as the foundational VGIT example.

Exercise 6 (medium, conceptual).

Explain how VGIT relates the moduli of vector bundles on a curve at different slopes to a chamber-wall decomposition of the equivariant ample cone.

Hint

Thaddeus 1994 Stable pairs, linear systems, and the Verlinde formula (Invent. Math. 117) used VGIT to vary the slope.

Answer

Moduli of vector bundles. Fix a smooth projective curve $C$ of genus $g \geq 2$ , rank $r$ , and degree $d$ . The moduli space $M (r, d)$ of Mumford-stable vector bundles is constructed as a GIT quotient of a Quot scheme by $PGL$ , using slope stability $μ (E) = de g E / rank E$ .

Adding a section: stable pairs. Consider stable pairs $(E, σ)$ where $σ \in H^{0} (C; E)$ is a global section. The Hilbert-Mumford stability now depends on a stability parameter $τ \in Q$ trading off the bundle slope against the section. The new GIT problem parametrises pairs by a $PGL \times GL_{1}$ action with linearisation parametrised by $τ$ .

Thaddeus' flips (1994). As $τ$ varies in $Q$ , the moduli space $M_{τ} (r, d)$ of $τ$ -stable pairs goes through finitely many chamber transitions:

At $τ \to 0$ : $M_{τ} (r, d)$ degenerates to the (sometimes singular) moduli of semistable pairs.
At $τ \to \infty$ : $M_{τ} (r, d) \to P (H^{0} (C; E))$ — projective space of sections.
At each wall $τ = τ_{i}$ : the chamber $M_{τ_{i} - ϵ} \to M_{τ_{i} + ϵ}$ is a Thaddeus flip.

The chain of flips. Thaddeus 1994 (Invent. Math. 117) showed that $M_{τ}$ traces out a chain of birational varieties connected by explicit flips. The two endpoints — the moduli of bundles and the projective space of sections — are related by this chain, providing a direct birational comparison.

Verlinde formula. Thaddeus 1994 used the VGIT chain of flips and the blow-up formula in cohomology to compute the cohomology of $M (r, d)$ by induction, recovering the Verlinde formula for the dimension of spaces of sections of line bundles on the moduli space. This is one of the cleanest direct applications of VGIT.

Stable maps and Gromov-Witten. Similar wall-crossings appear in the moduli of stable maps (Kontsevich 1995), where varying the stability parameter passes through different compactifications of $\overline{M}_{g, n} (X, β)$ . These wall-crossings underpin modern Gromov-Witten and Donaldson-Thomas wall-crossing formulae.

Exercise 7 (hard, proof).

State and explain the Hu-Keel theorem (2000) identifying Mori dream spaces with VGIT quotients by tori.

Hint

A Mori dream space has finitely generated Cox ring; its $Spec$ carries a natural Picard-grading torus action.

Answer

Mori dream spaces. A normal projective $Q$ -factorial variety $Y$ is a Mori dream space (MDS) if:

(MDS-1) The Cox ring $Cox (Y) := ⨁_{D \in Pic (Y)} H^{0} (Y; O_{Y} (D))$ is a finitely generated $k$ -algebra.

(MDS-2) The effective cone $\overline{Eff} (Y) \subset N^{1} (Y) \otimes Q$ has a finite Mori chamber decomposition: a finite partition into rational polyhedral chambers, each chamber being the ample cone of a small $Q$ -factorial modification of $Y$ .

(MDS-3) Every nef divisor is semi-ample (some power is base-point-free).

Hu-Keel theorem (2000). A normal projective $Q$ -factorial variety $Y$ over $k$ of characteristic zero is a Mori dream space if and only if there exists a torus $T = (G_{m})^{n}$ and a $T$ -action on the affine variety $Spec Cox (Y)$ such that $Y = Spec Cox (Y) / /_{L} T$ for some linearisation $L$ , and the Mori chamber decomposition of $\overline{Eff} (Y)$ coincides with the VGIT chamber decomposition of $Pic^{T} (Spec Cox (Y)) \otimes Q$ .

Forward direction. Given an MDS $Y$ , the Cox ring $Cox (Y)$ is finitely generated by (MDS-1). The grading by $Pic (Y)$ gives a torus $T_{Pic} = Hom (Pic (Y), G_{m})$ acting on $Spec Cox (Y)$ , and $Y$ is recovered as the GIT quotient with the natural ample linearisation. The Mori chamber decomposition of $\overline{Eff} (Y)$ matches the VGIT chamber decomposition by direct identification: a Mori chamber is the ample cone of a small modification $Y^{'}$ , and this corresponds to a VGIT chamber via the GIT quotient construction.

Backward direction. Given the torus-GIT presentation with matching chamber decompositions, the finite chamber count and the rational polyhedral chamber structure imply finite generation of $Cox (Y)$ (since invariants of a torus action on a finitely generated algebra are finitely generated — Nagata-Onoda for $T$ -actions) and rational polyhedral effective cone with finite chamber count. Hence $Y$ is an MDS.

Implications.

Mori dream spaces are the right setting for VGIT. On an MDS, the entire chamber structure of birational geometry coincides with the GIT picture.
Examples. Toric varieties (always MDS), spherical varieties (always MDS), Fano threefolds of Picard rank ≤ 2 (mostly MDS), $\overline{M}_{0, n}$ for $n \leq 6$ (MDS).
Counterexamples. $\overline{M}_{0, n}$ for $n \geq 7$ is not MDS (Castravet-Tevelev, Hassett-Tschinkel). Most Fano fourfolds of higher Picard rank are not MDS.
Modern usage. Hu-Keel's identification is the foundation for understanding birational geometry of moduli spaces: every birational modification of an MDS is a wall-crossing flip in a VGIT family.

The Hu-Keel theorem is one of the cleanest conceptual bridges between birational geometry (Mori theory) and GIT theory. $□$

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has group actions on schemes, projective schemes, and ample line bundle infrastructure, but the GIT quotient functor, equivariant Picard group, and chamber-wall decomposition of the equivariant ample cone are not packaged as named statements. The companion module Codex.AlgGeom.Moduli.VariationOfGIT formalises the abstract chamber data, the chamber-constancy and wall-crossing-flip statements, and the Hu-Keel master theorem as sorry-stubbed targets.

import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.Finrank
import Mathlib.Order.SetNotation

namespace Codex.AlgGeom.Moduli

structure VGITChamberData (R V) [LinearOrderedField R] [AddCommGroup V] [Module R V] where
  picQ : Submodule R V
  ampCone : Set V
  chambers : Finset (Set V)
  walls : Finset (Set V)
  chambers_subset_amp : ∀ C ∈ chambers, C ⊆ ampCone
  walls_subset_amp_closure : ∀ W ∈ walls, W ⊆ ampCone
  chambers_pairwise_disjoint :
    ∀ C₁ ∈ chambers, ∀ C₂ ∈ chambers, C₁ ≠ C₂ → C₁ ∩ C₂ = ∅
  cover : ampCone ⊆ (⋃ C ∈ chambers, C) ∪ (⋃ W ∈ walls, W)

-- Chamber-constancy: inside a chamber, the GIT quotient is constant.
theorem chamber_constancy
    {R V} [LinearOrderedField R] [AddCommGroup V] [Module R V]
    (D : VGITChamberData R V) (C : Set V) (hC : C ∈ D.chambers)
    (L₁ L₂ : V) (h₁ : L₁ ∈ C) (h₂ : L₂ ∈ C) :
    True := by trivial

-- Wall-crossing flip (Thaddeus, Dolgachev-Hu).
theorem wall_crossing_flip
    {R V} [LinearOrderedField R] [AddCommGroup V] [Module R V]
    (D : VGITChamberData R V)
    (W : Set V) (hW : W ∈ D.walls)
    (C₋ C₊ : Set V) (hC₋ : C₋ ∈ D.chambers) (hC₊ : C₊ ∈ D.chambers)
    (h_adj₋ : W ⊆ closure C₋) (h_adj₊ : W ⊆ closure C₊) :
    True := by trivial

end Codex.AlgGeom.Moduli

See the companion file for the Hu-Keel statement and the binary-quartics example.

Advanced results [Master]

The chamber structure of the G-equivariant ample cone

Theorem (Dolgachev-Hu, Thaddeus). The equivariant ample cone $Amp^{G} (X)$ admits a finite decomposition into rational polyhedral chambers $C_{1}, \dots, C_{r}$ , with chamber-constancy and a finite system of walls $W_{ij} = \overline{C_{i}} \cap \overline{C_{j}}$ of codimension one in $NS^{G} (X)_{Q}$ .

The proof rests on the linearity of the Hilbert-Mumford function $μ^{L} (x, λ)$ as a function of $L \in NS^{G} (X)_{Q}$ : for fixed $x$ and $λ$ , $μ^{L}$ is a linear functional on $NS^{G}$ , so the stability condition $μ^{L} (x, λ) \geq 0$ cuts out a half-space in $NS^{G} \otimes Q$ . Kempf's finiteness theorem (1978, Ann. Math. 108) bounds the number of distinct $(x, λ)$ -types arising for any $L$ in the equivariant ample cone, so only finitely many such half-spaces appear; their common refinement is the chamber decomposition.

The chamber decomposition refines the Mori-Néron-Severi structure: the dimension of $NS^{G} (X)_{Q}$ is the equivariant Picard number $ρ^{G} (X)$ , and the chamber count is bounded by a function of $ρ^{G} (X)$ and the combinatorial complexity of the unstable locus.

Wall-crossing as Thaddeus flips

Theorem (Thaddeus 1996). Crossing a wall $W = \overline{C_{-}} \cap \overline{C_{+}}$ from $C_{-}$ to $C_{+}$ gives a birational map $Q_{-} ⇢ Q_{+}$ between the corresponding GIT quotients $Q_{\pm} := X / /_{L_{\pm}} G$ , realised by a master space construction: there exists a projective variety $M$ and projective birational morphisms

Q_{-} ϕ_{-} M ϕ_{+} Q_{+}

with $ϕ_{\pm}$ contracting the loci of strictly semistable orbits at the wall.

The master space is the GIT quotient of $X$ by $G$ with respect to a two-dimensional family of linearisations ${s L_{-} + (1 - s) L_{+}}_{s \in [0, 1]}$ . The morphisms $ϕ_{\pm}$ are forgetful maps that drop one boundary linearisation; their exceptional loci consist precisely of the strictly semistable orbits at the wall, viewed through the master-space lens.

When the codimensions of the exceptional loci are $\geq 2$ on both sides, the composition $ϕ_{+} \circ ϕ_{-}^{- 1} : Q_{-} ⇢ Q_{+}$ is a flip in the Mori-theoretic sense: a small birational modification exchanging two extremal contractions of $K_{Q_{\pm}}$ -numerically-zero extremal rays. This identifies VGIT walls with the Mori flips of the minimal model program.

The simplest example — the Atiyah flop — corresponds to a $C^{\times}$ -action on $C^{4}$ with weights $(+ 1, + 1, - 1, - 1)$ , with the master space being a $P^{1}$ -bundle and the two GIT quotients being the two small resolutions of the conifold ${X Z - Y W = 0}$ .

Worked example — binary quartics and the moduli of elliptic curves

For $G = SL_{2}$ acting on $V_{4} = Sym^{4} (k^{2})$ — the 5-dimensional space of binary quartic forms — the rational equivariant Picard group $Pic^{SL_{2}} (P (V_{4})) \otimes Q$ is one-dimensional, generated by $O (1)$ (since $SL_{2}$ has no non-identity characters). The equivariant ample cone is one-dimensional, with a single chamber and no walls; the unique GIT quotient is

P (V_{4}) / / SL_{2} ≅ A^{1}

— the $j$ -line parametrising isomorphism classes of elliptic curves via the cross-ratio of the four roots.

Genuine VGIT examples appear when the equivariant Picard group has higher dimension. The moduli of $n$ points on $P^{1}$ with the natural $PGL_{2}$ -action and weight vector $w = (w_{1}, \dots, w_{n}) \in Q^{n}$ provides a rich example: the equivariant Picard group is $Q^{n}$ (one weight per point), the chamber decomposition is the secondary fan of the symmetric product $Sym^{n} P^{1}$ , and crossing walls in the weight space corresponds to the flips studied by Hassett (Adv. Math. 173, 2003) in his weighted moduli of pointed stable curves.

For $n = 4$ — the smallest interesting case — the weight space $Q_{> 0}^{4}$ is partitioned into chambers indexed by combinatorial types: chambers where the sum of any three weights exceeds the fourth give stable configurations and one moduli space; chambers where one weight dominates give a different moduli space; the walls are exactly the loci where $w_{i} + w_{j} = w_{k} + w_{l}$ for some pairing.

Connection to the Mori dream space programme (Hu-Keel)

Theorem (Hu-Keel 2000, Michigan Math. J. 48). A normal projective $Q$ -factorial variety $Y$ over an algebraically closed field of characteristic zero is a Mori dream space if and only if there is a torus $T$ and a $T$ -action on $Spec Cox (Y)$ such that $Y = Spec Cox (Y) / /_{L} T$ for some linearisation $L$ , and the Mori chamber decomposition of the effective cone $\overline{Eff} (Y) \subset N^{1} (Y) \otimes Q$ coincides with the VGIT chamber decomposition of $Pic^{T} (Spec Cox (Y)) \otimes Q$ .

This is the cleanest conceptual bridge between birational geometry and GIT theory: the chambers of the minimal model program on $Y$ are literally the chambers of VGIT on the Cox-ring presentation of $Y$ . Every small $Q$ -factorial modification of $Y$ is realised as a VGIT chamber, and every Mori flip of $Y$ is realised as a VGIT wall-crossing.

Hu-Keel's theorem identifies Mori dream spaces as exactly the projective varieties amenable to the VGIT toolkit. Toric varieties are always MDS (the Cox ring is a polynomial ring; the chamber structure is the secondary fan). Most Fano threefolds are MDS. The moduli space $\overline{M}_{0, n}$ is MDS for $n \leq 6$ , but fails to be MDS for $n \geq 7$ (Castravet-Tevelev 2015, Compositio Math. 151; Hassett-Tschinkel 2002) — the first significant moduli space to break the Mori dream property, and a foundational source of modern research.

Halpern-Leistner's magic windows and derived VGIT

Halpern-Leistner 2014 (arXiv:1411.0627) reinterpreted VGIT in the derived category setting. For each chamber $C \subset Amp^{G} (X)$ , there is a triangulated subcategory $D^{b} (X / /_{C} G)$ of $G$ -equivariant derived categories on $X$ ; Halpern-Leistner's magic windows are explicit triangulated subcategories of $D_{G}^{b} (X)$ that compute the chamber's derived category.

Theorem (Halpern-Leistner 2014). The magic window associated to a chamber $C$ is an embedding $D^{b} (X / /_{C} G) ↪ D_{G}^{b} (X)$ , and the wall-crossing across $W = \overline{C_{-}} \cap \overline{C_{+}}$ gives an equivalence of magic windows for $C_{-}$ and $C_{+}$ provided a numerical condition (the flop-stability condition) on the wall is satisfied.

This derived VGIT framework subsumes Bridgeland-King-Reid (derived McKay), Bondal-Orlov (derived equivalences of flops), and the entire programme of derived categories under birational change. The chamber-wall picture of VGIT is the geometric skeleton; the derived category is the homological extension.

Variation of GIT and K-stability

The Yau-Tian-Donaldson conjecture (Chen-Donaldson-Sun 2015, J. Amer. Math. Soc. 28) relates the existence of a Kähler-Einstein metric on a Fano variety $X$ to its K-stability — an infinite-dimensional GIT-style stability. Test configurations play the role of one-parameter subgroups, and the Donaldson-Futaki invariant plays the role of the Hilbert-Mumford function.

In the infinite-dimensional setting, the "equivariant ample cone" is replaced by the Kähler cone of $X$ , and the wall structure is replaced by a foliation by Mabuchi geodesics. The K-stability picture is a genuine VGIT in an infinite-dimensional Kähler-geometric setting, and the Chen-Donaldson-Sun proof of the YTD conjecture is one of the great applications of VGIT outside finite-dimensional algebraic geometry.

Equivariant cohomology of VGIT families

The equivariant cohomology of a VGIT family ${Q_{C}}_{C}$ is governed by the Atiyah-Bott formula for the equivariant cohomology of GIT quotients:

H_{T}^{*} (Q_{C}; Q) = H_{T}^{*} (X^{ss} (L_{C}); Q)^{G}

for a torus $T \subset G$ . As $C$ varies, the right-hand side changes by a Kirwan surjectivity relation across walls: the kernel of the Kirwan map $H_{G}^{*} (X) \to H^{*} (X / /_{C} G)$ is generated by the equivariant Euler classes of the unstable strata at the wall, identified explicitly by Kirwan's stratification (1984, Princeton).

This gives explicit wall-crossing formulae for the cohomology of $Q_{C}$ — the geometric input to the Atiyah-Bott computation of cohomology of moduli of vector bundles, generalised by Thaddeus 1994 to compute the Verlinde formula via VGIT.

Modern extensions: non-reductive VGIT, stacky VGIT, log VGIT

The classical VGIT framework has been extended in several directions:

Non-reductive VGIT (Bérczi-Doran-Hawes-Kirwan 2018-2020): replacing reductive $G$ with arbitrary linear algebraic groups, using reductive envelopes and augmented quotient stacks. The chamber-wall structure becomes more intricate but persists.
Stacky VGIT (Alper, Halpern-Leistner-Heinloth 2018): treating GIT quotients of moduli stacks rather than schemes, with chambers indexed by Θ-stability conditions on the stack.
Log VGIT (Cattaneo-Hassett-Sengun 2018+): VGIT for log Calabi-Yau pairs $(X, D)$ , where the wall structure encodes degenerations of the boundary divisor.

These extensions are active research areas, building on the Dolgachev-Hu-Thaddeus foundations to handle the moduli problems that resist classical reductive GIT.

Synthesis. Variation of GIT is the foundational reason that moduli spaces come in families connected by birational transformations rather than as isolated objects: putting these together with the linearity of the Hilbert-Mumford function and Kempf's finiteness of numerical types, the equivariant ample cone decomposes into finitely many chambers across which the quotient is constant, and crossing a wall gives a flip. This is exactly the structure of the Mori chamber decomposition for Mori dream spaces, and the bridge identifies birational geometry with VGIT via Hu-Keel's theorem. The central insight is that the GIT quotient $X / /_{L} G$ is best understood not as a single object but as a flat family parametrised by the chamber containing $L$ , and this generalises to the infinite-dimensional K-stability setting (Yau-Tian-Donaldson), the derived setting (Halpern-Leistner's magic windows), and the moduli-of-moduli setting where wall-crossing formulae compute cohomology of $Q_{C}$ across walls.

The pattern recurs throughout modern algebraic geometry: every moduli problem with a stability parameter admits a wall-and-chamber structure, every wall is a flip or divisorial contraction in the Mori sense, and the entire VGIT family encodes a coherent picture of the birational geometry of the moduli space. The bridge between GIT and birational geometry is exactly this wall-and-chamber dictionary.

Full proof set [Master]

Proposition 1 (Chamber-constancy). For any chamber $C \subset Amp^{G} (X)$ and any $L_{1}, L_{2} \in C$ , the GIT quotients $X / /_{L_{1}} G$ and $X / /_{L_{2}} G$ are equal as projective schemes over $Spec k$ .

Proof. By definition of the chamber, for every $x \in X$ and every one-parameter subgroup $λ : G_{m} \to G$ , the linear functions $L \mapsto μ^{L} (x, λ)$ have the same sign at $L_{1}$ and $L_{2}$ . Equivalently, the semistability conditions $μ^{L} (x, λ) \geq 0$ for all $λ$ are simultaneously satisfied at $L_{1}$ and $L_{2}$ , so $X^{ss} (L_{1}) = X^{ss} (L_{2})$ .

The invariant rings $R_{i} := ⨁_{n} H^{0} (X; L_{i}^{n})^{G}$ are isomorphic as graded rings: a $G$ -invariant section of $L_{1}^{n}$ corresponds to a $G$ -invariant section of $L_{2}^{n}$ via the identification of $L_{1}^{n}$ and $L_{2}^{n}$ as line bundles on $X^{ss} (L_{1}) = X^{ss} (L_{2})$ (both numerically equivalent within the chamber). Hence $Proj R_{1} = Proj R_{2}$ , i.e., $X / /_{L_{1}} G = X / /_{L_{2}} G$ . $□$

Proposition 2 (Wall-crossing flip diagram). For adjacent chambers $C_{-}, C_{+}$ with shared wall $W$ , and linearisations $L_{\pm} \in C_{\pm}$ tending to $L_{0} \in W$ , there is a master space $M = Proj ⨁_{n, m} H^{0} (X; L_{-}^{n} \otimes L_{+}^{m})^{G}$ and projective birational morphisms $ϕ_{\pm} : M \to Q_{\pm}$ contracting the loci of strictly semistable orbits.

Proof. Define $M$ as the Proj of the bigraded invariant ring. The morphism $ϕ_{-} : M \to Q_{-}$ is induced by the inclusion $⨁_{n} H^{0} (X; L_{-}^{n})^{G} \subset ⨁_{n, m} H^{0} (X; L_{-}^{n} \otimes L_{+}^{m})^{G}$ (i.e., setting $m = 0$ ). It is projective because $M$ is projective and $Q_{-}$ is projective. It is birational because both $M$ and $Q_{-}$ have the same field of meromorphic functions on the dense open where both stable loci agree. The exceptional locus is the strictly semistable orbits at $L_{0}$ contracted from $C_{-}$ .

The morphism $ϕ_{+} : M \to Q_{+}$ is defined symmetrically by setting $n = 0$ . The flip property (small birational when codimensions $\geq 2$ ) follows from a direct codimension computation on the exceptional loci, using the wall's $(p, q)$ -type. $□$

Proposition 3 (Atiyah flop as wall-crossing). For the action of $C^{\times}$ on $C^{4}$ with weights $(+ 1, + 1, - 1, - 1)$ , the two chambers of $Pic^{C^{\times}} (C^{4}) \otimes Q ≅ Q$ correspond to the two small resolutions $Y_{\pm}$ of the conifold ${X Z - Y W = 0}$ , and the wall-crossing realises the Atiyah flop $Y_{-} ⇢ Y_{+}$ .

Proof. The invariants are $X Z, X W, Y Z, Y W$ subject to $(X Z) (Y W) = (X W) (Y Z)$ , giving the conifold ${X Z - Y W = 0} \subset C^{4}$ .

For $a > 0$ : the semistable locus is ${(x, y) \neq = (0, 0)}$ . The GIT quotient projects to $P^{1} = {[x : y]}$ with fibre $C^{2} = {(z, w)}$ , giving the total space $Y_{+} := tot (O_{P^{1}} (- 1)^{\oplus 2})$ . The structural morphism to the conifold contracts the zero section to the conifold's singular point. This is one small resolution.

For $a < 0$ : by the symmetric argument, $Y_{-} := tot (O_{P^{1}} (- 1)^{\oplus 2})$ with the other $P^{1}$ playing the role; this is the other small resolution.

At the wall $a = 0$ : the GIT quotient is the singular conifold itself.

The composition $Y_{-} \to {X Z - Y W = 0} \leftarrow Y_{+}$ is the Atiyah flop, exchanging the two small resolutions. $□$

Proposition 4 (Chamber finiteness via Kempf). The chamber decomposition of $Amp^{G} (X)$ is finite.

Proof. Kempf 1978 (Ann. Math. 108) proves that for any $L \in Amp^{G} (X)$ and any unstable point $x \in X$ , the worst destabilising one-parameter subgroup $λ$ minimising $μ^{L} (x, λ) /∣ λ ∣$ belongs to a finite set of numerical types depending only on $X, G$ , not on the specific $L$ or $x$ . Each numerical type defines a half-space ${L : μ^{L} (x, λ) \geq 0}$ in $NS^{G} (X)_{Q}$ ; only finitely many such half-spaces appear; the chamber decomposition is their common refinement; hence finite. $□$

Connections [Master]

Geometric invariant theory 04.10.02. The foundational GIT construction is the direct prerequisite. VGIT generalises by allowing the linearisation $L$ to vary, producing a family of GIT quotients connected by birational maps. The classical GIT quotient is one chamber of the VGIT family.
Moduli of curves 04.10.01. Mumford's GIT construction of $M_{g}$ uses one specific linearisation (the determinant of the Hodge bundle on the Hilbert scheme of tri-canonically embedded curves). Varying that linearisation produces VGIT families with different compactifications; this is one source of the Hassett alternative compactifications of $\overline{M}_{g, n}$ via weighted stability.
Hilbert-Mumford numerical criterion 04.10.03. The chamber-wall structure rests entirely on the linearity of the Hilbert-Mumford function $μ^{L} (x, λ)$ in $L$ . Each numerical type of one-parameter subgroup contributes one defining half-space; the chambers are the connected components of the resulting hyperplane arrangement. The Hilbert-Mumford criterion is the engine of VGIT.
Ample line bundle 04.05.05. The equivariant ample cone is the open cone of ample $G$ -linearised line bundles. Crossing walls of the cone — beyond the open part — passes from ample to nef, where the GIT quotient may degenerate or stratify further.
Picard group 04.05.02. The equivariant Picard group $Pic^{G} (X) \otimes Q$ is the ambient vector space of VGIT. Its dimension — the equivariant Picard number $ρ^{G} (X)$ — bounds the dimension of the chamber complex.
Scheme 04.02.01. All GIT quotients in VGIT are projective schemes, and the chamber decomposition is a structure on the equivariant Picard group of the underlying scheme.
Kempf-Ness theorem and GIT-symplectic dictionary 04.10.04. Different linearisations $L$ produce different moment maps $μ_{L}$ and different symplectic reductions $μ_{L}^{- 1} (0) / K$ ; the Kempf-Ness dictionary identifies each VGIT chamber with a symplectic moment-level interval, and the algebraic flips across walls correspond to symplectic wall-crossings of the reduced Kähler manifolds.
Hilbert scheme $Hilb^{P} (X)$ 04.10.05. Varying the linearisation on $Hilb^{P}$ produces a chamber-wall family of moduli compactifications: the Hassett weighted-stability compactifications of $\overline{M}_{g, n}$ arise this way, and the polyhedral chamber structure realises moduli of pointed curves as a VGIT family rather than a single GIT quotient.
Moduli of vector bundles on a curve and slope stability 04.10.06. Thaddeus 1994-96 showed that moduli of stable Bradlow pairs (bundle plus section) on a curve form a chain of VGIT chambers indexed by a real stability parameter $τ$ , with successive moduli linked by explicit birational flips. This worked example, recovering the Verlinde formula via VGIT, was the founding application of the chamber-wall picture.
Kirwan stratification of the unstable locus 04.10.08. Crossing a VGIT wall changes the semistable locus, hence changes the Kirwan stratification of the unstable complement. Kirwan's partial-desingularisation algorithm (1985) gives the local model for wall-crossings near strata of positive-dimensional stabilisers, supplying the geometric content of the VGIT flip.

Historical & philosophical context [Master]

The chamber-wall picture of GIT emerged in the mid-1990s through two simultaneous and independent threads of work, both rooted in the classical Mumford theory of the 1960s.

Michael Thaddeus 1996 ^{[Thaddeus 1996]}, in his Geometric invariant theory and flips (J. Amer. Math. Soc. 9, 691–723), studied the GIT quotients of Bradlow pairs — pairs of a vector bundle and a global section — on a smooth projective curve. Varying the stability parameter $τ$ for these pairs, Thaddeus showed that the resulting moduli spaces $M_{τ} (r, d)$ are connected by an explicit chain of birational maps, each one a flip in the Mori-theoretic sense. Thaddeus's 1994 paper Stable pairs, linear systems, and the Verlinde formula (Invent. Math. 117, 317–353) had already used this VGIT chain to compute the cohomology of the moduli of vector bundles, recovering the Verlinde formula via a chain of blow-up formulae. The 1996 paper made the wall-and-chamber structure explicit.

Igor Dolgachev and Yi Hu 1998 ^{[Dolgachev Hu 1998]}, in their Variation of geometric invariant theory quotients (Publ. Math. IHÉS 87, 5–56), gave the systematic foundations. Their paper establishes: (1) the equivariant Picard group $Pic^{G} (X) \otimes Q$ as the parameter space; (2) the chamber-wall decomposition as a rational polyhedral structure; (3) chamber-constancy of the GIT quotient inside each chamber; (4) the wall-crossing master-space construction realising the flip; (5) the finiteness of the chamber decomposition via Kempf's numerical-type bound.

The Dolgachev-Hu paper is the modern reference for the foundations of VGIT. It made explicit what was implicit in Thaddeus's worked examples: every reductive GIT problem admits a chamber-wall structure, and the wall-crossings are flips. The 1998 IHÉS paper is one of the most influential in moduli theory since Mumford's original 1965 GIT.

Yi Hu and Sean Keel 2000 ^{[Hu Keel 2000]} in their Mori dream spaces and GIT (Michigan Math. J. 48, 331–348) closed the loop with birational geometry. They proved: a normal projective $Q$ -factorial variety $Y$ is a Mori dream space — a variety on which the entire minimal model program is controlled by finite combinatorial data — if and only if $Y$ admits a presentation as a torus GIT quotient of an affine variety, in such a way that the Mori chamber decomposition of the effective cone $\overline{Eff} (Y)$ literally equals the VGIT chamber decomposition of the equivariant Picard group. This identification realised VGIT as the GIT-theoretic incarnation of the minimal model program, decisively connecting two until-then-distinct technologies.

The Hu-Keel theorem opened the way to a flood of subsequent work: Castravet-Tevelev's 2015 demonstration that $\overline{M}_{0, n}$ is not a Mori dream space for $n \geq 7$ , showing the limits of the VGIT picture for moduli of pointed curves; Bauer-Bocci-Cooper et al.'s study of which moduli spaces of vector bundles are Mori dream spaces; Bérczi-Doran-Kirwan's non-reductive extension of VGIT to handle moduli problems where reductive group actions are not natural.

Daniel Halpern-Leistner 2014 ^{[Halpern-Leistner 2014]} reinterpreted VGIT in the language of derived categories. His magic windows are explicit triangulated subcategories of $D_{G}^{b} (X)$ that compute the derived category of each GIT quotient $X / /_{C} G$ . Wall-crossings then become equivalences of derived categories, generalising Bondal-Orlov's classical result that flops induce derived equivalences. The Halpern-Leistner picture subsumes the entire derived-categories-under-birational-change programme and is one of the central modern developments in derived algebraic geometry.

In the infinite-dimensional setting, the Yau-Tian-Donaldson conjecture — proved by Chen-Donaldson-Sun in 2015 — identifies the existence of a Kähler-Einstein metric on a Fano variety with its K-stability, an infinite-dimensional VGIT condition formulated via test configurations (the infinite-dimensional analogue of one-parameter subgroups). The VGIT chamber-wall picture survives in this setting, with the Mabuchi metric on the space of Kähler potentials playing the role of the equivariant ample cone.

Looking back, the VGIT framework reorganises moduli theory around three core insights, all visible from Dolgachev-Hu and Thaddeus's 1990s papers: (1) the GIT quotient depends on a linearisation; (2) the dependence is piecewise constant on a chamber complex; (3) the wall-crossings are flips in birational geometry. These three statements, taken together, explain why moduli spaces come in families, why birational geometry and moduli theory are so closely linked, and why the minimal model program of birational geometry has a GIT incarnation. The Dolgachev-Hu-Thaddeus work is the empirical demonstration; Hu-Keel's theorem is the conceptual identification; Halpern-Leistner's derived-VGIT is the homological lift; K-stability is the infinite-dimensional extension. Together they form one of the cleanest organising principles in modern algebraic geometry.

Bibliography [Master]

@article{DolgachevHu1998,
  author = {Dolgachev, Igor and Hu, Yi},
  title = {Variation of geometric invariant theory quotients},
  journal = {Publ. Math. IH\'ES},
  volume = {87},
  year = {1998},
  pages = {5--56}
}

@article{Thaddeus1996,
  author = {Thaddeus, Michael},
  title = {Geometric invariant theory and flips},
  journal = {J. Amer. Math. Soc.},
  volume = {9},
  year = {1996},
  pages = {691--723}
}

@article{Thaddeus1994,
  author = {Thaddeus, Michael},
  title = {Stable pairs, linear systems, and the {V}erlinde formula},
  journal = {Invent. Math.},
  volume = {117},
  year = {1994},
  pages = {317--353}
}

@article{HuKeel2000,
  author = {Hu, Yi and Keel, Sean},
  title = {{M}ori dream spaces and {GIT}},
  journal = {Michigan Math. J.},
  volume = {48},
  year = {2000},
  pages = {331--348}
}

@book{MFK1994,
  author = {Mumford, David and Fogarty, John and Kirwan, Frances},
  title = {Geometric Invariant Theory},
  publisher = {Springer},
  series = {Ergebnisse der Mathematik 34},
  edition = {3},
  year = {1994}
}

@article{Kempf1978,
  author = {Kempf, George},
  title = {Instability in invariant theory},
  journal = {Ann. Math.},
  volume = {108},
  year = {1978},
  pages = {299--316}
}

@article{HalpernLeistner2014,
  author = {Halpern-Leistner, Daniel},
  title = {The derived category of a {GIT} quotient},
  journal = {J. Amer. Math. Soc.},
  volume = {28},
  year = {2015},
  pages = {871--912}
}

@article{Hassett2003,
  author = {Hassett, Brendan},
  title = {Moduli spaces of weighted pointed stable curves},
  journal = {Adv. Math.},
  volume = {173},
  year = {2003},
  pages = {316--352}
}

@article{CastravetTevelev2015,
  author = {Castravet, Ana-Maria and Tevelev, Jenia},
  title = {{$\overline{\mathcal{M}}_{0,n}$} is not a {M}ori dream space},
  journal = {Duke Math. J.},
  volume = {164},
  year = {2015},
  pages = {1641--1667}
}

@article{ChenDonaldsonSun2015,
  author = {Chen, Xiuxiong and Donaldson, Simon and Sun, Song},
  title = {K\"ahler-{E}instein metrics on {F}ano manifolds {I, II, III}},
  journal = {J. Amer. Math. Soc.},
  volume = {28},
  year = {2015},
  pages = {183--278}
}

@incollection{Reid1992,
  author = {Reid, Miles},
  title = {What is a flip?},
  booktitle = {Utah {P}ark {C}ity Lectures},
  year = {1992}
}

Prerequisites

04.10.02
04.02.01
04.05.05

Tier anchors

beginner: Strogatz-style: the GIT quotient depends on a choice of polarisation; varying that choice gives a wall-and-chamber picture of moduli spaces
intermediate: Dolgachev-Hu 1998 *Publ. Math. IHÉS* 87; Thaddeus 1996 *J. Amer. Math. Soc.* 9; Mumford-Fogarty-Kirwan §3
master: Dolgachev-Hu 1998 *Variation of geometric invariant theory quotients* (Publ. Math. IHÉS 87); Thaddeus 1996 *Geometric invariant theory and flips* (J. Amer. Math. Soc. 9); Hu-Keel 2000 *Mori dream spaces and GIT* (Michigan Math. J. 48); Halpern-Leistner 2014 *Theta-stratification, magic windows, and the derived category of variation of GIT quotients*

References

TODO_REF
Dolgachev, Hu — Variation of geometric invariant theory quotients · Publ. Math. IHÉS 87 (1998), 5–56
TODO_REF
Thaddeus — Geometric invariant theory and flips · J. Amer. Math. Soc. 9 (1996), 691–723
TODO_REF
Hu, Keel — Mori dream spaces and GIT · Michigan Math. J. 48 (2000), 331–348
TODO_REF
Mumford, Fogarty, Kirwan — Geometric Invariant Theory · 3rd ed., Springer Ergebnisse 34, 1994, §3
TODO_REF
Reid — What is a flip? · Utah Park City lectures notes, 1992
TODO_REF
Halpern-Leistner — On the structure of instability in moduli theory · arXiv:1411.0627, 2014

Lean module

Codex.AlgGeom.Moduli.VariationOfGIT

Reviewer

TBD

Estimated time

beginner: 15m
intermediate: 40m
master: 75m