Tropicalisation.

Let $(X,\mathcal M_X)$ be a logarithmic scheme, where the sheaf of monoids $\mathcal M_X$ gives the divisorial logarithmic structure with respect to some simple normal crossing divisor $D\subset X$ . Explicitly, for an open subset $U \subseteq X$ , the sheaf $\mathcal M_X$ is given by

Then we can associate a fan $\Sigma _X$ to this in the following way. Recall that the characteristic sheaf $\overline {\mathcal M}_X$ for the divisorial logarithmic structure is defined by

$$\begin{align*}\overline{\mathcal M}_X = \mathcal M_{X}/\mathcal O_X^*. \end{align*}$$

The sheaf $\mathcal M_{X}$ records functions vanishing at most on D and $\overline {\mathcal M}_X$ records their vanishing orders. For each point $x\in X$ , there is an isomorphism $\overline {\mathcal M}_{X,x} \cong \mathbb N^k$ , where k is the number of components of D which contain x. We let

This will be contained in $\mathbb R^r_{\geq 0}$ , where r is the number of components of D, since D is a simple normal crossing divisor. We call $\Sigma _X$ the tropicalisation of X.

Subdivisions of the tropicalisation define expansions of X.

In the following, we will want to study possible monomial birational modifications of the scheme X around the divisor D. In the tropical language, these are expressed as subdivisions.

Definition 2.1.1. Let $\Upsilon $ be a fan, let $|\Upsilon |$ be its support and $\upsilon $ be a continuous map

$$\begin{align*}\upsilon\colon |\Upsilon| \longrightarrow \Sigma_X \end{align*}$$

such that the image of every cone in $\Upsilon $ is contained in a cone of $\Sigma _X$ and that is given by an integral linear map when restricted to each cone in $\Upsilon $ . We say that $\upsilon $ is a subdivision if it is injective on the support of $\Upsilon $ and the integral points of the image of each cone $\tau \in \Upsilon $ are exactly the intersection of the integral points of $\Sigma _X$ with $\tau $ .

A subdivision of the tropicalisation defines a birational modification of X in the following way (for a reference on toric geometry, see [Reference Fulton5]). The subdivision

has an associated toric variety $\mathbb A_{\Upsilon }$ , which comes with a $\mathbb G^r_m$ -equivariant birational map $\mathbb A_\Upsilon \to \mathbb A^r$ . Then we have an induced morphism of quotient stacks

$$\begin{align*}[\mathbb A_\Upsilon/\mathbb G_m^r] \longrightarrow [\mathbb A^r/\mathbb G_m^r] \end{align*}$$

and we may define the induced birational modification of X to be

Visualising the problem.

Here, we describe how to visualise the tropicalisation arising from the divisorial logarithmic structure on X associated to a simple normal crossing divisor $D\subset X$ . We explain this for the case which interests us here, that is, we assume that $X\to C$ is locally given by $\operatorname {\mathrm {Spec}} k[x,y,z,t]/(xyz-t)$ and the boundary divisor is .

Given a divisorial logarithmic structure on X, the tropicalisation is a fan or cone complex which for each defining function of the divisor records the degree of vanishing of this function in X. Here, the functions vanishing at D will be $x,y$ and z. As $X_0$ is made up of three components meeting in a point, in this case we may actually represent $\Sigma _X$ as a fan in $\mathbb R^3_{\geq 0}$ , given by the positive orthant and its faces, as can be seen in Figure 1. In this image the three half-lines correspond to the divisors $Y_1,\ Y_2$ and $Y_3$ in X. We may think of these half-lines as recording the orders of vanishing in $x,y$ and z respectively. The 2-dimensional faces spanned by two half-lines correspond to the orders of vanishing in both coordinates vanishing at $Y_i\cap Y_j$ , and the three-dimensional interior of the cone corresponds to the triple intersection point $Y_1\cap Y_2\cap Y_3$ , i.e. records orders of vanishing in all three variables. For convenience, we shall refer to this tropicalisation as $\operatorname {\mathrm {trop}}(X)$ in later sections.

Figure 1 Tropicalisation of X.

The dual complex of $X_0$ (see [Reference de Fernex, Kollár and Xu4] for a definition) can be visualised by taking a hyperplane slice through the cone in Figure 1; this yields a triangle with vertices corresponding to $Y_1,Y_2$ and $Y_3$ in $X_0$ , edges between these vertices corresponding to the lines $Y_i\cap Y_j$ , and 2-dimensional interior corresponding to the point $Y_1\cap Y_2\cap Y_3$ , as pictured in Figure 2. We shall abuse notation and refer to the dual complex of $X_0$ as $\operatorname {\mathrm {trop}}(X_0)$ .

Figure 2 Tropicalisation of $X_0$ .

Recall that $C\cong \mathbb A^1$ and it can be considered to be a logarithmic scheme with divisorial logarithmic structure given by $0\in C$ . The fan of $\mathbb A^1$ is a half-line with a distinguished vertex. The map $X\to C$ can be seen as a log smooth morphism of log schemes and, in this case, tropicalisation is functorial. Making a choice of point on this half-line therefore corresponds to choosing a height for the triangle within the cone $\mathbb R^3_{\geq 0}$ . Geometrically, we can think of changing the height of the triangle as making a finite base change on X.

Tropicalisation map.

We construct a tropicalisation map which takes points of $X^\circ $ to $\Sigma _X$ , as in Section 1.4 of [Reference Maulik and Ranganathan12]. We recall the details of this map here. We assume that $\mathcal {K}$ is a valued field extending k. First, we take a point of $X^\circ (\mathcal K)$ , given by some morphism $\operatorname {\mathrm {Spec}} \mathcal K \to X^\circ $ . By the properness of X, this extends to a morphism $\operatorname {\mathrm {Spec}} R \to X$ for some valuation ring R. Now, let $P\in X$ denote the image of the closed point by the second morphism. The stalk of the characteristic sheaf at P is given by $\mathbb N^r$ , where r is the number of linearly independent vanishing equations of D at the point P. For example, in our context, if $P\in Y_1\subset X_0$ , then $r = 1$ and $\mathbb N$ is generated by the function x; if $P\in Y_1\cap Y_2$ , then $r=2$ with $\mathbb N^2$ generated by the functions x and y; etc.

Each element of $\mathbb N^r$ corresponds to a function f on X in the neighbourhood of P up to multiplication by a unit and we may then evaluate f with respect to the valuation map associated to $\mathcal K$ . This determines an element of

This gives rise to a morphism

$$\begin{align*}\operatorname{\mathrm{trop}} \colon X^\circ(\mathcal K) \longrightarrow \Sigma_X \end{align*}$$

called the tropicalisation map. Now let the valuation map $\mathcal K\to \mathbb R$ be surjective and let $Z^\circ \subset X^\circ $ be an open subscheme. We denote by $\operatorname {\mathrm {trop}}(Z^\circ )$ the image of the map $\operatorname {\mathrm {trop}}$ restricted to $Z^\circ (\mathcal K)$ . Maulik and Ranganathan are then able to show, based on previous work of Tevelev [Reference Tevelev20] for the toric case and Ulirsch [Reference Ulirsch22] for logarithmic schemes, that given such an open subscheme $Z^\circ \subset X^\circ $ , the subset $\operatorname {\mathrm {trop}}(Z^\circ )$ gives rise to an expansion $X'$ of X in which the closure Z of $Z^\circ $ has the required transversality properties. This gives us a convenient dictionary to move back and forth between the geometric and combinatorial points of view.

The possible tropicalisations of such subschemes, corresponding to expansions on the geometric side, are captured on the combinatorial side by the notion of 1-complexes embedding into $\Sigma _X$ . See [Reference Maulik and Ranganathan12] for precise definitions.

Existence and uniqueness of transverse limits.

Maulik and Ranganathan introduce notion of algebraic transversality, which is equivalent to the admissible condition of Li and Wu (see Definition 5.3.2).

As mentioned above, for an open subscheme $Z^\circ \subset X^\circ $ , we may consider its image $\operatorname {\mathrm {trop}}(Z^\circ )$ under the tropicalisation map. Now recall from Section 2.1 that a subdivision of the tropicalisation $\Sigma _X$ defines an expansion of X. The expansion corresponding to the subdivision given by $\operatorname {\mathrm {trop}}(Z^\circ )$ in $\Sigma _X$ is in general not well behaved. Indeed, it may not be flat when embedded into a larger family or even representable as a scheme. It will therefore be necessary, for each possible $\operatorname {\mathrm {trop}}(Z^\circ )$ , to make a choice of polyhedral subdivision corresponding to an actual blow-up on $X\times \mathbb A^1$ . The choices and transversality conditions are set up so that the logarithmic Hilbert schemes they construct will satisfy the valuative criterion for properness.

Construction of the stacks of expansions.

Maulik and Ranganathan construct a moduli space of possible expansions arising from Tevelev’s procedure. Let us denote the set of isomorphism classes of 1-complexes which embed into $\Sigma _X$ by $|T(\Sigma _X)|$ . Some subtleties arise at this point, namely that in general the space constructed will not be representable as a logarithmic algebraic stack. We refer the reader to [Reference Maulik and Ranganathan12] for details. In brief, the object $|T(\Sigma _X)|$ is a logarithmic stack which cannot be thought of as a scheme. In order to solve this, it is necessary to make some choice of polyhedral subdivision of $|T(\Sigma _X)|$ . Through this procedure, they obtain a moduli space of tropical expansions T, which has the desired cone structure.

This operation results in nonuniqueness, as we are making a choice of polyhedral subdivision and there is in general no canonical choice.

Proper Deligne-Mumford stacks.

In order to construct the universal family $\mathfrak {Y} \subset T\times \Sigma $ , some additional choices must be made. Indeed, as mentioned above, $\operatorname {\mathrm {trop}}(Z^\circ )$ does not in general define a blow-up, so we must make adjustments to ensure representability and flatness when fitting the expansions together into one large family over a large base. Here this is resolved by adding distinguished vertices to the relevant complexes. These added vertices will be 2-valent vertices along edges of the 1-complexes parameterised by T and we call them tube vertices. Geometrically, they look like $\mathbb P^1$ -bundles over curves in $X_0$ (where we took X to be a family of surfaces). Again, this operation is not canonical and results in nonuniqueness.

The addition of these tube vertices in the tropicalisation means that there are more potential components in each expansion, which interferes with the previously set up uniqueness results. Indeed, recall that $\operatorname {\mathrm {trop}}(Z^\circ )$ gave us exactly the right number of vertices in the dual complex in order for each family of subschemes $Z^\circ \subset X^\circ $ to have a unique limit representative. Therefore, to reflect this, Donaldson-Thomas stability asks for subschemes to be DT stable if and only if they are tube schemes precisely along the tube components. We say that a 1-dimensional subscheme is a tube if it is the schematic preimage of a zero-dimensional subscheme in D. In the case of Hilbert schemes of points, this condition will translate simply to a zero-dimensional subscheme Z in a modification $X_0'$ of $X_0$ being DT stable if and only if no tube component contains a point of the support of Z and every other irreducible component of $X_0'$ excluding the original components of $X_0$ contains at least one point of the support of Z.

Maulik and Ranganathan define a subscheme to be stable if it is algebraically transverse and DT stable. In the setting of [Reference Li and Wu11] and [Reference Gulbrandsen, Halle and Hulek6], where the singular locus of $X_0$ is smooth, Li-Wu stability is equal to the stability of Maulik and Ranganathan because there are no tube components. For fixed numerical invariants the substack of stable subschemes in the space of expansions forms a Deligne-Mumford, proper, separated stack of finite type over C.

Comparison with the results of this paper.

The construction we present in this paper has the surprising property that we do not need to label any components as tubes in order for the stack of stable objects we define to be proper. This is an artifact of the specific choices of blow-ups to be included in our expanded degenerations. The work of Maulik and Ranganathan shows us that this is not expected in general. As mentioned in Section 1.3, we will discuss in an upcoming paper how to construct proper stacks of stable objects in cases where different choices of expansions are made and it becomes necessary for us as well to introduce an analogue of the Donaldson-Thomas stability condition for our setting.

First blow-up of the $Y_1$ component.

We start by blowing up $Y_{1} \times _{\mathbb A^1} V(t_1)$ inside $X\times _{\mathbb A^1} \mathbb A^{n+1}$ , where $V(t_i)$ denotes the locus where $t_i=0$ .

Notation. We name the space resulting from this blow-up $X_{(1,0)}$ to signify we have blown up the component $Y_1$ once and the component $Y_2$ zero times.

We can describe this blow-up locally in the following way. The ideal of the blow-up is $I_1=\langle x,t_1\rangle $ . Globally this will correspond to an ideal sheaf $\mathcal I_1$ . Then there is a surjective map of graded rings

$$\begin{align*}A[x_0^{(1)},x_1^{(1)}] \longrightarrow S_1 = \bigoplus_{n\geq 0} I_1^n \end{align*}$$

which maps

$$\begin{align*}x_0^{(1)}\longmapsto x \ \textrm{ and } \ x_1^{(1)}\longmapsto t_1, \end{align*}$$

where . This induces an embedding

and $\operatorname {\mathrm {Proj}}(S_1)$ , i.e. our blow-up, is cut out in $\mathbb P^1\times \operatorname {\mathrm {Spec}} A $ by the equations

$$ \begin{align*} x_0^{(1)}t_1 &= xx_1^{(1)} \\ x_0^{(1)} yz &= x_1^{(1)} t_2 \cdots t_{n+1}. \end{align*} $$

Proof. The locus $V(x,t_1)$ where x and $t_1$ vanish in $X\times _{\mathbb A^1} \mathbb A^{n+1}$ is a Weil divisor. This means that blowing $X\times _{\mathbb A^1} \mathbb A^{n+1}$ up along this divisor does nothing except where $V(x,t_1)$ intersects the singular locus of the total space. Let $X_{(1,0)}\to \mathbb A^{n+1}$ be the natural projection. Then the fibres above $(t_1,\ldots , t_{n+1})$ where $t_1$ is nonzero are still the same after the blow-up and so are the fibres where $t_1=0$ and all the other $t_i$ are nonzero because the total space is still smooth at all points of these fibres. This is because, in our étale local coordinates, the total space $X\times _{\mathbb A^1} \mathbb A^{n+1}$ is given by

$$\begin{align*}\operatorname{\mathrm{Spec}}[x,y,z,t_1,\dots,t_{n+1}]/(xyz-t_1\cdots t_{n+1}). \end{align*}$$

At a point $P\in X\times _{\mathbb A^1} \mathbb A^{n+1}$ where all but one of the $t_i$ are invertible, the dimension of the Zariski tangent space is equal to that of the above quotient ring.

At points where $t_1=0$ and at least one of the other $t_i$ is zero, however, singularities of the total space occur where more than one of the coordinates $x,y,z$ is zero, i.e. at the intersections of the $Y_i$ components. Our blow-up therefore causes a new component to appear around the $Y_1$ component.

Notation. We denote by $\Delta _1^{(1)}$ the new reducible exceptional component introduced by the blow-up which is described in the proof of Proposition 3.1.1 above.

This blow-up is represented in Figure 4, where the added red vertices in the tropical picture (on the right) correspond to the two irreducible components of $\Delta _1^{(1)}$ and the edge connecting them corresponds to the intersection of these irreducible components. In the geometric picture (on the left), we can see clearly that an exceptional has been added at the intersection of $Y_1$ with $Y_2$ and $Y_3$ in fibres where $t_1=t_i=0$ , as this is the intersection of $V(x,t_1)$ with the singular locus of $X\times _{\mathbb A^1} \mathbb A^{n+1}$ .

Figure 4 Geometric (left) and tropical (right) pictures of a fibre in $X_{(1,0)}$ where $t_1=t_i=0$ .

Further blow-ups of the $Y_1$ component.

Let $b_{(1,0)} \colon X_{(1,0)} \to X\times _{\mathbb A^1}\mathbb A^{n+1} $ be the map defined by the first blow-up given above. We then proceed to blow-up $b_{(1,0)}^*(Y_{1} \times _{\mathbb A^1} V(t_2))$ inside $X_{(1,0)}$ . We name the resulting space $X_{(2,0)}$ and the composition of both blow-ups is denoted $b_{(2,0)} \colon X_{(2,0)} \to X\times _{\mathbb A^1}\mathbb A^{n+1}$ . We continue to blow up each $b_{(k-1,0)}^*(Y_{1} \times _{\mathbb A^1} V(t_k))$ inside $X_{(k-1,0)}$ for each $k\leq n$ . The resulting space is denoted $X_{(n,0)}$ . Each blow-up adds an additional reducible exceptional component. On the tropical side, this corresponds to adding two vertices to the triangle connected by an edge, as can be seen in Figure 5. Finally, we denote by

$$\begin{align*}\beta^1_{(k,0)} \colon X_{(k,0)} \longrightarrow X_{(k-1,0)} \end{align*}$$

the morphisms corresponding to each individual blow-up. We therefore have the equality

$$\begin{align*}\beta^1_{(k,0)}\circ \cdots \circ \beta^1_{(1,0)} = b_{(k,0)} \end{align*}$$

Figure 5 Geometric (left) and tropical (right) pictures of a fibre in $X_{(2,0)}$ where $t_1=t_2=t_3=0$ .

Proposition 3.1.3. The fibre where $t_i=0$ for all $i\in \{1,\ldots , n+1\}$ has exactly n expanded $\Delta _1$ -components. The equations of the blow-ups in local coordinates are as follows:

(3.1.1) $$ \begin{align} x_0^{(1)}t_1 &= xx_1^{(1)}, \nonumber \\ x_1^{(k-1)}x_0^{(k)}t_k &= x_0^{(k-1)}x_1^{(k)}, \qquad \textrm{ for } \ 2\leq k\leq n, \\ x_0^{(n)} yz &= x_1^{(n)} t_{n+1}. \nonumber \end{align} $$

Proof. Locally, we may think of the ideal of the first blow-up as $\langle x,t_1 \rangle $ . It introduces projective coordinates $(x_0^{(1)}:x_1^{(1)})$ . We may think of the ideal of the second blow-up as $\langle x_0^{(1)}/x_1^{(1)},t_2 \rangle $ , introducing projective coordinates $(x_0^{(2)}:x_1^{(2)})$ , and so on. We therefore get equations

$$ \begin{align*} x_0^{(1)}t_1 &= x_1^{(1)}x, \\ x_1^{(1)}x_0^{(2)}t_2 &= x_0^{(1)}x_1^{(2)}, \\ &\dots, \\ x_1^{(n-1)}x_0^{(n)}t_n& = x_0^{(n-1)}x_1^{(n)}. \end{align*} $$

Combining these equations with the equation $xyz=t_1\cdots t_{n+1}$ yields (3.1.1). From there, it is easy to see that when all $t_i$ vanish, we see n expanded $\Delta _1$ -components.

We label by $\Delta _1^{(k)}$ the $\Delta _1$ -component resulting from the k-th blow-up. The exceptional coordinates $(x_0^{(k)}:x_1^{(k)})$ are nonzero on its interior.

In fibres of the construction where $\Delta _1^{(k)}$ , for some k, is not expanded out, i.e. not contracted by some map $\beta ^1_{(i,0)}$ , we will want to think of it in the following way.

The following proposition and Example 3.1.7 illustrate what is meant by this notion of equal components.

Proof. These properties can be shown by studying the local equations of the blow-ups. For 1., the coordinates introduced by the j-th blow-up are proportional to $yz$ . This follows from the equality

$$\begin{align*}x_0^{(j)} yz = x_1^{(j)} t_{j+1} \cdots t_{n+1}, \end{align*}$$

obtained from the above equations of the blow-ups, and from the assumption that $t_{j+1},\dots ,t_{n+1}\neq 0$ .

For 2., it follows from the equality

$$\begin{align*}x_0^{(j)}t_1\cdots t_j = xx_1^{(j)}, \end{align*}$$

obtained from the equations of the blow-ups, and from the assumption that $t_{1},\dots ,t_{j}\neq 0$ .

For 3., it follows from the equality

$$\begin{align*}x_0^{(i)}x_1^{(j)} = x_1^{(i)}t_{i+1}\cdots t_j x_0^{(j)} \end{align*}$$

obtained from the equations of the blow-ups, and from the assumption that $t_j\neq 0$ for all $i<j<k$ .

Blow-ups of the $Y_2$ component.

For the component $Y_2$ we can make similar definitions to the above. We blow up $b_{(n,0)}^*Y_{2} \times _{\mathbb A^1} V(t_{n+1})$ in $X_{(n,0)}$ and name the resulting space $X_{(n,1)}$ . Let $b_{(n,k)} \colon X_{(n,k)} \to X\times _{\mathbb A^1}\mathbb A^{n+1} $ be the composition of the n blow-ups of $Y_1$ and the first k blow-ups of $Y_2$ on $X_{(n,0)}$ . Similarly to the above, but with the order of the basis directions reversed, we blow up $b_{(n,k-1)}^*(Y_{2} \times _{\mathbb A^1} V(t_{n+2-k}))$ in $X_{(n,k-1)}$ for each $k\leq n$ .

Proposition 3.1.8. The equations of the blow-ups in local coordinates are as follows, where $(y_0^{(k)}:y_1^{(k)})$ are the coordinates of the $\mathbb P^1$ introduced by the k-th blow-up:

(3.1.2) $$ \begin{align} y_0^{(1)}t_{n+1} &= yy_1^{(1)}, \nonumber \\ y_1^{(k-1)}y_0^{(k)}t_{n+2-k}& = y_0^{(k-1)}y_1^{(k)} \ \textrm{ for } \ 2\leq k\leq n, \\ y_0^{(n)} xz& = y_1^{(n)} t_{1} \nonumber \\ x_0^{(k)} y_0^{(n+1-k)} z &= x_1^{(k)} y_1^{(n+1-k)}. \nonumber \end{align} $$

Notation. The components introduced by these new blow-ups are labelled $\Delta _2^{(k)}$ . To simplify notation, we will denote the base $\mathbb A^{n+1}\times _{\mathbb A^1} C$ by $C[n]$ , the expanded construction $X_{(n,n)}$ by $X[n]$ and the natural projection to the original family X by $\pi : X[n] \to X$ .

Figure 6 shows a fibre of $X[3]\to C[3]$ over a point $(t_1,t_2,t_3)\in C[3]$ where $t_1=t_2=0$ and $t_3\neq 0$ . The geometry of such a fibre is described in more detail in Example 3.1.16. We have blow-up morphisms

$$ \begin{align*} &\beta^1_{(i,j)} \colon X_{(i,j)} \longrightarrow X_{(i-1,j)}, \\ &\beta^2_{(i,j)} \colon X_{(i,j)} \longrightarrow X_{(i,j-1)}, \end{align*} $$

corresponding to each individual blow-up of a pullback of the $Y_1$ -component and $Y_2$ -component respectively. The composition of all the blow-up morphisms is denoted

As the following proposition shows, the spaces $X_{(i,j)}$ are well-defined, as the order in which we make the blow-ups, i.e. expand out the $\Delta _1$ or the $\Delta _2$ -components first, makes no difference. We can therefore express the space $X_{(m_1,m_2)}$ as the space $X_{(m_1,0)}$ on which we perform a sequence of blow-ups of the pullback of $Y_2$ or as the space $X_{(0,m_2)}$ on which we perform a sequence of blow-ups of the pullback of $Y_1$ , etc.

Figure 6 Geometric and tropical picture at $t_1=t_2=0$ in $X[2]$ .

Proposition 3.1.9. The following blow-up diagram commutes

Proof. We show that the space $X[1] = X_{(1,1)}$ can be constructed by first blowing up along $Y_1$ and then $Y_2$ or by reversing the order of these operations. Indeed, if we start by blowing up $Y_{1} \times _{\mathbb A^1} V(t_{1})$ in $X\times _{\mathbb A^1}\mathbb A^{n+1}$ , we obtain the étale local equations (3.1.1). This gives us the space $X_{(1,0)}$ . Then blowing up $b_{(1,0)}^*Y_{2} \times _{\mathbb A^1} V(t_{2})$ in $X_{(1,0)}$ yields the étale local equations (3.1.2) and by definition this gives us the space $X_{(1,1)}$ .

Now, if we start by blowing up $Y_{2} \times _{\mathbb A^1} V(t_{2})$ in $X\times _{\mathbb A^1}\mathbb A^{n+1}$ , we obtain étale local equations

$$ \begin{align*} y_0^{(1)}t_{n+1} &= yy_1^{(1)}, \\ y_0^{(1)} xz &= y_1^{(1)} t_{1} \end{align*} $$

and this yields the space $X_{(0,1)}$ . If we then blow up $b_{(0,1)}^*Y_{1} \times _{\mathbb A^1} V(t_{1})$ in $X_{(0,1)}$ , we shall obtain the equations

$$ \begin{align*} x_0^{(1)} y_0^{(1)} z &= x_1^{(1)} y_1^{(1)} \\ x_0^{(1)}t_1 &= xx_1^{(1)}, \\ x_0^{(1)} yz &= x_1^{(1)} t_{2}. \end{align*} $$

But these are exactly the equations (3.1.1) and (3.1.2), so the resulting space is again $X[1] = X_{(1,1)}$ . This argument can be easily generalised to $X[n]$ for any n.

Proposition 3.1.10. If we take $X\to C$ to be the étale local model

$$\begin{align*}\operatorname{\mathrm{Spec}} k[x,y,z,t]/(xyz-t) \longrightarrow \operatorname{\mathrm{Spec}} k[t], \end{align*}$$

the corresponding scheme $X[n]$ obtained after the sequence of blow-ups b is a subvariety of $(X\times _{\mathbb A^1} \mathbb A^{n+1})\times (\mathbb P^1)^{2n}$ cut out by the local equations (3.1.1) and (3.1.2).

We now extend the definition of $\Delta _1$ -components to the schemes $X[n]$ and fix some additional terminology.

Example 3.1.16 and Figure 6 illustrate how, in the $\pi ^*((Y_1\cap Y_2)^\circ )$ locus, the $\Delta _1$ and $\Delta _2$ -components are equal.

Description of fibres of $X[n]\to C[n]$ .

In order to understand what these blow-ups look like, we describe the fibres of the scheme $X[n]$ over $C[n]$ , where certain basis directions vanish.

Example 3.1.16. Two basis directions vanish. Here, we consider fibres where $t_i=t_j=0$ for some $i<j$ and no other $t_k = 0$ . The blow-ups of pullbacks of the $Y_1$ -component cause exactly one $\Delta _1$ -component to be expanded in such a fibre, and this expanded component is given by $\Delta _1^{(i)} = \ldots = \Delta _1^{(j-1)}$ . In this case, the singularities of the total space occurring at the intersection of $Y_1$ and $Y_2$ have already been resolved by expanding out this $\Delta _1$ -component. As the blow-ups of pullbacks of the $Y_2$ -component also cause one $\Delta _2$ -component to be expanded in this fibre, given by $\Delta _2^{(n+2-j)} = \ldots = \Delta _2^{(n+1-i)}$ , we therefore have

$$\begin{align*}\Delta_1^{(i)} = \ldots = \Delta_1^{(j-1)} = \Delta_2^{(n+2-j)} = \ldots = \Delta_2^{(n+1-i)} \end{align*}$$

in the $\pi ^*((Y_1\cap Y_2)^\circ )$ locus of the fibre. This can be easily deduced from studying the equations of the blow-ups. In the $\pi ^*((Y_1\cap Y_3)^\circ )$ locus of the fibre, we see a single expanded component of pure type given by $\Delta _1^{(i)} = \ldots = \Delta _1^{(j-1)}$ . Similarly, in the $\pi ^*((Y_2\cap Y_3)^\circ )$ locus of the fibre, we see a single expanded component of pure type given by $\Delta _2^{(n+1-j)} = \ldots = \Delta _2^{(n+1-i)}$ . Finally, the component $\Delta _1^{(k)}$ is equal to the union $Y_2\cup Y_3$ for $k<i$ and $\Delta _1^{(l)}$ is equal to the component $Y_1$ if $l>j-1$ . The situation for the $\Delta _2$ components is similar. This can be seen in Figure 6.

Before we continue we fix some terminology which will help us describe the expanded components.

Three basis directions vanish. When $t_i=t_j=t_k=0$ , where $i<j<k$ , and all other basis directions are nonzero, then in the locus $\pi ^*((Y_1\cap Y_2)^\circ )$ , we see exactly two expanded components, which are both of mixed type. Note that, more generally in any fibre of $X[n]$ , all expanded components in the $\pi ^*((Y_1\cap Y_2)^\circ )$ locus are of mixed type. This is because, in any fibre of $X[n]$ , we have that $\Delta _1^{(l)} = \Delta _2^{(n+1-l)}$ in the $\pi ^*((Y_1\cap Y_2)^\circ )$ locus for all l.

In the example given here, the two bubbles in the $\pi ^*((Y_1\cap Y_2)^\circ )$ locus can be described as follows. The bubble which intersects $Y_1$ nontrivially is given by $\Delta _1^{(i)} = \ldots = \Delta _1^{(j-1)}$ . By the above, each of these $\Delta _1$ -components is equivalent to a $\Delta _2$ -component in the $\pi ^*((Y_1\cap Y_2)^\circ )$ locus, so this bubble is equivalently given by $\Delta _2^{(n+1-j)} = \ldots = \Delta _2^{(n+1-i)}$ . The second bubble in this locus, which intersects $Y_2$ nontrivially, is given by

$$\begin{align*}\Delta_1^{(j)} = \ldots = \Delta_1^{(k-1)} = \Delta_2^{(n+2-k)} = \ldots = \Delta_2^{(n+1-j)}. \end{align*}$$

There is a single bubble expanded out in the $\pi ^*(Y_1\cap Y_2\cap Y_3)$ locus. This is a $\mathbb P^1\times \mathbb P^1$ , given by the meeting of the $\Delta _1^{(i)} = \ldots = \Delta _1^{(j-1)}$ and $\Delta _2^{(n+2-k)} = \ldots = \Delta _2^{(n+1-j)}$ components. Finally, in the $\pi ^*((Y_1\cap Y_3)^\circ )$ locus we see exactly two bubbles given by the two distinct expanded $\Delta _1$ -components and in the $\pi ^*((Y_2\cap Y_3)^\circ )$ locus we see also two bubbles given by the two distinct expanded $\Delta _2$ -components. This can be seen in Figure 7. The intersection of the two edges in the interior of the triangle in the tropical picture creates a new vertex, corresponding to the new bubble in the $\pi ^*(Y_1\cap Y_2\cap Y_3)$ locus. The other modified special fibres in $X[n]$ can be described similarly. See Figure 8 for a geometric picture of a fibre of $X[n]\to C[n]$ with more expanded components.

Figure 7 Geometric and tropical picture at $t_i = t_j = t_k =0$ in $X[n]$ .

Now, we note that there is a natural inclusion

(3.1.3)

which, in turn, induces a natural inclusion

Under these inclusions, we may consider $X[n]$ as the subvariety of $X[n+k]$ where all $t_i =1$ for $i>n+1$ .

The group action.

We may define a group action on $X[n]$ very similarly to [Reference Gulbrandsen, Halle and Hulek6]. Let $G\subset \operatorname {\mathrm {SL}}(n+1)$ be the maximal diagonal torus. We have $\mathbb G_m^n\cong G\subset \mathbb G_m^{n+1}$ , where we can view an element of G as an $(n+1)$ -tuple $(\sigma _1,\dots ,\sigma _{n+1})$ such that $\prod _i \sigma _i = 1$ . This acts naturally on $\mathbb A^{n+1}$ , which induces an action on $C[n]$ . The isomorphism $\mathbb G_m^n\cong G$ is given by

$$\begin{align*}(\tau_1, \ldots, \tau_{n})\longrightarrow (\tau_1, \tau_1^{-1}\tau_2, \ldots, \tau_{n-1}^{-1}\tau_{n}, \tau_n^{-1}). \end{align*}$$

We shall use the notation $(\tau _1, \ldots , \tau _{n})$ to describe elements of G throughout this work.

Proposition 3.1.18. There is a unique G-action on $X[n]$ such that $X[n]\to X\times _{\mathbb A^1}\mathbb A^{n+1}$ is equivariant with respect to the natural action of G on $\mathbb A^{n+1}$ .

In the étale local model, it is the restriction of the action on $(X\times _{\mathbb A^1}\mathbb A^{n+1})\times (\mathbb P^1)^{2n}$ which is trivial on X, which acts by

$$ \begin{align*} t_1 &\longmapsto \tau_1^{-1}t_1 \\ t_k &\longmapsto \tau_k^{-1}\tau_{k-1} t_k \\ t_{n+1} &\longmapsto \tau_{n} t_{n+1} \end{align*} $$

on the basis directions, and which acts by

$$ \begin{align*} (x_0^{(k)}:x_1^{(k)}) &\longmapsto (\tau_k x_0^{(k)}: x_1^{(k)}) \\ (y_0^{(k)}:y_1^{(k)}) &\longmapsto (y_0^{(k)}: \tau_{n+1-k} y_1^{(k)}). \end{align*} $$

on the $\Delta $ -components.

Note that the group action on the $(y_0^{(k)}:y_1^{(k)})$ coordinates follows immediately from the fact that $\Delta _1^{(k)} = \Delta _2^{(n+1-k)}$ in the $\pi ^*((Y_1\cap Y_2)^\circ )$ locus. Given the equations of the blow-ups above, there is no other possible choice of action such that the map $\pi :X[n] \to X\times _{\mathbb A^1}\mathbb A^{n+1}$ is G-equivariant (the equations must be invariant under group action). Note also that the natural inclusions

we described in the previous section are equivariant under the group action.

Lemma 3.1.19. We have the isomorphism

$$\begin{align*}H^0(C[n],\mathcal O_{C[n]})^G \cong k[t], \end{align*}$$

where $H^0(C[n],\mathcal O_{C[n]})^G$ denotes the space of G-invariant sections of $H^0(C[n],\mathcal O_{C[n]})$ .

Linearisations.

The following lemma gives a method to construct all the linearised line bundles we will need to vary the GIT stability condition.

Lemma 3.2.2. There exists a G-linearised ample line bundle $\mathcal L$ on $X[n]$ such that locally the lifts to this line bundle of the G-action on each $\mathbb P^1$ corresponding to a $\Delta _1^{(k)}$ and on each $\mathbb P^1$ corresponding to a $\Delta _2^{(n+1-k)}$ are given by

(3.2.1) $$ \begin{align} (x_0^{(k)};x_1^{(k)}) &\longmapsto (\tau_k^{a_k} x_0^{(k)}; \tau_k^{-b_k} x_1^{(k)}) \end{align} $$

(3.2.2) $$ \begin{align} (y_0^{(n+1-k)};y_1^{(n+1-k)}) &\longmapsto (\tau_k^{-c_k} y_0^{(n+1-k)}; \tau_k^{d_k} y_1^{(n+1-k)}) \end{align} $$

for any choice of positive integers $a_k,b_k,c_k,d_k$ .

Proof. Similarly to the proof of Lemma 1.18 in [Reference Gulbrandsen, Halle and Hulek6], we see that each locally free sheaf $\mathcal F_i^{(k)}$ on $X\times _{\mathbb A^1} \mathbb A^{n+1}$ has a canonical G-linearisation. There is an induced G-action on the projective product $\prod _{i,k}\mathbb P( \mathcal F_i^{(k)})$ , which is equivariant under the embedding

The G-action on each $\mathbb P( \mathcal F_i^{(k)})$ lifts to a G-action on the corresponding vector bundle, which gives us a canonical linearisation of the line bundle $\mathcal O_{\mathbb P( \mathcal F_i^{(k)})}(1)$ . Locally, the actions on $\mathcal O_{\mathbb P( \mathcal F_1^{(k)})}(1)$ and $\mathcal O_{\mathbb P( \mathcal F_2^{(k)})}(1)$ are given respectively by

$$\begin{align*} (x_0^{(k)}; \tau_k^{-1} x_1^{(k)}) \quad \text{and} \quad ( y_0^{(k)}; \tau_{n+1-k} y_1^{(k)}). \end{align*}$$

We therefore may define the lifts (3.2.1) and (3.2.2) on the line bundles $\mathcal O_{\mathbb P( \mathcal F_1^{(k)})}(a_k+b_k)$ and $\mathcal O_{\mathbb P( \mathcal F_{2}^{(n+1-k)})}(c_k+d_k)$ respectively. We then pull back each $\mathcal O_{\mathbb P( \mathcal F_1^{(k)})}(a_k+b_k)$ and $\mathcal O_{\mathbb P( \mathcal F_{2}^{(k)})}(c_{n+1-k}+d_{n+1-k})$ to $\prod _{i,k}\mathbb P( \mathcal F_i^{(k)})$ and form their tensor product to obtain a G-linearised line bundle, which we denote by $\mathcal L$ .

Each such line bundle $\mathcal L$ which can be constructed in this way will induce a G-linearised line bundle $\mathcal M$ on $H^m_{[n]}$ . This, in turn, will yield a GIT stability condition on $H^m_{[n]}$ .

Existence of limits under action of a 1-PS.

Let P be any point in $X[n]$ and let $p_n \colon X[n] \to C[n]$ be the projection to the base. As stated in [Reference Gulbrandsen, Halle and Hulek6], since the morphism $p_n \colon X[n] \to C[n]$ is proper, the limit $P_0$ of P under a 1-PS as defined above exists if and only if its projection onto the base, $p_n(P)\in C[n]$ , has a limit. The G-action on the base is a pullback of the action on $\mathbb A^{n+1}$ and the corresponding action of a 1-PS is

$$ \begin{align*} t_1 &\longmapsto \tau^{-s_1}t_1, \\ t_k &\longmapsto \tau^{s_{k-1}-s_k}t_k, \quad \textrm{for } 1<k\leq n, \\ t_{n+1} &\longmapsto \tau^{s_n}t_{n+1}. \end{align*} $$

The projection $p_n(P)$ of the point P to the base has a limit as $\tau $ tends to zero if and only if each power of $\tau $ in the action is nonnegative on the nonzero basis directions $t_i$ , i.e. if and only if

(4.2.1) $$ \begin{align} &0\geq s_1\geq \ldots \geq s_{n+1} \geq 0, \end{align} $$

where each inequality from left to right must hold if $t_1, \ldots , t_{n+1}$ is nonzero respectively. As a consequence, at a point $P\in X[n]$ , we look at which $t_i$ vanish at $p_n(P)$ and this determines a subset of the inequalities (4.2.1). Only Hilbert-Mumford invariants for 1-PS actions satisfying these inequalities need to be considered at point P. In particular, when $t_i\neq 0$ for all i, this implies that $s_i=0$ for all i, so the 1-PS are trivial and all points are trivially semistable.

Lifts of 1-PS action to the line bundle.

Let $\mathcal L$ be a line bundle as described in Lemma 3.2.2. Assume that locally the lifts to $\mathcal L$ of the G-action on each $\mathbb P^1$ corresponding to a $\Delta _1^{(k)}$ and on each $\mathbb P^1$ corresponding to a $\Delta _2^{(n+1-k)}$ are given by

$$ \begin{align*} (x_0^{(k)};x_1^{(k)}) &\longmapsto (\tau_k^{a_k} x_0^{(k)}; \tau_k^{-b_k} x_1^{(k)}) \\ (y_0^{(n+1-k)};y_1^{(n+1-k)}) &\longmapsto (\tau_k^{-c_k} y_0^{(n+1-k)}; \tau_k^{d_k} y_1^{(n+1-k)}) \end{align*} $$

for some choice of positive integers $a_k,b_k,c_k,d_k$ . Then the corresponding lifts of the 1-PS action to $\mathcal L$ are given by

$$ \begin{align*} (x_0^{(k)};x_1^{(k)}) &\longmapsto (\tau^{a_ks_k} x_0^{(k)}: \tau^{-b_ks_k}x_1^{(k)}) , \\ (y_0^{(n+1-k)};y_1^{(n+1-k)}) &\longmapsto (\tau^{-c_ks_{n+1-k}} y_0^{(k)}: \tau^{d_k s_{n+1-k}}y_1^{(k)}). \end{align*} $$

The Hilbert-Mumford invariants that interest us are the invariants relating to 1-PS subgroups of the induced action of G on $H^m_{[n]}$ with associated line bundle $\mathcal M$ , described in (4.3.1). In Lemma 4.3.1, we will see that these Hilbert-Mumford invariants can be seen as a sum of what is called a bounded and combinatorial weight. The bounded weight is given by the scheme structure of a length m zero-dimensional $[Z]\in H^m_{[n]}$ , whereas the combinatorial weight depends only on the support of Z as m points with multiplicity. In Lemma 4.3.3, we will see that in certain cases the combinatorial weight can be made to overpower the bounded weight.

Relationship between bounded and combinatorial weights.

The following lemmas describe how the Hilbert-Mumford invariant can be decomposed into a sum of invariants.

Lemma 4.3.1. Given a point $[Z]\in H^m_{[n]}$ and a 1-PS $\lambda _s$ given by $(s_1,\ldots ,s_{n})\in \mathbb Z^{n}$ , denote the limit of $\lambda _s(\tau )\cdot Z$ as $\tau $ tends to zero by $Z_0$ . The Hilbert-Mumford invariant can be decomposed into a sum

$$\begin{align*}\mu^{\mathcal M_l}(Z,\lambda_s) = \mu_b^{\mathcal M_1}(Z,\lambda_s) + l \cdot \mu_c^{\mathcal M_1}(Z,\lambda_s) \end{align*}$$

of the bounded weight $\mu _b^{\mathcal M_l}(Z,\lambda _s)$ , coming from the scheme structure of $Z_0$ , and the combinatorial weight $\mu _c^{\mathcal M_l} (Z,\lambda _s)$ , coming from the weights of the 1-PS action on $\mathcal L$ .

Proof. The Hilbert-Mumford invariant $\mu ^{\mathcal M_l}(Z,\lambda _s)$ is given by the negative of the weight of the $\mathbb G_m$ -action on the line bundle $\mathcal M_l$ at the point $Z_0$ . At the point $Z_0$ , the line bundle $\mathcal M_l$ is given by $\det (H^0(\mathcal O_{Z_0} \otimes \mathcal L^{\otimes l}))$ . We can write $Z_0$ as a union of length $n_P$ zero-dimensional subschemes $\bigcup _P Z_{0,P}$ supported at points P. Let $\mathcal L^{\otimes l}(P)$ denote the fibre of $\mathcal L^{\otimes l}$ at P. Following [Reference Gulbrandsen, Halle and Hulek6], there is an isomorphism

$$\begin{align*}H^0(\mathcal O_{Z_0} \otimes \mathcal L^{\otimes l}) \cong \bigoplus_P \big( H^0(\mathcal O_{Z_0,P}) \otimes \mathcal L^{\otimes l}(P) \big). \end{align*}$$

Then, by taking determinants, as in [Reference Gulbrandsen, Halle and Hulek6], we get

$$\begin{align*}\bigwedge^m H^0(\mathcal O_{Z_0} \otimes \mathcal L^{\otimes l}) \cong \Big( \bigwedge^m H^0(\mathcal O_{Z_0})\Big) \otimes \Big( \bigotimes_P \mathcal L^{\otimes l n_P}(P) \Big). \end{align*}$$

which allows us to write the invariant $\mu ^{\mathcal M_l}(Z,\lambda _s)$ as a sum of its bounded weight $\mu _b^{\mathcal M_l}(Z,\lambda _s)$ , coming from $\bigwedge ^m H^0(\mathcal O_{Z_0})$ in the above, and its combinatorial weight $\mu _c^{\mathcal M_l} (Z,\lambda _s)$ , coming from $\bigotimes _P \mathcal L^{\otimes ln_P}(P)$ . It is clear also that

$$\begin{align*}\mu_b^{\mathcal M_l}(Z,\lambda_s) = \mu_b^{\mathcal M_1}(Z,\lambda_s), \end{align*}$$

since the bounded weight does not depend on the value of l, and

$$\begin{align*}\mu_c^{\mathcal M_l}(Z,\lambda_s) = l \cdot \mu_c^{\mathcal M_1}(Z,\lambda_s). \end{align*}$$

Hence, we have

$$\begin{align*}\mu^{\mathcal M_l}(Z,\lambda_s) = \mu_b^{\mathcal M_1}(Z,\lambda_s) + l \cdot \mu_c^{\mathcal M_1}(Z,\lambda_s).\\[-39pt] \end{align*}$$

Note that, whereas the combinatorial weight depends on the choice of linearised line bundle, the bounded weight does not. Similarly to [Reference Gulbrandsen, Halle and Hulek6], we can show that the bounded weight, as its name suggests, can be given an upper bound in the sense of Lemma 4.3.2.

Terminology. In the proofs of the next results, we will say that a point of the support of a subscheme is on a certain side of a $\Delta $ -component to describe whether it lies on the $(0:1)$ or $(1:0)$ side of the corresponding $\mathbb P^1$ .

The following result is based on Lemma 2.3 of [Reference Gulbrandsen, Halle and Hulek6], with some slight modifications to suit our setting.

Lemma 4.3.2. Let $\mu _b^{\mathcal M_1}(Z,\lambda _s)$ be the bounded weight of $[Z]\in H_{[n]}^m$ and $s\in \mathbb Z^{n}$ such that the limit of $\lambda _s(\tau )\cdot Z$ as $\tau $ goes to zero exists. Then there exist $b_1,\dots ,b_n$ such that

$$\begin{align*}\mu_b^{\mathcal M_1}(Z,\lambda_s) = \sum_{i=1}^{n} b_is_i, \end{align*}$$

where $|b_i|\leq 2m^2$ for every i.

Let us discuss now how the bounded weight affects the overall stability condition. The following lemma is immediate from [Reference Gulbrandsen, Halle and Hulek6], but we recall their proof here for convenience.

Lemma 4.3.3. Let Z be a length m zero-dimensional subscheme in a fibre of $X[n]\to C[n]$ . Assume that, for all $s\in \mathbb Z^n$ such that the limit $\lambda _s(\tau )\cdot Z$ as $\tau $ tends to zero exists, the combinatorial weight can be written as

$$\begin{align*}\mu_c^{\mathcal M_1}(Z,\lambda_s) = \sum_{i=1}^{n} c_is_i, \end{align*}$$

where $c_is_i\geq 0$ with equality if and only if $s_i=0$ . Then Z is stable with respect to the G-linearised line bundle $\mathcal M_l$ on $H^m_{[n]}$ for some large enough l.

Proof. As we have shown that the bounded weight can be expressed as

$$\begin{align*}\mu_b^{\mathcal M_1}(Z,\lambda_s) = \sum_{i=1}^{n} b_is_i, \end{align*}$$

where $|b_i|\leq 2m^2$ , and recalling that the Hilbert-Mumford invariant can be expressed as

$$\begin{align*}\mu^{\mathcal M_l}(Z,\lambda_s) = \mu_b^{\mathcal M_1}(Z,\lambda_s) + l \cdot \mu_c^{\mathcal M_1}(Z,\lambda_s), \end{align*}$$

it is just a matter of choosing a big enough value of l to make the combinatorial weight overpower the bounded weight. This allows us effectively to treat the bounded weight as negligible and ignore it in our computations.

A criterion for positive combinatorial weights.

Let Z be a length m zero-dimensional subscheme in a fibre of $X[n]\to C[n]$ . With the following lemmas, we shall establish that if there is at least one point of the support of Z in the union $(\Delta _1^{(k)})^\circ \cup (\Delta _2^{(n+1-k)})^\circ $ for every k (where these $\Delta $ -components are not necessarily expanded out), then there exists a GIT stability condition which makes Z stable. For example, in Figure 6, there should be a point of the support of Z in at least one of the three expanded bubbles. In Figure 7, it would, for example, be enough for Z to have a point of its support in the bubble $\Delta _1^{(i)}=\Delta _2^{(k)}$ as well as in any union $(\Delta _1^{(k)})^\circ \cup (\Delta _2^{(n+1-k)})^\circ $ which equals a $Y_i$ component.

We start by showing that for such a subscheme Z there exists a G-linearised line bundle $\mathcal M$ on $H^m_{[n]}$ such that the corresponding combinatorial weight will be strictly positive. We will then use Lemma 4.3.3 to show that Z is stable in the corresponding GIT stability.

Proof. We will construct a G-linearised line bundle $\mathcal L$ on $X[n]$ as in Lemma 3.2.2, by specifying lifts of the G-action on each $\mathbb P(\mathcal F_1^{(k)})$ and $\mathbb P(\mathcal F_2^{(n+1-k)})$ to line bundles $\mathcal O_{\mathbb P( \mathcal F_1^{(k)})}(a_k+b_k)$ and $\mathcal O_{\mathbb P( \mathcal F_{2}^{(n+1-k)})} (c_k+d_k)$ for some chosen values $a_k,b_k,c_k,d_k\in \mathbb Z_{\geq 0}$ .

Let $k\in \{ 1, \dots , n \}$ . If there is some point of the support of Z, denoted P, in $(\Delta _1^{(k)})^\circ \subseteq \pi ^*(Y_1\cap Y_3)$ , and if $m'$ points of the support lie on the $(1:0)$ side of $\Delta _1^{(k)}$ , then we will want the lift of the G-action on $\mathbb P(\mathcal F_1^{(k)})$ to $\mathcal O_{\mathbb P( \mathcal F_1^{(k)})}(a_k+b_k)$ to be locally given by

$$\begin{align*}(x_0^{(k)}; x_1^{(k)}) \longmapsto (\tau_k^{m(m-m')} x_0^{(k)}; \tau_k^{-m(m'+1)} x_1^{(k)})\in \mathbb A^2. \end{align*}$$

This lift is therefore defined on $\mathcal O_{\mathbb P(\mathcal F_1^{(k)})}(m^2+m)$ , i.e. we have chosen $a_k= m(m-m')$ and $b_k = m(m'+1)$ . We will then choose $c_k = 0$ and $d_k = 1$ , so that the action on $\mathbb P( \mathcal F_{2}^{(n+1-k)})$ lifts to $\mathcal O_{\mathbb P( \mathcal F_{2}^{(n+1-k)})}(1)$ and it is locally given by

$$\begin{align*}(y_0^{(n+1-k)}; y_1^{(n+1-k)}) \longmapsto ( y_0^{(n+1-k)}; \tau_k y_1^{(n+1-k)})\in \mathbb A^2. \end{align*}$$

If there is no point of the support of Z in $\Delta _1^{(k)}$ , we set the lift of the G-action on $\mathbb P(\mathcal F_1^{(k)})$ to $\mathcal O_{\mathbb P( \mathcal F_1^{(k)})}(1)$ to be locally given by

$$\begin{align*}(x_0^{(k)}; x_1^{(k)}) \longmapsto (\tau_k x_0^{(k)}; x_1^{(k)})\in \mathbb A^2, \end{align*}$$

i.e. we have chosen $a_k=1$ and $b_k=0$ . In this case there must be at least one point of the support in $(\Delta _2^{n+1-k})^\circ $ . Let $m"$ be the number of points of the support on the $(1:0)$ side of $\Delta _2^{(n+1-k)}$ . We then set $c_k = m(m-m")$ and $d_k = m(m"+1)$ , i.e. we have a lift of the G-action on $\mathbb P( \mathcal F_{2}^{(n+1-k)})$ to $\mathcal O_{\mathbb P( \mathcal F_{2}^{(n+1-k)})}(m^2+m)$ , locally given by

$$\begin{align*}(y_0^{(n+1-k)}; y_1^{(n+1-k)}) \longmapsto (\tau_k^{-m(m-m")} y_0^{(n+1-k)}; \tau_k^{-m(m"+1)} y_1^{(n+1-k)})\in \mathbb A^2. \end{align*}$$

Repeating this process over all $k\in \{ 1, \dots , n \}$ will give us a description of $\mathcal L$ and we may form the G-linearised line bundle $\mathcal M$ from this line bundle in the way described at the start of this section. For more details on why this yields a positive combinatorial weight, see the proof of the following lemma. Note that this is not the only GIT stability condition for which Z is stable.

Lemma 4.3.7. Let Z be as in the statement of Lemma 4.3.6 and let $\mathcal M$ be a G-linearised line bundle constructed as in the proof of Lemma 4.3.6. Then, for any $s\in \mathbb Z^n$ , the combinatorial weight can be written

$$\begin{align*}\mu_c^{\mathcal M}(Z,\lambda_s) = \sum_{i=1}^{n} c_is_i, \end{align*}$$

where $c_is_i \geq 0$ with equality if and only if $s_i=0$ .

Proof. It is clear that the combinatorial weight may be written as a sum

$$\begin{align*}\mu_c^{\mathcal M}(Z,\lambda_s) = \sum_{i=1}^{n} c_is_i. \end{align*}$$

Now, let us take any $k\in \{ 1, \dots , n \}$ . First, let us assume that there is at least one point of the support in $(\Delta _1^{(k)})^\circ \subseteq \pi ^*(Y_1\cap Y_3)$ and denote by $m'$ the number of points of the support on the $(1:0)$ side of $\Delta _1^{(k)}$ . Then, if $s_k>0$ ,

$$ \begin{align*} &c_ks_k \geq (-m'm(m-m') + (m-m')m(m'+1))s_k -(m')s_k = (m^2-m')s_k \geq 0. \end{align*} $$

Here, $m^2$ corresponds to the weight coming from $\mathbb P( \mathcal F_{1}^{(k)})$ and $m'$ corresponds to the weight coming from $\mathbb P( \mathcal F_{2}^{(n+1-k)})$ . The value $m'$ arises from the fact that there are at most $m'$ points of the support on the $(0:1)$ side of $\Delta _2^{(n+1-k)}$ . And since $m'$ can be at most $m-1$ , we have that $m^2-m'>0$ .

Now, if $s_k<0$ , then

$$\begin{align*}c_ks_k \geq (-(m'+1)m(m-m') +(m-m'-1)m(m'+1))s_k +0 = -(m'+1)ms_k \geq 0, \end{align*}$$

where again the two terms correspond to the weights coming from $\mathbb P( \mathcal F_{1}^{(k)})$ and $\mathbb P( \mathcal F_{2}^{(n+1-k)})$ . As $(m'+1)m>0$ , this gives the desired answer.

Finally, if there is no point of the support in $(\Delta _1^{(k)})^\circ \subseteq \pi ^*(Y_1\cap Y_3)$ , we can make a very similar argument, as the weight coming from $\mathbb P( \mathcal F_{1}^{(k)})$ is overpowered by the weight coming from $\mathbb P( \mathcal F_{2}^{(n+1-k)})$ in the line bundle $\mathcal M$ we set up.

Nonseparated GIT quotient stacks.

Although the GIT quotients $I^m_{[n],\mathcal M}$ are projective and thus proper over $\mathbb A^1$ , their corresponding stack quotients are not necessarily proper. Indeed, the GIT quotient does not see the orbits of the group action themselves but the closures of these orbits. For example in Figure 9, the red pair of points and the blue pair of points are in the same orbit closure, so the GIT quotient considers them as equivalent, while the corresponding stack quotient regards them as belonging to separate orbits. This means that, in the stack, allowing for both pairs will break separatedness.

Figure 9 Nonseparatedness in GIT stable locus.

In the following sections, when studying quotient stacks, we will therefore want to consider the sublocus of the GIT stable locus containing only length m zero-dimensional subschemes which are smoothly supported in a given fibre of $X[n]\to C[n]$ . Building a compactification in which limits are represented by smoothly supported subschemes will also be useful for future applications as it allows us to break down the problem of a Hilbert scheme of m points on a singular surface into products of Hilbert schemes of fewer than m points on smooth components.

Patching together GIT stability conditions.

No single GIT quotient $I^m_{[n],\mathcal M}$ contains all desired limits as smoothly supported subschemes. Therefore in the stack construction, the stability condition we define will draw on these local quotients, but globally will not correspond to one single GIT stability condition. We now define a notion that we will use in the following sections.

Making the expanded degenerations large enough.

Finally, if we construct a unique GIT quotient in which not all limit subschemes are smoothly supported then the limits given by orbit closures containing only subschemes with singular support will not lie in a fibre of the expected base codimension. This gives an intuition that the degeneration we have chosen is too small. That being said, it can be useful to think about this GIT quotient if what we are trying to do is simply to resolve singularities in a way that preserves some good properties of the space, e.g. in the context of constructing minimal models for type III degenerations of Hilbert schemes of points on K3 surfaces.

The stack $\mathfrak {C}$ .

In the following we define the stack $\mathfrak {C}$ identically to the stack of expanded degenerations defined by Li and Wu. For convenience, we recall the details of this construction here.

Let us consider $\mathbb A^{n+1}$ with its natural torus action $\mathbb G_m^{n}$ as defined above. We then impose some additional relations given by a collection of isomorphisms which we describe in the following. As before, we label elements of the base as $(t_1,\dots , t_{n+1})$ . We start by defining the set

Let $I\subseteq [n+1]$ and $I^\circ = [n+1]-I$ be the complement of I. For $|I|= r+1$ , let

$$\begin{align*}\operatorname{\mathrm{ind}}_{I}\colon [r+1] \longrightarrow I \subset [n+1] \end{align*}$$

and

$$\begin{align*}\operatorname{\mathrm{ind}}_{I^\circ}\colon [n-r] \longrightarrow I^\circ \subset [n+1] \end{align*}$$

be the order preserving isomorphisms. Let

$$\begin{align*}\mathbb A^{n+1}_{I} = \{ (t)\in \mathbb A^{n+1} |\ t_i = 0,\ t_i\in I \} \subset \mathbb A^{n+1} \end{align*}$$

and

$$\begin{align*}\mathbb A^{n+1}_{U(I)} = \{ (t)\in \mathbb A^{n+1} |\ t_i \neq 0,\ t_i\in I^\circ \} \subset \mathbb A^{n+1}. \end{align*}$$

Then we have the isomorphism

$$\begin{align*}\widetilde{\tau}_I \colon (\mathbb A^{r+1}\times \mathbb G_m^{n-r}) \longrightarrow \mathbb A^{n+1}_{U(I)} \end{align*}$$

given by

$$\begin{align*}(a_1, \ldots, a_{r+1},\sigma_1,\dots, \sigma_{n-r}) \longmapsto (t_1,\dots, t_{n+1}), \end{align*}$$

where

$$ \begin{align*} & t_k = a_l, \ \mathrm{ if } \ \operatorname{\mathrm{ind}}_{I}(l)=k, \\ & t_k = \sigma_l, \ \mathrm{ if } \ \operatorname{\mathrm{ind}}_{I^\circ}(l)=k. \end{align*} $$

Then, given $I,I'\subset [n+1]$ such that $|I|=|I'|$ , we define an isomorphism

$$\begin{align*}\widetilde{\tau}_{I,I'} = \widetilde{\tau}_{I} \circ \widetilde{\tau}_{I'}^{-1} \colon \mathbb A^{n+1}_{U(I')} \longrightarrow \mathbb A^{n+1}_{U(I)}. \end{align*}$$

Recall from Section 3.1 that we had natural inclusions (3.1.3)

which called standard embeddings in [Reference Li and Wu11].

Finally, we define $\mathfrak {U}^{n}$ to be the quotient $[\mathbb A^{n+1}/\!\!\sim ]$ by the equivalences generated by the $\mathbb G_m^{n}$ -action and the equivalences $\widetilde {\tau }_{I,I'}$ for pairs $I,I'$ with $|I|=|I'|$ . We can define open immersions

$$ \begin{align*} \mathfrak{U}^{n} \longrightarrow \mathfrak{U}^{n+1}, \end{align*} $$

induced by the standard embeddings. Let be the direct limit over n and let .

The stack $\mathfrak {X}$ .

Let $X[n]\to C[n]$ be as in Section 3 and recall that $\pi \colon X[n] \to X$ is the projection to the original family. Let

be the standard embedding. Then the induced family (τ¯ _I ^* X[n], τ¯ _I ^* π) is isomorphic to $(X[m],\pi )$ over $C[m]$ . The equivalences on $\mathfrak {U}^{n}$ lift to C-isomorphisms of fibres.

We define $\mathfrak {X}^{n}$ to be the quotient $[X[n]/\!\!\sim ]$ by the equivalences generated by the $\mathbb G_m^{n}$ -action and equivalences lifted from $\mathfrak {U}^{n}$ . There are natural immersions of stacks

$$ \begin{align*} \mathfrak{X}^{n} \longrightarrow \mathfrak{X}^{n+1}, \end{align*} $$

induced by the immersions $\mathfrak {U}^{n} \to \mathfrak {U}^{n+1}$ . Finally, we define $\mathfrak {X} = \lim \limits _{\to } \mathfrak {X}^{n}$ to be the direct limit over n. It is an Artin stack.

Tropical interpretation.

In [Reference Maulik and Ranganathan12], the stack of expansions is given as a cone complex, constructed by making a choice of polyhedral subdivision of the moduli of indistinguishable m points on $\operatorname {\mathrm {trop}}(X)$ (for the case of Hilbert schemes of points). For a Hilbert scheme of m points, the choice of polyhedral subdivision we make is the quotient of the fan of $\mathbb A^{2m+1}$ identifying cones of the same dimension. Note that the subdivision chosen in [Reference Li and Wu11] for a Hilbert scheme of m points is the quotient of the fan of $\mathbb A^{m+1}$ . This is because when the singular locus of $X_0$ is smooth, each of the m points can tend into the singular locus of $X_0$ , which is made up of a single codimension 1 stratum in $X_0$ . We therefore require m blow-ups. In our case, the singular locus of $X_0$ has a second deeper stratum in codimension 2 given by the triple intersection point. We therefore need m more blow-ups to allow all m points to fall into this deeper stratum.

Intersecting with the smoothly supported locus.

We start by examining some stability conditions on the scheme $H^m_{[n]}$ . We have defined several G-linearised line bundles $\mathcal M$ on this space. As before, let us denote by $H^{m,ss}_{[n],\mathcal M}$ and $H^{m,s}_{[n],\mathcal M}$ the corresponding GIT semistable and stable loci respectively. As discussed in Section 5.1, considering the GIT stable locus does not give us a separated quotient stack, among other reasons because it contains some subschemes which are not smoothly supported.

Proof. Recall that the relative Hilbert scheme of m points on $X[n] \to C[n]$ , which we denoted $H^m_{[n]}$ , is the scheme which represents the functor

$$\begin{align*}h\colon {\underline{\rm k}\text{-}\underline{\rm Sch}}^{\operatorname{\mathrm{op}}} \longrightarrow {\underline{\rm Sets}}, \end{align*}$$

where ${\underline{\rm k}\text{-}\underline{\rm Sch}}^{\operatorname {\mathrm {op}}}$ is the category of k-schemes. This functor associates to any k-scheme B the set of flat families over B of subschemes of fibres of $X[n]$ over $C[n]$ . Restricting $X[n]$ to the smooth locus of its fibres yields a family of open subschemes $X[n]^{\operatorname {\mathrm {sm}}}$ over $C[n]$ , and we can similarly define a Hilbert functor $h_{\operatorname {\mathrm {sm}}}$ on this family. There is a morphism from the corresponding Hilbert scheme to $H^m_{[n]}$ which is clearly a monomorphism and it is étale since deformations of smoothly supported subschemes are smoothly supported. We could also note that the complement of $H^m_{[n],\operatorname {\mathrm {sm}}}$ in $H^m_{[n]}$ is closed by the valuative criterion since the limit of any subscheme with part of its support in the singular locus of a fibre must also have part of its support in the singular locus of a fibre.

We remark that since the smooth locus of the fibres of $X[n]\to C[n]$ is G-invariant, restricting the functor to this locus preserves the G-invariance. The intersections of the semistable and stable loci with the loci of smoothly supported subschemes therefore yield G-invariant open subschemes.

Notation. We denote by $H^{m, s}_{[n],\mathcal M,\operatorname {\mathrm {sm}}} $ and $ H^{m, ss}_{[n],\mathcal M,\operatorname {\mathrm {sm}}}$ the loci of GIT stable and semistable subschemes which are smoothly supported.

We have the following inclusions:

$$\begin{align*}H^{m, s}_{[n],\mathcal M,\operatorname{\mathrm{sm}}} \subset H^{m, ss}_{[n],\mathcal M,\operatorname{\mathrm{sm}}} \subset H^{m, ss}_{[n],\mathcal M}. \end{align*}$$

Li-Wu stability.

We recall here the notion of stability used in [Reference Li and Wu11], in order to compare it with the GIT stability and construct an appropriate stability condition for this case.

Definition 5.3.2. Let $X[n]_0$ be a fibre of $X[n]\to C[n]$ over a closed point and let D denote the singular locus of $X[n]_0$ . A subscheme Z in $X[n]_0$ is said to be admissible if the morphism

$$\begin{align*}\mathcal I_Z \otimes \mathcal O_D \to \mathcal O_D \end{align*}$$

is injective, where $\mathcal I_Z$ is the ideal sheaf of Z, i.e. when Z is normal to D. A family Z in $X[n]\to C[n]$ is admissible if it is admissible in every fibre over a closed point. A subscheme Z in $\mathfrak {X}$ is Li-Wu stable (LW stable) if and only if it is admissible and has finite automorphism group, which means here that the stabiliser of Z with respect to the torus action we defined on the blow-ups must be finite.

For a length m zero-dimensional scheme Z in $\mathfrak {X}$ , the admissible condition will mean that none of the points in the support of Z lie in the singular locus of the given fibre. Denote the LW stable locus by $H^m_{[n],\operatorname {\mathrm {LW}}}$ .

Lemma 5.3.3. There is a G-equivariant open immersion

(5.3.1) $$ \begin{align} H^{m, s}_{[n],\mathcal M,\operatorname{\mathrm{sm}}}\subset H^m_{[n],\operatorname{\mathrm{LW}}} \end{align} $$

as subschemes of $H^m_{[n]}$ for all G-linearised line bundles $\mathcal M$ on $H^m_{[n]}$ .

The above inclusion no longer holds for the GIT semistable locus. This is a strict inclusion as the LW stability is clearly a weaker condition than the GIT strict stability with smooth support.

Modified GIT stability.

As stated above, we only want to allow length m zero-dimensional subschemes to be stable if their support lies in the smooth locus of a fibre. However, restricting the GIT stability condition to this locus makes the space of stable subschemes no longer universally closed. Indeed, there is no single GIT condition which can represent all desired length m zero-dimensional subschemes as smoothly supported subschemes. We must therefore define a modified GIT stability condition which patches together several GIT stability conditions in order to obtain the desired stable locus.

We may write $H^m_{[n],\operatorname {\mathrm {SWS}}}$ as the union

over all choices of G-linearised line bundle $\mathcal M$ . The G-equivariant open immersion (5.3.1) implies that $H^m_{[n],\operatorname {\mathrm {SWS}}} \subset H^m_{[n],\operatorname {\mathrm {LW}}}$ is also a G-eeqiuvariant open immersion in $H^m_{[n]}$ . We will now want to compare these stability conditions on the stack $\mathfrak {X}$ , so we will need to extend our definition of WS stability to this stack.

Given a C-scheme S, an object of $\mathfrak {X}(S)$ is a pullback family $\xi ^* X[n]$ for a morphism

$$\begin{align*}\xi \colon S \to C[n]. \end{align*}$$

Now we describe WS stability on the stack $\mathfrak {X}$ .

Stacks of stable objects.

Let us denote by $\mathfrak {M}^m_{\operatorname {\mathrm {SWS}}}$ and $\mathfrak {M}^m_{\operatorname {\mathrm {LW}}}$ the stacks of SWS and LW stable length m zero-dimensional subschemes in $\mathfrak {X}$ respectively. Let S be a C-scheme. An object of $\mathfrak {M}^m_{\operatorname {\mathrm {SWS}}}(S)$ is defined to be a pair $(Z,\mathcal X)$ , where $\mathcal X\in \mathfrak {X}(S)$ and Z is an S-flat SWS stable family in $\mathcal X$ . Similarly, an object of $\mathfrak {M}^n_{\operatorname {\mathrm {LW}}}(S)$ is a pair $(Z,\mathcal X)$ , where $\mathcal X\in \mathfrak {X}(S)$ and Z is an S-flat LW stable family in $\mathcal X$ .

As is briefly discussed in Section 1.3, we can also make constructions, equivalent to some of the constructions of Maulik and Ranganathan, which require choices of representatives of limit subschemes and labelling of components as tube components. The construction we make here requires the minimal amount of choice (the only choice was in choosing to blow up $Y_1$ and $Y_2$ but not $Y_3$ at the very start).

Equivalences of tropicalisations.

Let $\eta $ and $\eta _0$ be the generic and closed points of , for a discrete valuation ring R with uniformising parameter u. Note that in the above proofs we have abused notation slightly: when considering pairs $(Z_\eta ,\mathcal X_\eta )\in \mathfrak {M}^m_{\operatorname {\mathrm {SWS}}}(\eta )$ (or $\mathfrak {M}^m_{\operatorname {\mathrm {LW}}}(\eta )$ ), the object we refer to as $\operatorname {\mathrm {trop}}(Z_\eta )$ is actually the image of $\xi _*(Z_\eta )$ under the tropicalisation map, where $\xi \colon S \to C[n]$ .

Similarly, if $\mathcal X_{\eta _0}$ is pulled back from some modified special fibre along a morphism $\xi $ , then we will write $\operatorname {\mathrm {trop}}(\mathcal X_{\eta _0})$ to mean the tropicalisation of the pushforward along $\xi $ . Notice that the morphism $\xi $ determines the tropicalisation. The expanded components of $\mathcal X_{\eta _0}$ are blow-ups along the vanishing of x or y with some power of u, therefore for different choices of powers of u, the resulting expanded components are not the same and will correspond to different tropical pictures. They may, however, get mapped to the same component in $X[n]$ . For example, let $X_0'$ be the fibre given by $t_1=t_2=0$ in $X[1]$ and let $\mathcal X_{\eta _0}$ and $\mathcal X^{\prime }_{\eta _0}$ be the pullbacks of $X[1]$ over the closed point $\eta _0$ of S under the morphisms $\xi = (u,u^3)$ and $\xi '= (u^2,u^2)$ from S to $C[1]$ . The tropicalisations of $\mathcal X_{\eta _0}$ and $\mathcal X^{\prime }_{\eta _0}$ are different, as shown in Figure 10, although the corresponding fibre $X_0'$ of $X[1]$ is the same.

Figure 10 Tropicalisations identified by the isomorphisms of the stack $\mathfrak {X}$ .

This phenomenon of several different tropical pictures corresponding to the same geometric fibre in $X[n]$ can then be be interpreted as coming from the fact that $X[n]$ can be seen as a sublocus of a larger $X[n']$ , for $n'>n$ , and from the equivalences on the stack $\mathfrak {X}$ given by the isomorphisms set up in Section 5.2. We may take $n'$ to be such that there is a morphism $S\to C[n']$ given by $(u,\dots ,u)$ . Over $\eta _0$ this yields the fibre of maximal base codimension of $X[n']$ , which we denote by $X[n']_0$ . Then the pushforward of $\mathcal X_{\eta _0}$ and $\mathcal X^{\prime }_{\eta _0}$ by $\xi $ and $\xi '$ correspond to contracting different components of $X[n']_0$ , so to different fibres of $X[n']$ . In the example above, we may consider $\mathcal X_{\eta _0}$ and $\mathcal X^{\prime }_{\eta _0}$ to be the pullbacks of $X[3]$ along the morphisms $\xi = (u,u^3,1,1)$ and $\xi ' = (1,u^2,u^2,1)$ restricted to the generic point $\eta _0$ . These will correspond to fibres $t_1=t_2=0$ and $t_2=t_3=0$ of $X[3]$ respectively. As these two fibres are identified by the isomorphisms of Section 5.2, so are $\mathcal X_{\eta _0}$ and $\mathcal X^{\prime }_{\eta _0}$ . We may therefore define the corresponding equivalence class for the tropicalisations.

By convention, for a scheme S and an object $\mathcal X\in \mathfrak {X}(S)$ , if $\operatorname {\mathrm {trop}}(\mathcal X)$ is not well-defined (for example, if we have only specified a family in $X[n]$ from which it is pulled back but not a morphism $S\to C[n]$ ), we may take $\operatorname {\mathrm {trop}}(\mathcal X)$ to be any element of the equivalence class $[\operatorname {\mathrm {trop}}(\mathcal X)]$ . Implicitly, we have used this assumption to draw the Figures in Section 3.1.

Existence and uniqueness of limits for special objects.

We need to establish some definitions before we prove the following auxiliary result on existence and uniqueness of limits for special elements, i.e. when the fibre $\mathcal X_\eta $ over the generic point of S is a modified special fibre itself.