2.1 Birational geometry

Consider a projective morphism $f\colon X\to Y$ of normal varieties over ${{\Bbbk}}$ , where Y is affine. The relative Picard group is $\operatorname{Pic}(X/Y):=\operatorname{Pic}(X)/f^*\operatorname{Pic}(Y)$ , and we set $\operatorname{Pic}(X/Y)_{{{\mathbb{Q}}}}:=\operatorname{Pic}(X/Y)\otimes_{{{\mathbb{Z}}}} {{\mathbb{Q}}}$ . A line bundle $L\in \operatorname{Pic}(X/Y)_{{{\mathbb{Q}}}}$ is nef (over Y) if $\deg L\vert_\ell\geq 0$ for every proper curve $\ell$ in X, and it is semi-ample (over Y) if $L^{m}$ is basepoint-free for some $m\geq 1$ . The stable base locus of L is defined to be the intersection of the base loci of the linear series $\vert L^{m}\vert$ for all $m\geq 1$ , and we say that L is movable if its stable base locus is of codimension at least two in X. Every semi-ample line bundle is nef, but the converse is not true in general.

The nef cone of X over Y is the closed convex cone $\operatorname{Nef}(X/Y)$ in $\operatorname{Pic}(X/Y)_{{{\mathbb{Q}}}}$ generated by line bundles on X that are nef over Y. The relative version of Kleiman’s ampleness criterion [Reference KleimanKle66, IV, § 4] implies that the relative ample cone $\operatorname{Amp}(X/Y)$ is the interior of $\operatorname{Nef}(X/Y)$ . The movable cone $\operatorname{Mov}(X/Y)$ is the closed convex cone in $\operatorname{Pic}(X/Y)_{{{\mathbb{Q}}}}$ obtained as the closure of the cone generated by all movable divisor classes. Note that the nef cone is contained in the movable cone.

Let $\tau\colon X\to X_0$ be a projective, surjective morphism over Y satisfying $\tau_*(\mathcal{O}_X)=\mathcal{O}_{X_0}$ , so $\tau$ has connected fibres. We say that $\tau$ is of fibre type if $\dim X_0\lt\dim X$ . Otherwise, $\tau$ is birational, and there are two cases: either the exceptional locus of $\tau$ , denoted by $\operatorname{Exc}(\tau)$ , contains a divisor, in which case $\tau$ is a divisorial contraction; or it does not, in which case $\tau$ is a small contraction. In the latter case, let L be a line bundle on X such that $L^{-1}$ is $\tau$ -ample. The flip of $\tau$ with respect to L is a commutative diagram

(2.1)

where $\tau^\prime$ is a small contraction, $\psi$ is an isomorphism in codimension one, and the strict transform of L along $\psi$ is $\tau^\prime$ -ample. If, in addition, the canonical class $K_X$ satisfies $K_X\cdot \ell=0$ for each curve $\ell$ contracted by $\tau$ , then (2.1) is the flop of the curve class $\ell$ [Reference Kollár and MoriKM98, Definition 6.10].

Let $L\in \operatorname{Pic}(X/Y)_{{{\mathbb{Q}}}}$ be such that the section ring

\[ R(X,L):=\bigoplus_{m\geq 0} f_* L^{m} \]

is a finitely generated $\mathcal{O}_Y$ -algebra. Then $X(L):= \operatorname{Proj}_Y R(X,L)$ fits into a commutative diagram

(2.2)

where $\psi_L$ is regular on the complement of the stable base locus of L in X. We do not assume in general that f is birational, nor do we assume that L is big, that is, $\psi_L$ need not be birational either. However, if L is movable, then the rational map $\psi_L$ is an isomorphism in codimension one. If L is movable and X(L) is ${{\mathbb{Q}}}$ -factorial, then we call X(L) a ${{\mathbb{Q}}}$ -factorial small birational model of X over Y. When X and X(L) are ${{\mathbb{Q}}}$ -factorial, we identify $\operatorname{Pic}(X(L)/Y)_{{{\mathbb{Q}}}}$ with $\operatorname{Pic}(X/Y)_{{{\mathbb{Q}}}}$ by taking strict transform along the birational map $\psi_L$ ; this in turn identifies $\operatorname{Mov}(X(L)/Y)$ with $\operatorname{Mov}(X/Y)$ . Let $\psi_L^*\operatorname{Amp}(X(L)/Y)$ and $\psi_L^*\operatorname{Nef}(X(L)/Y)$ denote the cones in $\operatorname{Pic}(X/Y)_{{\mathbb{Q}}}$ obtained by taking the strict transform along $\psi_L$ of all classes on X(L) that are relatively ample and nef, respectively.

Given $L, L^\prime\in \operatorname{Pic}(X/Y)_{{{\mathbb{Q}}}}$ with finitely generated section rings, we say that L is Mori equivalent to $L^\prime$ if there is an isomorphism $\varphi\colon X(L)\to X(L^\prime)$ such that the rational maps $\psi_L, \psi_{L^\prime}$ satisfy $\varphi\circ \psi_L = \psi_{L^\prime}$ . A Mori chamber is a Mori equivalence class whose interior is open in $\operatorname{Pic}(X/Y)_{{{\mathbb{Q}}}}$ . These chambers are typically studied under the additional assumption that $\operatorname{Pic}(X/Y)_{{\mathbb{Q}}}$ is isomorphic to the Néron–Severi space $N^1(X/Y):=\operatorname{Pic}(X/Y)_\mathbb{Q}/\!\equiv$ of numerical equivalence classes, where $L\equiv L^\prime$ if and only if $\deg(L\vert_\ell) = \deg(L^{\prime}\vert_\ell)$ for every proper curve $\ell$ in X.

To see how the isomorphism $\operatorname{Pic}(X/Y)_{{\mathbb{Q}}}\cong N^1(X/Y)$ arises in the case of interest to us, recall first the following fundamental and well-known result.

Proof. Suppose that $L^m\in \operatorname{Pic}(X/Y)$ is a basepoint-free line bundle over Y. The induced morphism $h\colon X\to \vert L^m\vert\cong \mathbb{P}_Y^N$ satisfies $h^*(\mathcal{O}(1))\cong L^m$ . For a proper curve $\ell $ in X, we have

\[ L\cdot \ell:= \frac{1}{m} \deg\big(h^*(\mathcal{O}(1))\vert_\ell\big) = \frac{1}{m}\deg\big(\mathcal{O}(1)\vert_{h_*[\ell]}\big) \]

where $h_*[\ell]$ is the pushforward of the curve class of $\ell$ . Thus, L is nef over Y, and $\ell$ is contracted by h if and only if $L\cdot \ell=0$ . Finite generation of R(X,L) is the relative version of a theorem of Zariski (see [Reference OhtaOht22, Lemma 6.11]), and the image of h is $\operatorname{Proj}_Y R(X,L)$ .

2.2 GIT quotients and the linearisation map

Let G be a reductive algebraic group acting on an affine variety V with coordinate ring ${{\Bbbk}}[V]$ . Let $G^\vee$ denote the character group of G. For $\theta\in G^\vee$ , we say that $f\in {{\Bbbk}}[V]$ is $\theta$ -semi-invariant if $f(g.v) = \theta(g)f(v)$ for all $v\in V$ and $g\in G$ , and we write ${{\Bbbk}}[V]_\theta$ for the space of $\theta$ -semi-invariant functions. A point $v\in V$ is $\theta$ -semi-stable if there exist $j\gt0$ and $f\in {{\Bbbk}}[V]_{j\theta}$ such that $f(v)\neq 0$ . The $\theta$ -semi-stable locus $V^\theta\subseteq V$ is the G-invariant, open subset of $\theta$ -semi-stable points. A point $v\in V^\theta$ is $\theta$ -stable if the stabiliser $G_v$ is finite and the orbit $G\cdot v$ is closed in $V^\theta$ . A character $\theta\in G^\vee$ is effective if $V^\theta$ is non-empty, and an effective character $\theta$ is generic if every $\theta$ -semi-stable point of V is $\theta$ -stable. The $\theta$ -semi-stable locus is unchanged if we replace $\theta$ by a positive multiple, so the definitions extend to any fractional character $\theta\in G^\vee_{{{\mathbb{Q}}}}:= G^\vee\otimes_{{{\mathbb{Z}}}} {{\mathbb{Q}}}$ .

For any effective $\theta\in G^\vee_{{{\mathbb{Q}}}}$ , the GIT quotient

\[X_{\theta{}}:= V{{/\!\!/\!}}_\theta \, G:= \operatorname{Proj} \bigg(\bigoplus_{j\geq 0} {{\Bbbk}}[V]_{j\theta}\bigg) \]

is the categorical quotient of the $\theta$ -semi-stable locus $V^\theta$ by the action of G. Note that $X_{\theta{}}$ is projective over the affine quotient

\[Y:= V{{/\!\!/\!}}_0 \, G = \operatorname{Spec} {{\Bbbk}}[V]^G. \]

If $\theta$ is generic, then $X_{\theta{}}$ is the geometric quotient of $V^\theta$ by G.

The set of effective fractional characters is a closed, convex cone in $G^\vee_{{{\mathbb{Q}}}}$ that admits a wall-and-chamber structure as follows. Fractional characters $\theta, \theta'\in G^\vee_{{{\mathbb{Q}}}}$ are GIT-equivalent if $V^{\theta}=V^{\theta'}$ . The GIT-equivalence classes form the relative interiors of a finite collection of rational polyhedral cones in $G^\vee_{{{\mathbb{Q}}}}$ , and the collection of all such cones, called GIT cones, forms a fan, called the GIT fan, whose support is the convex cone of effective fractional characters in $G^\vee_{{{\mathbb{Q}}}}$ . The set of generic stability parameters $\theta$ in $G^\vee_{{\mathbb{Q}}}$ decomposes as the union of (GIT) chambers, each of which is the interior of a top-dimensional cone in the GIT fan. As shown by Ressayre [Reference RessayreRes98], it can happen that the interior of a top-dimensional GIT cone is not a chamber. However, in this paper we work only with stability parameters $\theta$ lying in the closure of the union of all GIT chambers, and we reserve the phrase GIT wall for any codimension-one face of the closure $\overline{C}$ of some GIT chamber C. The characterisation of the GIT fan via GIT-equivalence was established by Ressayre [Reference RessayreRes00] (see Halic [Reference HalicHal04] for affine V), building on the earlier work of Dolgachev and Hu [Reference Dolgachev and HuDH98], and Thaddeus [Reference ThaddeusTha96]. Those papers assume that V is normal, but in fact, GIT-equivalence is unaffected by passing to the normalisation of V; explicitly, if $\nu\colon \widetilde{V}\to V$ is the normalisation, then ${\widetilde{V}}^\theta = \nu^{-1}(V^\theta)$ for any $\theta\in G^\vee_{{{\mathbb{Q}}}}$ .

Let C be a GIT chamber and fix $\theta\in C$ , so $\theta$ is generic. For $\chi\in G^\vee$ , consider the G-equivariant line bundle $\chi\otimes \mathcal{O}_{V^\theta}$ on the $\theta$ -stable locus in V given by equipping the trivial line bundle with the action of G on each fibre given by $\chi$ ; explicitly, the action of G on $V^\theta$ lifts to the action on $V^\theta\times \mathbb{A}^1$ such that the dual action on functions is $g\cdot (f,t) = (g\cdot f, \chi^{-1}(g) t)$ . It follows that the space of sections is isomorphic to the space ${{\Bbbk}}[V^\theta]_\chi$ of $\chi$ -semi-invariant functions on $V^\theta$ . By descent [Reference NevinsNev08], $\chi\otimes \mathcal{O}_{V^\theta}$ descends to a line bundle on $X_{\theta{}}$ if the stabiliser of each $x \in V^{\theta}$ is in the kernel of $\chi$ . Since all stabilisers are finite, and there are only finitely many conjugacy classes of such stabilisers by [Reference LunaLun73, Corollaire 3], there is some multiple $j \chi\in G^\vee$ of $\chi$ that descends to a line bundle on X that we denote $L_{j\chi}$ . We define $L_{\chi}:= ({1}/{j})L_{j \chi}\in \operatorname{Pic}(X_{\theta{}}/Y)_{{{\mathbb{Q}}}}$ .

Definition 2.3. Let C be a GIT chamber. For $\theta\in C$ and $X_{\theta{}}=V{{/\!\!/\!}}_\theta \, G$ , the linearisation map for C is the $\mathbb{Q}$ -linear map

(2.3) \begin{equation} L_C\colon G^\vee_{\mathbb{Q}} \longrightarrow \operatorname{Pic}(X_{\theta{}}/Y)_{{{\mathbb{Q}}}} \end{equation}

determined by setting $L_C(\chi):= L_\chi$ for all $\chi\in G^\vee$ .

2.3 Variation of GIT quotient

Let C be a GIT chamber and let $\theta\in C$ . In addition, let $\theta_0$ be a general point in any face of the closure $\overline{C}$ . The G-equivariant inclusion of the $\theta$ -stable locus into the $\theta_0$ -semi-stable locus of V fits into a commutative diagram of varieties

(2.4)

where $\pi_0$ is a good categorical quotient, $\pi$ is a geometric quotient and $\tau$ is a projective morphism; the morphism $\tau$ is said to be induced by variation of GIT quotient. The G-equivariant line bundle $\theta_0\otimes \mathcal{O}$ on $V^{\theta_0}$ descends to the polarising ample bundle $\mathcal{O}(1)$ on $X_{\theta{_0}}$ , and its restriction to $V^\theta$ descends to the line bundle $L_C(\theta_0)$ on $X_{\theta{}}$ . Commutativity of diagram (2.4) gives

(2.5) \begin{equation} L_C(\theta_0) = \tau^*\big(\mathcal{O}(1)\big). \end{equation}

Let $C_-$ and $C_+$ be adjacent GIT chambers separated by a wall. Let $\theta_-\in C_-$ , $\theta_+\in C_+$ and let $\theta_0$ be a general point in the wall $\overline{C_-}\cap \overline{C_+}$ . The morphisms $\tau_-\colon X_{\theta{_-}}\rightarrow X_{\theta{_0}}$ and $\tau_+\colon X_{\theta{_+}}\rightarrow X_{\theta{_0}}$ obtained by VGIT as in (2.4) fit into a commutative diagram

(2.6)

of varieties over $Y=V{{/\!\!/\!}}_0 \, G$ . Let us assume that $X_{\theta{_\pm}}$ are normal.

The next result generalises a result of Thaddeus [Reference ThaddeusTha96, Theorem 3.3], though the necessary assumption on the dimension of the unstable locus is missing from that statement. For this, we take the Stein factorisations of the morphisms $\tau_\pm$ from diagram (2.6) to obtain a commutative diagram

(2.7)

where the morphisms $\widetilde{\tau}_\pm$ have connected fibres. Note that the target $\widetilde{X}_{\theta_0}$ is just the normalisation of $X_{\theta_0}$ since $\tau_{\pm}$ are birational by Lemma 2.4(i) and $X_{\theta{_\pm}}$ are normal. In other words, $\widetilde{\tau}_{\pm}$ are the functorial maps induced by normalisation.

Proposition 2.7. Assume that $X_{\theta{_+}}$ and $X_{\theta{_-}}$ are normal, that $\operatorname{Uns}(\tau_+)\subseteq X_{\theta{_+}}$ and $\operatorname{Uns}(\tau_-)\subseteq X_{\theta{_-}}$ have codimension at least two, and that $\tau_+$ and $\tau_-$ contract at least one curve. Then (2.7) is a flip with respect to the line bundle $L_{C_-}(\theta_+)$ on $X_{\theta{_-}}$ , and

(2.8) \begin{equation} L_{C_-}(\eta) \cong \psi^*\big(L_{C_+}(\eta)\big) \,\text{for all }\eta\in G^\vee_{\mathbb{Q}}. \end{equation}

If, in addition, $X_{\theta{_-}}$ is ${{\mathbb{Q}}}$ -factorial and $L_{C_-}$ is surjective, then $X_{\theta{_+}}$ is ${{\mathbb{Q}}}$ -factorial.

Combining Lemma 2.4(ii) with our assumption on the unstable locus implies that the complement of $V^{\theta_0\mathrm{\hbox{-}st}}/G$ is of codimension at least two in both $X_{\theta{_+}}$ and $X_{\theta{_-}}$ . Since $X_{\theta{_+}}$ and $X_{\theta{_-}}$ are normal, line bundles on both $X_{\theta{_+}}$ and $X_{\theta{_-}}$ are uniquely determined, up to isomorphism, by their restriction to $V^{\theta_0\mathrm{\hbox{-}st}}/G$ . By restricting all three maps from (2.6) to the isomorphisms over this locus, we see that both $L_{C_+}(\eta)$ on $X_{\theta{_+}}$ and $L_{C_-}(\eta)$ on $X_{\theta{_-}}$ are obtained by descent from $\eta\otimes \mathcal{O}$ on $V^{\theta_0\mathrm{\hbox{-}st}}$ . Therefore, isomorphism (2.8) holds. In particular, the strict transform of $L_{C_-}(\theta_+)$ along $\psi$ is the line bundle $L_{C_+}(\theta_+)$ on $X_{\theta{_+}}$ .

The polarising ample bundle $L_{C_+}(\theta_+)$ on $X_{\theta{_+}}$ is $\widetilde{\tau}_+$ -ample, so to prove that (2.7) is a flip, we need only show that $L_{C_-}(\theta_+)^{-1}$ is $\widetilde{\tau}_-$ -ample. For this, the ample bundle $L_0:=\mathcal{O}(1)$ on $\widetilde{X}_{\theta_0}$ satisfies $L_{C_-}(\theta_0) = (\widetilde{\tau}_-)^*(L_0)$ by (2.5). By choosing alternative characters $\theta_+\in C_+$ and $\theta_-\in C_-$ if necessary, we may assume that $\theta_0 = {1}/{2}(\theta_++\theta_-)$ , in which case

\[L_{C_-}(\theta_+)\otimes L_{C_-}(\theta_-) = L_{C_-}(\theta_+ + \theta_-) = L_{C_-}(2\theta_0) = (\widetilde{\tau}_-)^*(L_0)^{2}. \]

The set of curve classes contracted by $\widetilde{\tau}_-$ is non-empty by assumption. The line bundles $L_{C_-}(\theta_-)$ and $\widetilde{\tau}_-^*(L_0)$ have positive and zero degree respectively on all such curves. It follows that

\[L_{C_-}(\theta_+)^{-1} = L_{C_-}(\theta_-)\otimes \tau_-^*(L_0)^{-2} \]

has positive degree on all such curves, so it is $\widetilde{\tau}_-$ -ample as required.

For the final statement, let $D_+$ be a Weil divisor on $X_{\theta{_+}}$ . Then $(\psi^{-1})_*D_+$ is a Weil divisor on $X_{\theta{_-}}$ , and since $X_{\theta{_-}}$ is ${{\mathbb{Q}}}$ -factorial, the divisor $m(\psi^{-1})_*D_+$ is Cartier for some $m\gt0$ . Since $L_{C_-}$ is surjective, there exists $\eta\in G^\vee_{{\mathbb{Q}}}$ such that $L_{C_-}(\eta) \cong \mathcal{O}_{X_{\theta{_-}}}(m(\psi^{-1})_*D_+)$ . Now (2.8) gives

\[ \mathcal{O}_{X_{\theta{_+}}}(mD_+) \cong (\psi^{-1})^*\mathcal{O}_{X_{\theta{_-}}}(m(\psi^{-1})_*D_+) \cong (\psi^{-1})^* L_{C_-}(\eta) \cong L_{C_+}(\eta) \]

which lies in $\operatorname{Pic}(X_{\theta{_+}}/Y)$ , so $mD_+$ is Cartier.

3.1 GIT regions

As before, let G denote a reductive algebraic group acting on an affine variety V. For $\theta\in G^\vee_{\mathbb{Q}}$ , write $X_{\theta{}}:=V{{/\!\!/\!}}_\theta \, G$ , and let $f\colon X_{\theta{}}\to Y:= X_0$ denote the projective morphism obtained by VGIT.

To formulate our key condition, let $C_-$ and $C_+$ be GIT chambers separated by a wall $\overline{C_-}\cap \overline{C_+}$ . We delete this separating wall if and only if the morphisms $\tau_-$ and $\tau_+$ from diagram (2.6) are both small. The result is an a priori coarser wall-and-chamber decomposition of the GIT fan.

Example 3.2. The Cox construction of the first Hirzebruch surface $\mathbb{F}_1:=\mathbb{P}_{\mathbb{P}^1}(\mathcal{O}\oplus \mathcal{O}(1))$ passes via the action of the torus $G=({{\Bbbk}}^\times)^2$ on $\mathbb{A}^4$ with weights $\left[\begin{smallmatrix}1 & -1 & 1 & 0 \\0 & 1 & 0 & 1\end{smallmatrix}\right]$ . There are two GIT chambers

$C_-:= \operatorname{Amp}(\mathbb{F}_1)= \left\{\alpha \left[\begin{smallmatrix}1\\0\end{smallmatrix}\right]+\beta \left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]\,\middle|\, \alpha,\beta>0\right\}\,\text{and}\,C_+:=\left\{\alpha \left[\begin{smallmatrix}-1\\1\end{smallmatrix}\right]+\beta \left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]\,\middle|\, \alpha,\beta>0\right\}.$

For $\theta_-\in C_-$ and $\theta_+\in C_+$ , we have $X_{\theta{_-}}\cong \mathbb{F}_1$ and $X_{\theta{_+}}\cong \mathbb{P}^2$ . The VGIT morphism $\tau_-\colon \mathbb{F}_1\to \mathbb{P}^2$ contracts the $(-1)$ -curve, whereas $\tau_+$ is an isomorphism. Thus, $R_{C_-} = \overline{C_-}$ and $R_{C_+} = \overline{C_+}$ .

The linearisation map $L_{C_-}$ in Example 3.2 is an isomorphism that identifies $R_{C_-}$ with the movable (in fact, the nef) cone of $X_{\theta{_-}}$ for $\theta_-\in C_-$ . Example 3.8 below illustrates that even when $L_{C_-}$ is an isomorphism, it need not identify $R_{C_-}$ with $\operatorname{Mov}(X_{\theta{_-}})$ for $\theta\in C_-$ .

3.2 The new GIT condition and two key examples

Define a GIT wall to be a flipping wall if it satisfies the assumptions of Proposition 2.7, that is, the wall separates two GIT chambers $C_\pm$ such that, in the notation of diagram (2.6), both $X_{\theta{_-}}$ and $X_{\theta{_+}}$ are normal, the unstable loci $\operatorname{Uns}(\tau_+)\subseteq X_{\theta{_+}}$ and $\operatorname{Uns}(\tau_-)\subseteq X_{\theta{_-}}$ have codimension at least two, and the morphisms $\tau_+$ and $\tau_-$ both contract at least one curve. For any wall in the boundary of $R_C$ , let $C_-$ denote the unique chamber in $R_C$ that contains the wall in its closure. Let $\theta_-\in C_-$ and let $\theta_0$ be general in the wall. We say that the wall, when approached from the chamber $C_-$ in $R_C$ , is small, divisorial or of fibre type if the induced morphism $\tau_-\colon X_{\theta{_-}} \to X_{\theta{_0}}$ is of the same type.

• – Condition 3.3 holds for the chamber $C_-$ in Example 3.2, while Condition 3.3(1) fails for $C_+$ . Many similar examples are described in Example 3.5 below.

• – If Condition 3.3(1) holds, it can happen that (2) fails to hold; see Example 3.6 below.

• – If Condition 3.3(1) and (2) both hold, it can happen that (3) fails to hold; see Example 3.8 below.

Note, in addition, that Condition 3.3(2) implies that $X_{\theta{}}$ is normal for every generic $\theta\in R_C$ .

1. (i) For the chamber $C=\operatorname{Amp}(X)$ , the linearisation map $L_C \colon G^\vee_{{\mathbb{Q}}}\to \operatorname{Pic}(X)_{{\mathbb{Q}}}$ is an isomorphism by [Reference Hu and KeelHK00, Proof of Proposition 2.11], so Condition 3.3(1) holds, while [Reference Hu and KeelHK00, Proposition 1.11] shows that conditions (2) and (3) also hold.

2. (ii) For any chamber $C^\prime$ that does not lie in $\operatorname{Mov}(X)$ , the kernel of $L_{C^\prime}$ has dimension at least one because the rank of $\operatorname{Pic}(X_{\theta{^\prime}})_{{\mathbb{Q}}}$ for $\theta^\prime\in C^\prime$ drops by one as we cross each boundary wall of the movable cone. In particular, Condition 3.3 fails for $C^\prime$ .

Analogous statements hold for the action induced by the $\operatorname{Pic}(X/Y)$ -grading on the Cox ring for any relative Mori dream space $X\to Y$ ; this requires a choice of line bundles on X that provide a basis for $\operatorname{Pic}(X/Y)_{{{\mathbb{Q}}}}$ (see Grab [Reference GrabGra19] or Ohta [Reference OhtaOht22] for details).

Example 3.6 (A local del Pezzo by quiver GIT). Let Z be the two-point blow-up of $\mathbb{P}^2$ . The total space $X:= \text{tot}(\omega_Z)$ of the canonical bundle on Z is smooth with trivial canonical class, the anticanonical ring $R:= \bigoplus_{k\geq 0} H^0(Z,\omega_Z^{-\otimes k})$ is Gorenstein [Reference Goto and WatanabeGW78, Example 5.1.13], and the morphism

\[f\colon X=\text{tot}(\omega_Z) \longrightarrow Y:= \operatorname{Spec} R \]

that contracts the zero section is a projective crepant resolution. In fact, f is a morphism of toric varieties: for the lattice $M={{\mathbb{Z}}}^3$ , we have that $R\cong {{\mathbb{C}}}[\sigma^\vee\cap M]$ , where $\sigma\subseteq N\otimes_{{{\mathbb{Z}}}} {{\mathbb{Q}}}$ is the strongly convex rational polyhedral cone obtained as the cone over the pentagon in Figure 1(a); the basic triangulation of the pentagon that determines the fan $\Sigma$ of X is also shown in Figure 1(a). The cones $\ell_1, \ell_2, \ell_3\in \Sigma$ each determine a $(-1,-1)$ -curve in X, and for $1\leq i\leq 3$ , flopping the curve defined by $\ell_i$ produces a projective crepant resolution $X_i\to Y$ . A curve in each of $X_1$ and $X_3$ can be flopped to produce the projective crepant resolution $X_4\to Y$ as shown in Figure 1(b).

Figure 1.

(a) Slice of the fan defining X. (b) Flops linking crepant resolutions.

The morphism $f\colon X\to Y$ is a relative Mori dream space, but our interest here lies with a GIT quotient construction that differs from the construction via the Cox ring of X as in Example 3.5. For this, list the torus-invariant prime Weil divisors $D_1, \dots,D_5$ on Y, one for each lattice point $v_1, \dots, v_5$ on the boundary of the pentagon as above. Define four reflexive sheaves of rank one on Y, namely $E_0:= \mathcal{O}_Y$ , $E_1:=\mathcal{O}_Y(D_1)$ , $E_2:=\mathcal{O}_Y(D_3)$ and $E_3:= \mathcal{O}_Y(D_1+D_5)$ , and set $\mathscr{E}:=\{E_0, E_1, E_2, E_3\}$ . Following [Reference Craw and Quintero VélezCQV12, Definition 2.2], the quiver of sections of $\mathscr{E}$ , denoted by Q, is shown in Figure 2(a): the vertex set corresponds to the collection $\mathscr{E}$ ; and each arrow is labelled by a Weil divisor where, for example, the label 12 is shorthand for the divisor $D_1+D_2$ . The algebra $\operatorname{End}_{R}(\bigoplus_{0\leq i\leq 3}E_i)$ can be presented as the quotient of the path algebra of Q by a two-sided ideal of relations determined by the labelling of arrows by divisors [Reference Craw and Quintero VélezCQV12, Lemma 2.5].

Figure 2.

(a) Quiver of sections Q on Y. (b) Graph indicating chambers that lie adjacent.

To reconstruct the morphism f by quiver GIT, let $\mathbb{N}(Q)$ denote the semigroup generated by the columns of the matrix

\[ \left[\begin{smallmatrix} -1 & -1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\ 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 2 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \end{smallmatrix}\right]\!; \]

each column corresponds to an arrow, where the top four entries record the head and tail, while the bottom five entries record the labelling divisor. Consider the toric variety $V:= \operatorname{Spec} {{\Bbbk}}[ \mathbb{N}(Q)]$ defined by the semigroup algebra of $\mathbb{N}(Q)$ . The algebraic torus $G:=\operatorname{Spec} {{\Bbbk}}[\operatorname{Wt}(Q)]$ with character lattice $\operatorname{Wt}(Q):=\{\theta\in {{\mathbb{Z}}}^4 \mid \sum_i \theta_i=0\}$ acts on V, where the weights of the action are recorded in the top four rows of the matrix. Following [Reference Craw and Quintero VélezCQV12, Proposition 2.14], the affine quotient satisfies $V{{/\!\!/\!}}_0 T \cong Y$ , while for each $\theta\in \operatorname{Wt}(Q)$ , variation of GIT gives a projective, birational, toric morphism $f_\theta\colon X_{\theta{}}:= V{{/\!\!/\!}}_\theta G\to Y$ . The GIT chamber

\[ C:=\big\{ \theta\in G^\vee_{{\mathbb{Q}}} \mid \theta_1 \lt0;\; \theta_2\lt0,\; \theta_1+\theta_2+\theta_3\gt 0\big\} \]

gives $X\cong X_{\theta{}}$ and $f=f_\theta$ for $\theta\in C$ . The linearisation map $L_{C}$ is an isomorphism that identifies C with the ample cone of X, so Criterion 3.3(1) is satisfied.

The defining inequalities of the chambers $C_1, C_2, C_3$ that lie adjacent to C are shown in Table 1, together with the unique chamber $C_4$ that lies adjacent to both $C_1$ and $C_3$ as in Figure 2(b).

Table 1.

Defining inequalities for the GIT chambers in Example 3.6.

Crossing the wall from C to $C_1$ induces the flop $X\dashrightarrow X_1$ , and symmetrically, crossing from C to $C_3$ induces the flop $X\dashrightarrow X_3$ . In addition, for $i\in \{1,3\}$ , one can cross a wall from $C_i$ to $C_4$ to induce the flop $X_i\dashrightarrow X_4$ . However, crossing the wall separating $C_-:=C$ from $C_+:= C_2$ does not induce the flop $X\dashrightarrow X_2$ as one might expect after comparing Figures 1(a) and 2(b). Rather, the morphism $\tau_-$ from diagram (2.6) contracts the curve in X determined by $\ell_2$ , whereas $\tau_+$ is an isomorphism. In particular, $V{{/\!\!/\!}}_{\theta_2}G$ for $\theta_2\in C_2$ is not ${{\mathbb{Q}}}$ -factorial, and $X_2\not\cong V{{/\!\!/\!}}_{\theta_2}G$ . Therefore, the wall separating C from $C_2$ is not a flipping wall despite being an internal wall of $R_C$ , so Condition 3.3(2) fails even though Condition 3.3(1) holds.

Example 3.8. Consider the action of the algebraic torus $({{\Bbbk}}^\times)^2$ on $\mathbb{A}^5$ with weights $\left[\begin{smallmatrix}0 & 0 & 1 & 1 & 1 \\1 & 1 & 0 & -1 & -1\end{smallmatrix}\right]$ . For the character $L_- := \left[\begin{smallmatrix}1\\1\end{smallmatrix}\right]$ , the Cox construction gives $X_-=\mathbb{A}^5{{/\!\!/\!}}_{L_-} ({{\Bbbk}}^\times)^2$ , where we identify the character lattice of $({{\Bbbk}}^\times)^2$ with $\operatorname{Pic}(X_-)$ . For $L_+ := \left[\begin{smallmatrix}2\\-1\end{smallmatrix}\right]$ , variation of GIT quotient determines a flop $\psi\colon X_-\dashrightarrow X_+:=\mathbb{A}^5{{/\!\!/\!}}_{L_+} ({{\Bbbk}}^\times)^2$ of smooth projective toric threefolds. The ample cones of $X_-$ and $X_+$ satisfy

\[\operatorname{Amp}(X_-)=\left\{\alpha \left[\begin{smallmatrix}1\\0\end{smallmatrix}\right]+\beta \left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]\;\middle|\;\alpha,\beta>0\right\}\,\text{and}\,\psi^*\operatorname{Amp}(X_+)=\left\{\alpha \left[\begin{smallmatrix}1\\0\end{smallmatrix}\right]+\beta \left[\begin{smallmatrix}1\\-1\end{smallmatrix}\right]\;\middle|\;\alpha,\beta>0\right\}.\]

Our interest lies in an alternative GIT construction introduced in [Reference Craw and SmithCS08]. For this, consider the globally generated line bundles , and set $\mathcal{L}:=\{L_0, L_1, L_2\}$ . Let $\mathbb{N}(\mathcal{L})$ denote the semigroup generated by the columns of the matrix

\[ \left[\begin{smallmatrix} -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \\ 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 2 & 1 & 0 & 2 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 2 & 0 & 1 & 2 & 0 & 1 \end{smallmatrix}\right]\!, \]

where, for $0\leq i,j\leq 2$ , each column with $-1$ in row i and $+1$ in row j corresponds to a given torus-invariant divisor defining a section of $L_j\otimes L_i^{-1}$ ; see [Reference Craw and SmithCS08, § 3]. The semigroup algebra of $\mathbb{N}(\mathcal{L})$ defines the toric variety $V:= \operatorname{Spec} {{\Bbbk}}[ \mathbb{N}(\mathcal{L})]$ , and the top three rows of the above matrix encode the weights of an action on V by the algebraic torus G of rank two whose character lattice is $G^\vee=\{(\theta_i)\in {{\mathbb{Z}}}^3 \mid \theta_0+\theta_1+\theta_2=0\}$ . There are two GIT chambers,

\[ C_-:=\{\theta\in G^\vee_{\mathbb{Q}} \mid \theta_1\gt0, \theta_2\gt0\} \,\text{and}\, C_+:= \{\theta\in G^\vee_{\mathbb{Q}} \mid \theta_1+\theta_2\gt0, \theta_2\lt0\}. \]

Observe that the GIT quotient $X_\eta$ is non-empty if and only if $\eta\in \overline{C_-}\cup \overline{C_+}$ .

We claim that the GIT regions satisfy $R_{C_-}= \overline{C_-}\cup \overline{C_+}=R_{C_+}$ . To see this, fix $\theta_-:=(-2,1,1)\in C_-$ and $\theta_+:=(-1,2,-1)\in C_+$ . Applying [Reference Craw and SmithCS08, Corollary 4.10, Theorem 4.15] shows that $X_-\cong X_{\theta{_-}}$ , and moreover, that $L_{C_-}$ identifies $G^\vee$ with the index 2 sublattice of $\operatorname{Pic}(X_-)$ spanned by $L_1$ and $L_2$ . It follows that $L_{C_-}$ is an isomorphism of rational vector spaces that identifies $C_-$ with $\operatorname{Amp}(X_-)$ . More generally, for $\eta\in \overline{C_-}\cup \overline{C_+}$ , the columns of the matrix are chosen to ensure that the $\eta$ -graded piece ${{\Bbbk}}[V]_\eta$ of the coordinate ring of V is isomorphic to the vector space $H^0(X_-,L_{C_-}(\eta))$ , so $X_\eta$ is isomorphic to $\operatorname{Proj}$ of the section ring of $L_{C_-}(\eta)$ . In particular, $X_{\theta{_+}}\cong X_+$ and the rational map $X_{\theta{_-}}\dashrightarrow X_{\theta{_+}}$ induced by crossing the wall separating $C_-$ and $C_+$ is the flop $\psi\colon X_-\dashrightarrow X_+$ . Thus, both VGIT morphisms $\tau_\pm$ are small, so the claim follows.

Note, however, that while $L_{C_-}$ is an isomorphism, it fails to identify $C_+$ with $\operatorname{Amp}(X_+)$ , so $L_{C_-}$ does not identify $R_{C_-}$ with $\operatorname{Mov}(X_{\theta{_-}})$ . In fact, the cone $C_+$ is generated by the vectors $(-1,1,0)$ and $(0,1,-1)$ , so the cone $L_{C_-}(C_+)$ is generated by $L_{C_-}((-1,1,0))= L_0^{-1}\otimes L_1\cong L_1= \begin{bmatrix}2\\0\end{bmatrix}$ and $L_{C_-}((0,1,-1)) = L_1 \otimes L_2^{-1}= \begin{bmatrix}2\\-1\end{bmatrix}$ , giving

$L_{C_-}(C_+)=\left\{\alpha \left[\begin{smallmatrix}2\\0\end{smallmatrix}\right]+\beta \left[\begin{smallmatrix}2\\-1\end{smallmatrix}\right]\;\middle|\; \alpha,\beta>0\right\}.$

It follows that $L_{C_-}$ identifies the character $\eta=(0,1,-1)$ in the boundary of $R_{C_-}$ with the ample bundle $\begin{bmatrix}2\\-1\end{bmatrix}$ on $X_+$ , so the induced VGIT morphism $X_{\theta{_+}}\to X_{\eta}$ is an isomorphism rather than a divisorial or fibre-type contraction.

3.3 The main result

We now establish the main geometric consequences of Condition 3.3. For C a GIT chamber, $\theta \in C$ , and $\eta \in \overline{C}$ , let $X_\theta \to \widetilde{X}_\eta$ be the Stein factorisation of the VGIT morphism $X_\theta \to X_\eta$ .

1. (i) the isomorphism $L_{C}$ identifies C and $\overline{C}$ with $\operatorname{Amp}(X/Y)$ and $\operatorname{Nef}(X/Y)$ , respectively;

2. (ii) each $L\in \operatorname{Pic}(X/Y)$ that is nef over Y is also semi-ample over Y;

3. (iii) for $\eta\in \overline{C}$ , the section ring $R(X,L_C(\eta))$ is a finitely generated $\mathcal{O}_Y$ -algebra, and moreover, $\widetilde{X}_\eta$ is isomorphic to $\operatorname{Proj} R(X,L_C(\eta))$ ; and

4. (iv) $\operatorname{Pic}(X/Y)_{{\mathbb{Q}}}\cong N^1(X/Y)$ .

For part (ii), any class in the interior of $\operatorname{Nef}(X/Y)$ is ample and hence semi-ample. For L in the boundary of $\operatorname{Nef}(X/Y)$ , it follows that $\eta:=L_{C}^{-1}(L)$ lies in the boundary of C. After multiplying by some $m\gt0$ if necessary, (2.5) shows that $L_C(m\eta) = \tau^*(\mathcal{O}(1))$ for the morphism $\tau\colon X\to X_{m\eta}$ over Y induced by VGIT. Thus, $L^m=\tau^*\big(\mathcal{O}(1)\big)$ is basepoint-free, so L is semi-ample over Y.

For part (iii), let $\eta\in \overline{C}$ . Parts (i) and (ii) imply that $L_C(\eta)$ is semi-ample, so the section ring $R(X,L_C(\eta))$ is finitely generated by a theorem of Zariski [Reference LazarsfeldLaz04, Example 2.1.30], and hence the model $X(L_C(\eta))$ is well defined. Since $\eta$ lies in a face of $\overline{C}$ , the VGIT morphism $\tau\colon X\to X_\eta$ satisfies $L_C(\eta)\cong\tau^*(\mathcal{O}_{X_\eta}(1))$ by (2.4). The Stein factorisation $\widetilde{\tau}\colon X_\theta \to \widetilde{X}_\eta$ then gives $L_C(\eta)\cong\widetilde{\tau}^*(\mathcal{O}(1))$ , where $\mathcal{O}(1)$ is ample on $\widetilde{X}_\eta$ . Since $\widetilde{\tau}_*(\mathcal{O}_X)\cong \mathcal{O}_{\widetilde{X}}$ , we have for each $m\geq 0$ that

(3.1) \begin{equation} f_*L_C(\eta)^m \cong (f_\eta)_*\big(\widetilde{\tau}_*(\widetilde{\tau}^*(\mathcal{O}(m))\big) \cong (f_\eta)_*\big(\mathcal{O}(m)\otimes \widetilde{\tau}_*(\mathcal{O}_X)\big) \cong (f_\eta)_* \mathcal{O}(m) \end{equation}

on Y, where $f_\eta\colon \widetilde{X}_\eta\to Y$ satisfies $f=f_\eta\circ \tau$ . Therefore, the section rings of $L_C(\eta)$ on X and $\mathcal{O}(1)$ on $\widetilde{X}_\eta$ are isomorphic, so $\widetilde{X}_\eta\cong \operatorname{Proj} R(X,L_C(\eta))$ as required.

Part (iv) follows by combining part (ii) with Corollary 2.2.

Proof. Conditions 3.3(2) and (3) are independent of the choice of chamber $C'\subset R_C$ , so it suffices to prove Condition 3.3(1) holds for C’. There are only finitely many GIT chambers [Reference ThaddeusTha96, Theorem 2.4], so we proceed by induction. We know Condition 3.3(1) holds for C. For the induction step, let $C_+$ and $C_-$ be adjacent chambers in $R_C$ separated by a GIT wall, where Condition 3.3(1) holds for $C_-$ . Let $\theta_+\in C_+$ and $\theta_-\in C_-$ . Since the wall separating $C_+$ and $C_-$ lies in the interior of $R_C$ , Condition 3.3(2) gives that $X_{\theta_+}$ is normal, and Proposition 2.7 implies that the birational map $\psi \colon X_{\theta{_-}}\dashrightarrow X_{\theta{_+}}$ from diagram (2.4) is an isomorphism in codimension one such that

(3.2) \begin{equation} L_{C_-}(\eta) \cong\psi^*L_{C_+}(\eta)\,\text{ for all }\eta\in G^\vee_{\mathbb{Q}}. \end{equation}

Since $X_{\theta_-}$ is ${{\mathbb{Q}}}$ -factorial and $L_{C_-}$ is surjective, $X_{\theta_+}$ is ${{\mathbb{Q}}}$ -factorial by Proposition 2.7. Pushforward along $\psi$ identifies the class groups of $X_{\theta_\pm}$ , and hence ${{\mathbb{Q}}}$ -factoriality implies that $\psi^*$ identifies the rational Picard groups $\operatorname{Pic}(X/Y)_{{\mathbb{Q}}}\cong \operatorname{Pic}(X_{\theta{_-}}/Y)_{{\mathbb{Q}}}\cong \operatorname{Pic}(X_{\theta{_+}}/Y)_{{\mathbb{Q}}}$ . Since $L_{C_-}$ is an isomorphism, so is $L_{C_+}$ by (3.2). This shows that Condition 3.3(1) holds for $C_+$ , so it holds for each chamber in $R_C$ by induction. This completes the proof of the first statement, while the second statement follows by applying the proof of Lemma 3.9 verbatim to each chamber C’ in $R_C$ .

By taking the composition of the linearisation map $L_{C}$ with the isomorphism from Lemma 3.9(iv), we may identify the target of $L_{C}$ with $N^1(X/Y)$ whenever Condition 3.3 holds.

1. (i) the linearisation map $L_{C}\colon G^{\vee}_{\mathbb{Q}} \rightarrow N^1(X/Y)$ is an isomorphism that identifies the GIT decomposition of the region $R_C$ with the Mori chamber decomposition of $\operatorname{Mov}(X/Y)$ ; and

2. (ii) for any $\eta\in R_C$ , the section ring $R(X,L_C(\eta))$ is a finitely generated $\mathcal{O}_Y$ -algebra, and the model $\operatorname{Proj} R(X,L_C(\eta))$ is the variety $\widetilde{X}_\zeta$ appearing in the Stein factorisation of $X_{\theta'} \to X_\zeta$ for $\theta'$ in a GIT chamber C’ whose boundary contains $\zeta$ .

In particular, every small birational model of X over Y can be obtained as a GIT quotient of the form $\widetilde{X}_\eta$ for some $\eta$ in the interior of $R_C$ .

Conversely, for a given GIT set-up, suppose that there is a chamber C with $X=X_{\theta{}}$ for $\theta\in C$ , such that $L_C$ is an isomorphism of fans between $R_C$ and $\operatorname{Mov}(X/Y)$ . If, in addition, each VGIT morphism for an interior wall of $R_C$ has unstable locus of codimension at least two, then all parts of Condition 3.3 are satisfied.

For the general case, we noted in Remark 3.11 that the linearisation maps $L_{C'}$ for all $C'\subset R_C$ are compatible with taking the strict transform along the appropriate birational map $\psi_{C'}$ . Thus, for the chamber C from Condition 3.3 and for any other chamber $C'\subset R_C$ with $\theta'\in C'$ , we obtain $L_C\big(\overline{C^\prime}\big) = \psi_{C'}^*L_{C^\prime}\big(\overline{C^\prime}\big) = \psi_{C'}^*\big(\operatorname{Nef}(X_{\theta{^\prime}}/Y)\big)$ by (3.2) and Lemma 3.10. Applying $L_C$ to the obvious decomposition $R_C = \bigcup_{C^\prime\subset R_C} \overline{C^\prime}$ gives

(3.3) \begin{equation} L_C(R_C) = \bigcup_{C^\prime\subset R_C} \psi_{C'}^* \big(\operatorname{Nef}(X_{\theta{^\prime}}/Y)\big). \end{equation}

Any wall in the boundary of $L_C(R_C)$ therefore lies in the boundary of $\operatorname{Nef}(X_{\theta{^\prime}}/Y)$ for some $C'\subset R_C$ and $\theta'\in C'$ . Condition 3.3(3) ensures that if $\theta_0$ is general in the corresponding boundary wall of $R_C$ , then the Stein factorisation of the induced VGIT morphism $\tau\colon X_{\theta{^\prime}}\to X_{\theta{_0}}$ is either of fibre type or it is a divisorial contraction, so $L_C$ sends the boundary of $R_C$ into the boundary of $\operatorname{Mov}(X/Y)$ . It follows from (3.3) that $L_C$ maps GIT chambers in $R_C$ to the open ample cones in $\operatorname{Mov}(X/Y)$ of small birational models of X over Y. More generally, it maps GIT cones in $R_C$ to cones of the Mori decomposition of $\operatorname{Mov}(X/Y)$ . This proves (i).

For (ii), let $\eta\in R_C$ and let $C^\prime\subset R_C$ denote any chamber such that $\eta\in \overline{C'}$ . Lemma 3.10 implies that the codomain of the Stein factorisation of $X_{\theta'}\to X_\eta$ for $\theta'\in C'$ satisfies $\widetilde{X}_\eta\cong X(L_{C'}(\eta))$ . Since both C and $C^\prime$ are contained in $R_C$ , the linearisation maps $L_{C'}$ and $L_C$ are compatible with taking the strict transform along the appropriate birational map, giving $\widetilde{X}_\eta \cong X(L_{C'}(\eta))\cong X(L_{C}(\eta))$ .

For the final statement, X is ${{\mathbb{Q}}}$ -factorial and hence so is $X_{\eta}$ for every generic $\eta\in R_C$ by Lemma 3.10. Let X’ be a small birational model of X over Y. Then there is a line bundle L’ in the interior of $\operatorname{Mov}(X/Y)$ satisfying $X(L')\cong X'$ . The character $\theta':=L_C^{-1}(L')$ lies in the interior of $R_C$ and satisfies $\widetilde{X}_{\theta'}\cong X'$ as required.

This result allows us to draw conclusions about the GIT quotients directly from known results in birational geometry as follows.

1. (i) $X_{\theta{}}$ is isomorphic to $X_{\theta{'}}$ over Y if and only if $\theta, \theta'$ lie in the same GIT cone;

2. (ii) if $\theta$ is generic and $\theta'$ in the interior of $R_C$ , then the normalisation $\widetilde{X}_{\theta'}$ of $X_{\theta{'}}$ is the (not necessarily $\mathbb{Q}$ -factorial) small birational model of $X_{\theta{}}$ over Y given by the line bundle $L_C(\theta')$ ;

3. (iii) for any $\theta$ in the interior of $R_C$ , the variety $X_{\theta{}}$ is ${{\mathbb{Q}}}$ -factorial if and only if $\theta$ is generic;

4. (iv) the dimension of $N^1(\widetilde{X}_{\eta}/Y)$ equals the dimension of the minimal GIT cone containing $\eta$ , where for any GIT chamber C’ with $\eta\in \overline{C'}$ and $\theta'\in C'$ , we write $X_{\theta'}\to \widetilde{X}_{\eta}$ for the Stein factorisation of the VGIT morphism $X_{\theta'}\to X_{\eta}$ .

For (ii), the interior of $R_C$ is identified with the interior of $\operatorname{Mov}(X/Y)$ , so the GIT quotients $X_{\theta{'}}$ and $X_{\theta{}}$ are isomorphic in codimension one over Y. Moreover, if $C' \subseteq R_C$ is a GIT chamber with $\theta' \in \overline{C'}$ , then the codomain of the Stein factorisation $X_{\theta{''}} \to \widetilde{X}_{\theta'}$ of the VGIT morphism for $\theta'' \in C'$ is just the normalisation of $X_{\theta{'}}$ , as noted after (2.7). The result then follows from Theorem 3.12(ii).

For (iii), one direction was noted in Lemma 3.10 while for the other, it is well known that the base of a flip is not ${{\mathbb{Q}}}$ -factorial. Indeed, if $X_{\theta{_0}}$ were ${{\mathbb{Q}}}$ -factorial, then, in the notation of Proposition 2.7, any Cartier divisor D satisfying $\mathcal{O}_{X_{\theta{_-}}}(D)=L_{C_-}(\theta_+)$ would define a Weil divisor $(\tau_-)_*(D)$ on $X_{\theta{_0}}$ , making mD and $(\tau_-)^*(\tau_-)_*(mD)$ linearly equivalent for some $m\gt0$ . However, the intersection numbers of these divisors with respect to a curve contracted by $\tau_-$ are negative and zero, respectively.

For (iv), let F be the minimal GIT cone containing $\eta$ , let $C'\subset R_C$ be a chamber containing F in its closure, and let $\theta'\in C'$ . Lemma 3.10 implies that $L_{C}$ identifies F with the minimal face of $\operatorname{Nef}(X_{\theta{^\prime}}/Y)$ containing $L_C(\theta)$ . By (2.3), every line bundle in the interior of this face is the pullback of an ample bundle via the Stein factorisation $\widetilde{\tau}\colon X_{\theta'}\to \widetilde{X}_\eta$ of the VGIT morphism $\tau\colon X_{\theta{^\prime}}\to X_\eta$ , so $\dim L_C(F) \leq \dim (\widetilde{\tau})^*N^1(\widetilde{X}_{\eta}/Y)$ . On the other hand, $L_C(F)$ is dual via the intersection pairing to the face $\sigma$ of the Mori cone of curves generated by the numerical classes of curves contracted by $\widetilde{\tau}$ , so $\dim L_C(F) = \dim (\sigma^\perp)$ for $\sigma^\perp =\{L\in N^1(\widetilde{X}_{\theta'}/Y) \mid \deg L\vert_\ell =0 \;\forall \; \ell\in \sigma\}$ . The pullback via $\widetilde{\tau}$ of any class in $N^1(\widetilde{X}_\eta/Y)$ has degree zero on each generator of $\sigma$ , so $(\widetilde{\tau})^*N^1(\widetilde{X}_\eta/Y)\subseteq \sigma^\perp$ and hence $\dim (\widetilde{\tau})^*N^1(\widetilde{X}_\eta/Y)\leq \dim (\sigma^\perp) = \dim L_C(F)$ . The map $\tau^*$ is injective, so $\dim F = \dim L_C(F) = \dim N^1(\widetilde{X}_\eta/Y)$ as required.

3.4 Relative Mori dream spaces

Example 3.5 illustrates that the GIT construction of any Mori dream space via its Cox ring gives rise to a GIT chamber that satisfies our Condition 3.3. The next result provides a partial converse, but we emphasise that even for a Mori dream space, we are typically interested in applying our Condition 3.3 for new GIT descriptions that do not involve the Cox ring directly.

1. (i) X is normal and ${{\mathbb{Q}}}$ -factorial;

2. (ii) $\operatorname{Pic}(X/Y)_{{\mathbb{Q}}}\cong N^1(X/Y);$

3. (iii) the relative nef cone $\operatorname{Nef}(X/Y)$ is generated by finitely many semi-ample line bundles; and

4. (iv) there exist $k\geq 0$ and ${{\mathbb{Q}}}$ -factorial varieties $X=X_0, X_1, \dots, X_k$ , each projective over Y, as well as birational maps $\psi_i\colon X\dashrightarrow X_i$ over Y for $0\leq i\leq k$ that are isomorphisms in codimension one, such that

   (3.4) \begin{equation} \operatorname{Mov}(X/Y) = \bigcup_{0\leq i\leq k} \psi_i^*\;\operatorname{Nef}(X_i/Y),\end{equation}
   where each cone in this description is generated by finitely many semi-ample line bundles.

3.5 Strong convexity and injectivity

In practice, injectivity of $L_C$ can often follow from the following simple criterion.

A cone is called strongly convex if the origin is a face. Given a fan, the origin is a cone if and only if all cones are strongly convex.

Given a chamber C, let $K_C$ be the intersection of all supporting hyperplanes of the walls of C. The following lemma is standard.

In particular, $K_C$ does not depend on the choice of chamber, and it is zero if and only if all cones are strongly convex. Let us simply call it K from now on.

Proof. By definition, if $\theta$ is a non-trivial effective character, then any G-fixed point is $\theta$ -unstable. So $V^{\theta}\neq V=V^0$ , and hence $\{0\}$ is a GIT cone.

In particular, if V is conical and the G action is equivariant, then the origin is a G-fixed point.

Putting these together, we conclude that the following useful criterion for Condition 3.3 holds.

1. (1) the closed cone $\overline{C}$ is strongly convex, the map $L_C$ is surjective, and for $\theta\in C$ the GIT quotient $X:=X_{\theta{}}$ is a ${{\mathbb{Q}}}$ -factorial, normal variety;

2. (2) for any GIT chambers $C_\pm$ in $R_C$ sharing a wall, the VGIT morphisms $\tau_{\pm}$ from (2.6) both contract a curve and they each map their unstable locus to the singular locus of $X_{\theta{_0}}$ which is of codimension at least two; and

3. (3) each boundary wall of $R_C$ is either divisorial or of fibre type.

Then Condition 3.3 is satisfied.

3.6 Minimal models of Gorenstein singularities

In this section we assume, in addition, that Y has Gorenstein singularities and that $f\colon X \to Y$ is a projective crepant resolution, or more generally, a projective ${{\mathbb{Q}}}$ -factorial terminalisation, that is, a projective crepant birational morphism from a variety with ${{\mathbb{Q}}}$ -factorial terminal singularities.

Suppose that there exists another projective ${{\mathbb{Q}}}$ -factorial terminalisation $f'\colon X'\to Y$ . Then X and X’ are birational minimal models over Y, so by [Reference Kollár and MoriKM98, Theorem 3.52] there is a movable line bundle L on X such that $X'\cong X(L)$ , and the morphism $f'=f_{L}$ fits into a commutative diagram (2.2). In particular, if there is a GIT construction such that $X\cong X_{\theta{}}$ for $\theta\in C$ as in Condition 3.3, then Theorem 3.12 implies that there is a chamber $C^\prime$ in the GIT region $R_C$ such that $X^\prime \cong X_{\theta{^\prime}}$ for $\theta^\prime\in C^\prime$ . More generally, we have the following corollary.

1. (i) If a projective, crepant morphism $g \colon Z \to Y$ is dominated by a projective ${{\mathbb{Q}}}$ -factorial terminalisation $f'\colon X'\to Y$ , then there exist a chamber $C^\prime$ in $R_C$ and $\eta\in \overline{C'}$ such that $Z\cong X_{\eta}$ and $g=f_\eta$ . Moreover, Z has terminal singularities if and only if $\eta$ lies in the interior of $R_C$ .

2. (ii) Conversely, for all $\eta\in R_C$ , the morphism $f_\eta\colon X_\eta\to Y$ is projective and crepant.

For (ii), each $\eta\in R_C$ lies in the closure of some chamber $C^\prime\subseteq R_C$ , so for $\theta^\prime\in C^\prime$ the morphism $f_{\theta^\prime}\colon X_{\theta{^\prime}}\to Y$ factors via $f_\eta\colon X_\eta\to Y$ . Since $f_{\theta^\prime}$ is crepant, so too is $f_\eta$ .

3.7 Weakening the hypotheses

We conclude this section by noting that our results hold in a slightly more general context.

Rather than assume that $L_C$ is an isomorphism in Condition 3.3(1), suppose instead that there exists an affine subspace $\Lambda \subseteq G^\vee_{{\mathbb{Q}}}$ that is not contained in the linear span of any GIT wall, such that the restriction of $L_C$ to $\Lambda$ is an affine isomorphism $\Lambda\cong \operatorname{Pic}(X/Y)_{{{\mathbb{Q}}}}$ . If one replaces the study of chambers C in $G^\vee_{{\mathbb{Q}}}$ by the open cones $C\cap \Lambda$ in $\Lambda$ , and the closed GIT region $R_C$ by $R_C\cap \Lambda$ , then the proof of Theorem 3.12 applies verbatim if one assumes the analogue of Condition 3.3 in this context.

One might hope to weaken Condition 3.3(1) further to require only surjectivity of $L_C$ . However, as Corollary 3.23 shows, if $L_C$ is not injective, then there are two cases to consider. Firstly, if $\overline{C}$ is not strongly convex, then it is natural to study a transverse slice $\Lambda$ to the maximal linear subspace in $\overline{C}$ , as described above. The second case is where C has a wall for which the VGIT morphism does not contract a curve. In this latter case, one might hope to restrict attention to an affine subspace that intersects none of these walls (such walls are called ‘fake’ or ‘type $\operatorname{0}$ ’ in the literature); this is precisely the situation described in Example 3.28 above.

4.1 Quiver varieties

Choose an arbitrary finite graph with vertices $0,\dots,r$ and let H be the set of pairs consisting of an edge, together with an orientation on it. Let $\operatorname{tl}(a)$ and $\operatorname{hd}(a)$ denote the tail and head respectively of the oriented edge $a \in H$ . Let $a^*$ denote the same edge, but with opposite orientation. We fix an orientation of the graph, that is, a subset $\Omega \subset H$ such that $\Omega \cup \Omega^* = H$ and $\Omega \cap \Omega^* = \emptyset$ . Then $\epsilon : H \rightarrow \{ \pm 1 \}$ is defined to take value 1 on $\Omega$ and $-1$ on $\Omega^*$ .

Fix collections $V_0, \dots, V_r$ and $W_0, \dots, W_r$ of finite-dimensional vector spaces over ${{\Bbbk}}$ and set

$$\mathbf{v} = (\dim V_0, \dots, \dim V_r), \, \mathbf{w} = (\dim W_0, \dots, \dim W_r).$$

The group $G(\mathbf{v}) := \prod_{k = 0}^r \operatorname{GL}(V_k)$ acts naturally on the space

$$\mathbf{M}(\mathbf{v},\mathbf{w}) := \bigg(\bigoplus_{a \in H} \operatorname{Hom}_{{{\Bbbk}}}(V_{\operatorname{tl}(a)}, V_{\operatorname{hd}(a)})\bigg) \oplus \bigg(\bigoplus_{k = 0}^r \big(\operatorname{Hom}_{{{\Bbbk}}}(V_k,W_k) \oplus \operatorname{Hom}_{{{\Bbbk}}}(W_k,V_k)\big)\bigg).$$

This action of $G(\mathbf{v})$ is Hamiltonian for the natural symplectic structure on $\mathbf{M}(\mathbf{v},\mathbf{w})$ and, after identifying the dual of $\mathfrak{g}(\mathbf{v}) := \operatorname{Lie} G(\mathbf{v})$ with $\mathfrak{g}(\mathbf{v})$ via the trace pairing, the corresponding moment map ${{\mu}} \colon \mathbf{M}(\mathbf{v},\mathbf{w}) \rightarrow \mathfrak{g}(\mathbf{v})$ satisfies

$${\mathbf{\mu}}(B,i,j) = \bigg( \sum_{\operatorname{hd}(a) = k} \epsilon(a) B_a B_{a^*} + i_k j_k \bigg)_{k = 0}^r$$

where $i_k \in \operatorname{Hom}_{{{\Bbbk}}}(W_k,V_k), j_k \in \operatorname{Hom}_{{{\Bbbk}}}(V_k,W_k)$ and $B_a \in \operatorname{Hom}_{{{\Bbbk}}}(V_{\operatorname{tl}(a)},V_{\operatorname{hd}(a)})$ . Though one can talk about arbitrary stability conditions in this context, as was done in [Reference NakajimaNak09], it is easier in our case to apply the trick of Crawley-Boevey [Reference Crawley-BoeveyCB01] and reduce to the case where each $W_k = 0$ by introducing a framing vertex.

The set H associated to the graph can be thought of as the arrow set of a quiver. We frame this quiver by adding an additional vertex $\infty$ , as well as $\mathbf{w}_i$ arrows from vertex $\infty$ to vertex i and another $\mathbf{w}_i$ arrows from vertex i to vertex $\infty$ . This framed (doubled) quiver is denoted by $Q = (Q_0,Q_1)$ , where $Q_0 = \{ \infty, 0, \dots, r \}$ . Each dimension vector $\mathbf{v} = (\dim V_0, \dots, \dim V_r)$ for the original graph determines a dimension vector $\alpha:= (1,\mathbf{v}) = (1,\dim V_0, \dots, \dim V_r)$ for Q. We may identify $\mathbf{M}(\mathbf{v},\mathbf{w})$ with the space

\[\operatorname{Rep}(Q,\alpha):= \bigoplus_{a\in Q_1} \operatorname{Hom}_{{{\Bbbk}}}({{\Bbbk}}^{\alpha_{\operatorname{tl}(a)}},{{\Bbbk}}^{\alpha_{\operatorname{hd}(a)}}) \]

of representations of Q of dimension vector $\alpha$ in such a way that the $G(\mathbf{v})$ -action on $\mathbf{M}(\mathbf{v},\mathbf{w})$ corresponds to the action of the group $G(\alpha):=\big(\prod_{i\in Q_0} \operatorname{GL}(\alpha_i)\big)/{{\Bbbk}}^\times$ on $\operatorname{Rep}(Q,\alpha)$ by conjugation and, moreover, the above map ${\mathbf{\mu}}$ corresponds to the moment map $\mu$ induced by this $G(\alpha)$ -action on $\operatorname{Rep}(Q,\alpha)$ .

From now on, we assume that $\mathbf{v}$ satisfies $v_i\neq 0$ for all $0\leq i\leq r$ , and $\mathbf{w} \neq 0$ . This is equivalent to choosing a dimension vector $\alpha$ for Q with component $\alpha_{\infty} = 1$ and $\alpha_i \neq 0$ for all i (and Q must have at least one arrow from some vertex i to vertex $\infty$ ). Then the rational vector space

\[\Theta := \big\{\theta \in \operatorname{Hom}(\mathbb{Z}^{Q_0},\mathbb{Q}) \mid \theta(\alpha)=0\big\} \]

satisfies $G(\alpha)^{\vee} \otimes_{\mathbb{Z}} \mathbb{Q} = \Theta$ , where $\chi_\theta(g) = \prod_{i \in Q_0} \det(g_i)^{\theta_i}$ for $g\in G(\alpha)$ . For $\theta\in \Theta$ , after replacing $\theta$ by a positive multiple if necessary, the (Nakajima) quiver variety associated to $\theta$ is the categorical quotient

\[ \mathfrak{M}_\theta(\mathbf{v},\mathbf{w}) := \mu^{-1}(0){{/\!\!/\!}}_{\chi_\theta} G(\alpha)= \mu^{-1}(0)^{\theta}/\!\!/G(\alpha) = \operatorname{Proj}\bigoplus_{k\geq 0} {{\Bbbk}}[\mu^{-1}(0)]^{\chi_{k\theta}}, \]

where $\mu^{-1}(0)^{\theta}$ denotes the locus of $\chi_\theta$ -semi-stable points in $\mu^{-1}(0)$ and ${{\Bbbk}}[\mu^{-1}(0)]^{\chi_{k\theta}}$ is the $\chi_{k\theta}$ -semi-invariant slice of the coordinate ring of the locus $\mu^{-1}(0)$ . Note that ${{\Bbbk}}^\times$ acts on $\mathbf{M}(\mathbf{v},\mathbf{w})$ by scaling, and this action descends to an action on $\mathfrak{M}_\theta(\mathbf{v},\mathbf{w})$ .

For a more algebraic description of $\mathfrak{M}_\theta(\mathbf{v},\mathbf{w})$ , extend $\epsilon$ to Q by setting $\epsilon(a) = 1$ if $a \colon \infty \rightarrow i$ and $\epsilon(a) = -1$ if $a \colon i \rightarrow \infty$ . The preprojective algebra $\Pi$ is the quotient of the path algebra ${{\Bbbk}} Q$ by the relation

(4.1) \begin{equation} \sum_{a \in Q_1} \epsilon(a) aa^* = 0.\end{equation}

Given $\theta\in \Theta$ , we say that a $\Pi$ -module M of dimension vector $\alpha$ is $\theta$ -semi-stable if $\theta(N)\geq 0$ for all submodules $N\subseteq M$ , and it is $\theta$ -stable if $\theta(N)\gt0$ for all proper non-zero submodules. Two $\theta$ -semi-stable $\Pi$ -modules are S-equivalent if their composition series agree in the abelian category of $\theta$ -semi-stable $\Pi$ -modules. A finite-dimensional $\Pi$ -module is said to be $\theta$ -polystable if it is a direct sum of $\theta$ -stable $\Pi$ -modules. King [Reference KingKin94] proved that a $\Pi$ -module M of dimension vector $\alpha$ is $\theta$ -semi-stable (respectively, $\theta$ -stable) if and only if the corresponding point of $\mu^{-1}(0)$ is $\chi_\theta$ -semi-stable (respectively, $\chi_\theta$ -stable) in the sense of GIT. In fact [Reference KingKin94, Propositions 3.2, 5.2] establishes that the quiver variety $\mathfrak{M}_\theta(\mathbf{v},\mathbf{w})$ is the coarse moduli space of S-equivalence classes of $\theta$ -semi-stable $\Pi$ -modules of dimension vector $\alpha$ , where the closed points of $\mathfrak{M}_\theta(\mathbf{v},\mathbf{w})$ are in bijection with the $\theta$ -polystable representations of $\Pi$ of dimension $\alpha$ . We write $\mathfrak{M}_\theta(\mathbf{v},\mathbf{w})^s$ for the (possibly empty) open subset of $\mathfrak{M}_\theta(\mathbf{v},\mathbf{w})$ parametrising $\theta$ -stable representations.

4.2 Kirwan surjectivity

Recall that $\mathbf{w}\neq 0$ and $v_i \neq 0$ for all i, with $\alpha:= (1,\mathbf{v})\in \mathbb{N}^{Q_0}$ . In this case, there exist $\theta$ that are non-degenerate in the sense of [Reference McGerty and NevinsMN18, Definition 3.1], but $\mu^{-1}(0)^{\theta}$ can be empty. Let us first recall the condition on $\alpha$ which excludes this.

Associated to the quiver Q is a root system R with positive roots $R^+ = R \cap {{\mathbb{Z}}}_{\ge 0}^{Q_0}$ . We set $R^+_{\theta} = \{ \gamma \in R^+ \, | \, \theta(\gamma) = 0 \}$ .

1. (i) $\alpha \in R^+$ ;

2. (ii) each $\theta\in \Theta$ is effective, that is, $\mu^{-1}(0)^{\theta} \neq \emptyset$ ;

3. (iii) $G(\alpha)$ acts freely on $\mu^{-1}(0)^{\theta}$ for $\theta$ in some dense open subset of $\Theta$ .

Moreover, when these conditions hold, the GIT chambers are precisely the interiors of the top-dimensional cones of the GIT fan.

In the preceding paragraph, we noted that each general $\theta \in \Theta$ is generic. This is precisely the final statement.

By [Reference Bellamy and SchedlerBS21, Theorem 1.15], the smooth locus of $\mathfrak{M}_{\theta}(\mathbf{v},\mathbf{w})$ equals the canonically polystable locus, which is the locus where the decomposition of polystable representations into a sum of stable ones is of generic type. In particular, if the stable locus is non-empty, then the smooth locus equals the stable locus $\mathfrak{M}_\theta(\mathbf{v},\mathbf{w})^s$ .

For $\theta \ge \theta_0$ , the morphism $f_{\theta} \colon \mathfrak{M}_{\theta}(\mathbf{v},\mathbf{w}) \to \mathfrak{M}_{\theta_0}(\mathbf{v},\mathbf{w})$ need not be surjective. However, it is birational onto its image [Reference Bellamy and CrawBC20, Theorem A.1]. Hence, it is not of fibre type. If the stable locus with respect to $\theta_0$ is empty then it can happen that $f_{\theta}$ is an isomorphism onto (the normalisation of) its image, showing that ‘fake’ walls (also known as walls of ‘type $\operatorname{0}$ ’; see § 6.3) exist.

As a consequence of Kirwan surjectivity, established in [Reference McGerty and NevinsMN18], we note the following theorem.

The main result of [Reference McGerty and NevinsMN18] says that the map $G^{\vee} \to H^2(\mathfrak{M}_{\theta}(\mathbf{v}, \mathbf{w}),\mathbb{Z})$ is surjective. Since $\mathfrak{M}_{\theta}(\mathbf{v}, \mathbf{w})$ is smooth, [Reference NakajimaNak01, Theorem 7.3.5] says that the cycle map $\operatorname{Pic}(\mathfrak{M}_{\theta}(\mathbf{v},\mathbf{w})) \to H^2(\mathfrak{M}_{\theta}(\mathbf{v}, \mathbf{w}),\mathbb{Z})$ is an isomorphism. We note that the paper [Reference NakajimaNak01] assumed that the quiver Q has no loops. However, the proof of [Reference NakajimaNak01, Theorem 7.3.5] does not require this restriction.

The map $L_C$ is obtained from $\ell_C$ by tensoring by $\mathbb{Q}$ , so is also surjective under the hypotheses of the theorem.

4.3 Applying the main result

Let $\langle - , -\rangle$ denote the Ringel form on $\mathbb{Z}^{Q_0}$ and $( - , - )$ its symmetrisation. For any $\gamma\in R$ , define $p(\gamma) := 1 - {1}/{2}(\gamma,\gamma)$ .

Definition 4.4. Define $\Sigma_{\theta}$ to be the set of $\gamma \in R^+_{\theta}$ such that

\[p(\gamma) \gt p\big(\beta^{(1)}\big) + \cdots + p\big(\beta^{(k)}\big) \]

for every proper decomposition $\gamma = \beta^{(1)} + \cdots + \beta^{(k)}$ with $\beta^{(i)} \in R^+_{\theta}$ .

Crawley-Boevey [Reference Crawley-BoeveyCB01, Theorem 1.2] showed that $\alpha \in \Sigma_{0}$ if and only if there exists a simple (= 0-stable) $\Pi$ -module of dimension vector $\alpha$ . More generally, it is shown in [Reference Bellamy and SchedlerBS21, Theorem 1.3] that there exists a $\theta$ -stable $\Pi$ -module of dimension $\alpha$ if and only if $\alpha \in \Sigma_{\theta}$ . Since $\alpha=(1,\mathbf{v})$ is indivisible, every $\theta$ -semi-stable representation will be $\theta$ -stable if $\theta(\beta) \neq 0$ for all roots $\beta \lt \alpha$ . That is, the GIT walls are contained in the union of hyperplanes $\beta^\perp$ . For a precise description of the GIT walls, see § 4.5 below.

The morphism $\tau \colon \mathfrak{M}_{\theta}(\mathbf{v},\mathbf{w}) \to \mathfrak{M}_{\theta_0}(\mathbf{v},\mathbf{w})$ is a surjective birational morphism because $\alpha \in \Sigma_0$ . Moreover, since $\theta_0 \in \overline{C} \smallsetminus C$ , there is a $\theta$ -stable $\Pi$ -module M of dimension $\alpha$ that is not $\theta_0$ -stable. Therefore, the image under $\tau$ of the corresponding point $[M]\in \mathfrak{M}_\theta(\mathbf{v},\mathbf{w})$ is strictly $\theta_0$ -polystable. The previous paragraph implies that $\tau[M]$ lies in the singular locus of $\mathfrak{M}_{\theta_0}(\mathbf{v}, \mathbf{w})$ , so $\tau$ contracts at least one curve by Zariski’s Main Theorem [Reference HartshorneHar77, III, Corollary 11.4].