Symmetric products of a semistable degeneration of surfaces

Abstract

We explicitly construct a V-normal crossing Gorenstein canonical model of the relative symmetric products of a local semistable degeneration of surfaces without a triple point by means of toric geometry. Using this model, we calculate the stringy E-polynomial of the relative symmetric product. We also construct a minimal model of degeneration of Hilbert schemes explicitly.

1 Introduction

Let S be a smooth (quasi-)projective algebraic surface. A theorem of Fogarty [8] says that the Hilbert scheme ${{\mathrm{Hilb}}}^n(S)$ of 0-dimensional subschemes on S of length n is a smooth (quasi-)projective algebraic variety of dimension 2n. This construction gives a very nice and interesting way to produce higher dimensional algebraic varieties. For example, if S is a K3 surface (resp. an abelian surface), then ${{\mathrm{Hilb}}}^n(S)$ (resp. the albanese fiber of ${{\mathrm{Hilb}}}^{n+1}(S)$) gives an example of higher dimensional irreducible symplectic compact Kähler manifold [2]. Besides the holomorphic symplectic geometry, the Hilbert scheme of points on a surface is related to many branches of mathematics, such as differential geometry, singularity theory, and representation theory.

As Hilbert schemes behave nicely in family, it is quite natural to think of the relative Hilbert scheme ${{\mathrm{Hilb}}}^n(\mathscr {S}/B)$ for a flat family of surfaces $\pi : \mathscr {S}\rightarrow B$. If the family $\pi$ degenerates at some point $b\in B$, one naturally expects that the family of Hilbert schemes ${{\mathrm{Hilb}}}^n(\mathscr {S}/B)\rightarrow B$ also degenerates at b. In this setting, one of the fundamental questions is to ask how much singular the induced degeneration of Hilbert schemes is. Of course, it will depend on the singularity of the degeneration of the original family. To get a modest degeneration of Hilbert schemes, it is natural to assume that the family of surfaces $\pi :\mathscr {S}\rightarrow B$ is semistable. In such a situation, another natural question is to find a good birational model of a degeneration of Hilbert schemes that is semistable (or very near to semistable) and minimal [9, 14], and to understand the behavior of the family.

However, even if the family $\pi :\mathscr {S}\rightarrow B$ is a semistable degeneration, and hence $\mathscr {S}$ is smooth, it seems difficult, at least to the author, to study the relative Hilbert scheme ${{\mathrm{Hilb}}}^n(\mathscr {S}/B)$ directly by the ring theoretic approach as in [8] or [4], in contrast to Ran’s work [18] on the case of semistable degeneration of curves in this direction. In fact, our relative Hilbert scheme can be seen as a closed subscheme ${{\mathrm{Hilb}}}^n(\mathscr {S}/B)\subset {{\mathrm{Hilb}}}^n(\mathscr {S})$, while ${{\mathrm{Hilb}}}^n(\mathscr {S})$ can be very singular for $\dim \mathscr {S}\geqslant 3$ and n large.

In this article, we will focus on the relative symmetric product ${{\mathrm{Sym}}}^n(\mathscr {S}/B)$ rather than the relative Hilbert scheme, and study its singularity and birational geometry. For that purpose, we start from a local model of semistable degeneration of surfaces $\mathscr {S}\rightarrow B$ and describe the symmetric product ${{\mathrm{Sym}}}^n(\mathscr {S}/B)$ as a quotient of certain affine toric variety by an action of the symmetric group (§1). This description leads us to a Gorenstein canonical model with only quotient singularities for a degeneration without a triple point (§2, Theorem 2.10). The Gorenstein canonical model is obtained as a quotient by a natural action of the symmetric group of the total space of a rank two toric vector bundle over a toric variety associated with the Coxeter complex of a root system of type A (Proposition 2.5). It is noteworthy that such a toric variety was studied by several authors from an interest of combinatorics and representation of symmetric groups [6, 17, 21]. This toric-quotient description also enables us to calculate the stringy E-polynomial of the Gorenstein canonical model, which encodes cohomological information of the degeneration (§3, Theorem 3.14, Proposition 3.17). In the last section, we discuss an explicit construction of a $\mathbb Q$-factorial terminal minimal model of the relative symmetric product (Theorem 4.1), which turns out to be a V-normal crossing degeneration of Hilbert schemes of general fibers. This gives a good birational model of a degeneration of Hilbert schemes in the case where the singular fiber of the original semistable degeneration of surfaces has no triple point. We also relate our minimal model to the relative Hilbert scheme explicitly in the case where the length $n=2$.

2 Local description of symmetric product

1.1 Let $S_2=S_3=\mathbb C^3$ and define $p _d:S_d\rightarrow B=\mathbb C$ by

$$\begin{aligned} p _d(x_1,x_2,x_3)= {\left\{ \begin{array}{ll} x_1x_2 &{} \text{ if } \quad d=2,\\ x_1x_2x_3 &{} \text{ if } \quad d=3. \end{array}\right. } \end{aligned}$$

The origin of $S_2$ is the local model at the general point of the singular locus of the singular fiber of semistable degeneration of surfaces, while the origin of $S_3$ is the maximally degenerate point. Let us denote the n-fold self-products of $S_d$ relative to $p _d$ by

$$\begin{aligned} \widetilde{X}^{(n)}_d= (S_d/B)^n = S_d \times _B S_d \times _B \cdots \times _B S_d. \end{aligned}$$

and let $\tilde{\pi }^{(n)}_d: \widetilde{X}^{(n)}_d\rightarrow B$ be the natural morphism. Then, the symmetric group $\mathfrak S_n$ acts on $\widetilde{X}^{(n)}_d$ and $\tilde{\pi }^{(n)}_d$ is $\mathfrak S_n$-invariant. The quotient variety

$$\begin{aligned} \pi ^{(n)}_d: X^{(n)}_d=\widetilde{X}^{(n)}_d/\mathfrak S_n \rightarrow B \end{aligned}$$

is the nth relative symmetric product of $p_d:S_d\rightarrow B$.

The fiber $(\pi ^{(n)}_d)^{-1}(b)$ for $b\in B,\, b\ne 0$ is just ${{\mathrm{Sym}}}^n(\mathbb C^*\times \mathbb C)$ for $d=2$ and ${{\mathrm{Sym}}}^n((\mathbb C^*)^2)$ for $d=3$. It is also easy to see the combinatorics of the fiber over the origin. It has $\left( {\begin{array}{c}n+d-1\\ d\end{array}}\right)$ components each of which corresponds to a partition of n,

$$\begin{aligned} \mathbf a=\left( a_i\; |\; 1\leqslant i\leqslant d,\; \sum a_i = n\right) . \end{aligned}$$

The component $X^{(n)}_{\mathbf a}$ is an image of a birational morphism

$$\begin{aligned} {{\mathrm{Sym}}}^{\mathbf a}(\mathbb C^2) = \prod _{i=1}^d {{\mathrm{Sym}}}^{a_i}(\mathbb C^2) \rightarrow X^{(n)}_{\mathbf a}. \end{aligned}$$

The morphism has finite fibers over the double curves except for the case $a_i=n$ for some i. It implies that $X^{(n)}_{\mathbf a}$ is non-normal for general $\mathbf a$. The geometry of the intersection of these components seems difficult to describe directly.

In contrast, it is easy to describe the product variety $\widetilde{X}^{(n)}_d$ by affine equations. Let $(z_{11},z_{12},z_{13},\ldots , z_{n1},z_{n2},z_{n3})$ be the coordinate of $(\mathbb C^3)^n=\mathbb C^{3n}$. Then, $\widetilde{X}^{(n)}_d$ is nothing but the complete intersection

$$\begin{aligned} z_{11}\cdots z_{1d}=z_{21}\cdots z_{2d}=\cdots =z_{n1}\cdots z_{nd}\quad (d=2,3) \end{aligned}$$

(1)

of dimension $2n+1$ and the projection $\tilde{\pi }^{(n)}_d: \widetilde{X}^{(n)}_d\rightarrow B=\mathbb C$ is the function defined by the value of these monomials. If $d=2$, the variety split into a product of the closed subvariety $\widetilde{X}^{(n) \prime }_2\subset \mathbb C^{2n}$ defined by the same Eq. (1) and $\mathbb C^n$.

Proposition 1.2

$X^{(n)}_d$ is $\mathbb Q$-Gorenstein, i.e., the canonical divisor of $X^{(n)}_d$ is $\mathbb Q$-Cartier.

Proof

The locus F of points with non-trivial stabilizers with respect to the action of $\mathfrak S_n$ is the union of linear subspaces defined by $z_{i1}=z_{j1},\, z_{i2}=z_{j2}$, and $z_{i3}=z_{j3}$ for $i\ne j$. Therefore, the codimension of F inside $\widetilde{X}^{(n)}_d$ is two. This implies that the quotient map $\widetilde{X}^{(n)}_d\rightarrow X^{(n)}_d$ is étale in codimension 1 and the proposition follows from Proposition 5.20 of [12]. $\square$

1.3.   For later use, we give a description of $\widetilde{X}^{(n)}_d$ as a toric variety. Let us first consider the case in which $d=2$. Let $M=\mathbb Z^{n+1}$ and $N={{\mathrm{Hom}}}_{\mathbb Z}(M,\mathbb Z)$ its dual. We denote by $[a_1\, a_2\, \ldots a_l]_{lr}$ the sequence $a_1, a_2, \ldots , a_l$ recurred r times. For example,

$$\begin{aligned}{}[0\, 0\,1\, 1]_{12} = ( 0, 0, 1,1, 0, 0, 1,1, 0, 0, 1,1). \end{aligned}$$

Using this notation, we define $C^{(n)}_2$ as the $(n+1)\times 2^n$ matrix whose rows are given by

$$\begin{aligned} \begin{aligned} {[}&1\, 0] _{2^n}\\ [&0\, 1]_{2^n}\\ [&0\, 0\, 1\, 1]_{2^n}\\&\vdots \\ [&0^{2^{n-1}}\, 1^{2^{n-1}}]_{2^n}\; , \end{aligned} \end{aligned}$$

where “ $0^r$ ” means 0 repeated r-times and the same for “ $1^r$ ”. For example,

$$\begin{aligned} C^{(3)}_2= \begin{pmatrix} 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \end{pmatrix} . \end{aligned}$$

Under the standard identification $N\cong \mathbb Z^{n+1}$, we define $\sigma ^{(n)}_2\subset N\otimes \mathbb R$ to be a rational polyhedral convex cone generated by the column vectors in $C^{(n)}_2$. The cone $\sigma ^{(n)}_2$ is a non-simplicial cone of maximal dimension for $n\geqslant 2$.

Proposition 1.4

The affine variety $\widetilde{X}^{(n)\prime }_2\subset \mathbb C^{2n}$ defined above is the affine toric variety $X(\sigma ^{(n)}_2)={{\mathrm{Spec}}}\mathbb C[\sigma ^{(n)\vee }_2\cap M]$.

This proposition is implicitly given in [23], §4. Here we give a proof of slightly different flavor.

Lemma 1.5

Let $\sigma _1,\sigma _2,\bar{\sigma }$ be strictly convex rational polyhedral cones on lattices $N_1, N_2, \bar{N}$, respectively. Assume that we have surjective homomorphisms $h_1:N_1\rightarrow \bar{N}$ and $h_2:N_2\rightarrow \bar{N}$ such that $\bar{\sigma }=h_{1,\mathbb R}(\sigma _1)=h_{2,\mathbb R}(\sigma _2)$. Let $\pi _i:X(\sigma _i)\rightarrow X(\bar{\sigma })\; (i=1,2)$ be the corresponding toric morphisms of affine toric varieties. Then, the fiber product $X(\sigma _1)\times _{X(\bar{\sigma })}X(\sigma _2)$ is an affine toric variety corresponding to the cone

$$\begin{aligned} \sigma _1\times _{\bar{N}} \sigma _2 =\{(v_1,v_2)\in (N_1\oplus N_2)_\mathbb R\, |\,v_i\in \sigma _i\,(i=1,2),\; h_1(v_1)=h_2(v_2)\} \end{aligned}$$

on the lattice $N_1\times _{\bar{N}}N_2=\{(v_1,v_2)\in N_1\oplus N_2\,|\, h_1(v_1)=h_2(v_2)\}$.

Proof

Let $M_1,M_2, \bar{M}$ be dual lattices of $N_1,N_2,\bar{N}$, respectively. Since $h_i\; (i=1,2)$ is surjective, $\bar{M}$ is a direct summand of $M_i$. The tensor product

$$\begin{aligned} \mathbb C[\sigma _1^{\vee }\cap M_1] \underset{\mathbb C[\bar{\sigma }^{\vee } \cap \bar{M}]}{\otimes } \mathbb C[\sigma _2^{\vee }\cap M_2] \end{aligned}$$

has a basis consisting of monomials $(m_1,m_2)\, (m_i\in M_i)$ subject to a relation

$$\begin{aligned} (m_1,m_2)=(m_1',m_2') \text{ if } \text{ and } \text{ only } \text{ if } m_1-m_1'=m_2'-m_2\in \bar{M}. \end{aligned}$$

This implies that the tensor ring is the monoid ring corresponding to a cone C that is the image of the product cone $\sigma _1^{\vee }\times \sigma _2^{\vee }$ under the surjective homomorphism to the fiber co-product

$$\begin{aligned} M_1\oplus M_2 \rightarrow M_1+_{\bar{M}}M_2=(M_1\oplus M_2)/\bar{M}. \end{aligned}$$

In particular, C is spanned by the vectors $(m_1,m_2)\in M_1+_{\bar{M}}M_2$ with $m_i$ a generator of a ray of $\sigma _i^{\vee }$ for each $i=1,2$. Passing to the dual, the dual cone $C^{\vee }$ is cut out by positive half-planes defined by $(m_1,m_2)$ as above on the fiber product of lattices $N_1\times _{\bar{N}}N_2$. This immediately implies that $C^{\vee }$ is nothing but the fiber product of cones $\sigma _1\times _{\bar{N}} \sigma _2$. $\square$

Proof of Proposition 1.4

Let $C\rightarrow B=\mathbb C$ be a family of curves defined by $(x_1,x_2)\mapsto x_1x_2$. This is a toric morphism corresponding to a surjective homomorphism of lattices

$$\begin{aligned} (1\; 1):N_C=\mathbb Z^2\rightarrow N_B=\mathbb Z \end{aligned}$$

that is compatibile with the cones $\sigma _C=\mathbb R_{\geqslant 0} \, d_1 +\mathbb R_{\geqslant 0} \, d_2$ and $\sigma _B=\mathbb R_{\geqslant 0}$, where $d_1=\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $d_2=\begin{pmatrix} 0 \\ 1 \end{pmatrix}$. Noting that $\widetilde{X}^{(n)\prime }_2$ is an n-fold fiber product $(C/B)^n$, the cone $\sigma ^{(n)}_2$ is nothing but the fiber product of cones

$$\begin{aligned} (\sigma _C/N_B)^n = \sigma _C \times _{N_B} \cdots \times _{N_B} \sigma _C \end{aligned}$$

on the lattice $(N_C/N_B)^n$ by the lemma above. If we take a basis $(N_C/N_B)^n\cong \mathbb Z^{n+1}$ as

$$\begin{aligned} \begin{aligned} e_0&= (d_1,d_1,\ldots , d_1),\quad e_1 = (d_2,d_1,\ldots , d_1) \\ e_j&= (0, \ldots , \underset{\begin{array}{c} \wedge \\ (j+1) \end{array}}{d_2-d_1},0,\ldots , 0) \end{aligned} \end{aligned}$$

one immediately sees that the cone $(\sigma _C/N_B)^n$ is generated by the column vectors of the matrix $C^{(n)}_2$ defined above. $\square$

1.6.    Let $\widetilde{M}=\mathbb Z^2 \otimes \mathbb Z^n$. The symmetric group $\mathfrak S_n$ acts on $\widetilde{M}$ by the permutation representation on the second factor. Define a $(n+1)\times 2n$ integral matrix $P^{(n)}_2$ by

$$\begin{aligned} P^{(n)}_2=(e_0\quad e_1\quad e_0+e_1-e_2\quad e_2\quad \cdots \quad e_0+e_1-e_n\quad e_n), \end{aligned}$$

(2)

where $e_0,\ldots , e_n$ is the standard basis of $M=\mathbb Z^{n+1}$. The cone $\sigma ^{(n)\vee }_2$ is nothing but the image under the surjective linear map $P^{(n)}_2:\widetilde{M}\otimes \mathbb R \rightarrow M\otimes \mathbb R$ of the cone spanned by the standard simplex in $\widetilde{M}$. Therefore, we have an induced action of $\mathfrak S_n$ on M and its dual N. More precisely, $\mathfrak S_n$ acts on M by permuting n pairs of vectors

$$\begin{aligned} (e_0\quad e_1 \quad | \quad e_0+e_1-e_2\quad e_2\quad |\quad \cdots \quad |\quad e_0+e_1-e_n\quad e_n), \end{aligned}$$

so that the action of $\mathfrak S_n$ on N is represented by matrices

$$\begin{aligned} (1\; 2)= \left( \begin{array}{c@{\quad }c@{\quad }c|c} 1 &{} 1 &{} -1 &{} \\ 0 &{} 0 &{} 1 &{} \\ 0 &{} 1 &{} 0 &{} \\ \hline &{} &{} &{} I_{n-2} \end{array} \right) \; \text{ and } \; (k\;\; k+1)= \left( \begin{array}{c|c@{\quad }c|c} I_k &{} &{} &{}\\ \hline &{} 0 &{} 1 &{}\\ &{} 1 &{} 0 &{} \\ \hline &{} &{} &{} I_{n-k-1} \end{array} \right) \end{aligned}$$

(3)

for $k>1$. The cone $\sigma ^{(n)}_2$ and its dual $\sigma ^{(n)\vee }_2$ are invariant under the action of $\mathfrak S_n$. Let $\sigma$ be a cone in $(N\oplus \mathbb Z^n)\otimes \mathbb R$ spanned by $\sigma ^{(n)}_2$ and the standard basis of $\mathbb Z^n$. Then, the associated affine toric variety $X(\sigma )$ is nothing but $\widetilde{X}^{(n)}_2$. The action of $\mathfrak S_n$ on $\widetilde{X}^{(n)}_2$ coincides with the action induced by the diagonal $\mathfrak S_n$-action on N and $\mathbb Z^n$.

1.7.   We also have a similar description of $\widetilde{X}^{(n)}_3$ as a toric variety. Let $M=\mathbb Z^{2n+1}$ and $N={{\mathrm{Hom}}}_{\mathbb Z}(M,\mathbb Z)\cong \mathbb Z^{2n+1}$. Let $C^{(n)}_3$ be the $(2n+1)\times 3^n$ matrix whose rows are given by

$$\begin{aligned} \begin{aligned} {}[&1\, 0\, 0]_{3^n}\\ [&0\, 1\, 0]_{3^n}\\ [&0\, 0\, 1]_{3^n}\\ [&0\, 0\, 0\, 1\, 1\, 1\, 0\, 0\, 0]_{3^n}\\ [&0\, 0\, 0\, 0\, 0\, 0\, 1\, 1\, 1]_{3^n}\\&\vdots \\ [&0^{3^{n-1}}\, 1^{3^{n-1}}\, 0^{3^{n-1}} ]_{3^n}\\ [&0^{3^{n-1}}\, 0^{3^{n-1}}\, 1^{3^{n-1}} ]_{3^n}. \end{aligned} \end{aligned}$$

and $\sigma ^{(n)}_3\subset N\otimes \mathbb R$ the cone generated by the column vectors of $C^{(n)}_3$. Then, the associated toric variety $X(\sigma ^{(n)}_3)={{\mathrm{Spec}}}\mathbb C[\sigma ^{(n)\vee }_3\cap M]$ is nothing but $\widetilde{X}^{(n)}_3$. The proof of this claim is completely parallel to the case of $\widetilde{X}^{(n)\prime }_2$; it is a direct consequence of Lemma 1.5.

3 Gorenstein canonical orbifold model

In this section, we construct a Gorenstein canonical model of $X^{(n)}_2$ with only quotient singularities. From now on, we concentrate on the case $d=2$. We suppress the subscript and ${\hbox {write }\widetilde{X}^{(n)}}$, ${\widetilde{X}^{(n)\prime }}$ and ${X^{(n)}}$ instead of ${\widetilde{X}^{(n)}_2}$, ${\widetilde{X}^{(n)\prime }_2}$ and ${X^{(n)}_2}$, respectively, for better readability.

Proposition 2.1

There is an $\mathfrak S_n$-equivariant small projective toric resolution

$$\begin{aligned} \tilde{\mu }^{(n)\prime }: \widetilde{Z}^{(n)\prime }\rightarrow \widetilde{X}^{(n)\prime }. \end{aligned}$$

Proof

$\widetilde{X}^{(n)\prime }$ is the affine closed subvariety in $\mathbb C^{2n}$ defined by

$$\begin{aligned} z_{11}z_{12}=z_{21}z_{22}=\cdots =z_{n1}z_{n2}. \end{aligned}$$

(4)

The n-plane $\Sigma ^{(n)}$ defined by $z_{11}=z_{21}=\cdots =z_{n1}=0$ is an $\mathfrak S_n$-invariant non-Cartier Weil divisor on $X^{(n)\prime }$. Let

$$\begin{aligned} \tilde{f} ^{(n)}:\widetilde{W}^{(n)\prime } \rightarrow \widetilde{X}^{(n)\prime } \end{aligned}$$

be the blowing-up of $\widetilde{X}^{(n)\prime }$ along $\Sigma ^{(n)}$. $\widetilde{W}^{(n)\prime }$ is the closed subvariety of $\mathbb C^{2n}\times \mathbb P^{n-1}$ defined by

$$\begin{aligned} z_{i1}y_j-z_{j1}y_i=0\quad (i\ne j) \end{aligned}$$

along with (4), where $[y_1:\cdots :y_n]$ is the homogeneous coordinate of $\mathbb P^{n-1}$. Let $P_i=[0:\cdots :0: \overset{\begin{array}{c} i\\ \vee \end{array}}{1} : 0 : \cdots : 0]\in \mathbb P^{n-1}$ and $U_i=(y_i\ne 0)\subset \mathbb P^{n-1}$. Then, it is easily checked using coordinate that there is a natural isomorphism of affine varieties

$$\begin{aligned} \widetilde{W}^{(n)\prime }\cap (\mathbb C^{2n}\times U_i)\cong \widetilde{X}^{(n-1)\prime }\times \mathbb C. \end{aligned}$$

Moreover,

$$\begin{aligned} D_i = \widetilde{W}^{(n)\prime }\cap (\mathbb C^{2n}\times \{P_i\}) \quad (i=1,2,\ldots , n) \end{aligned}$$

is a non-$\mathbb Q$-Cartier Weil divisor of $\widetilde{W}^{(n)\prime }$ that is the strict transform of the divisor on $\widetilde{X}^{(n)\prime }$ defined by

$$\begin{aligned} z_{11}=\cdots =z_{i-1,1}=z_{i2}=z_{i+1,1}=\cdots =z_{n1}=0. \end{aligned}$$

$D_i$ is identified with $\Sigma ^{(n-1)}\times \mathbb C\subset \widetilde{X}^{(n-1)\prime }\times \mathbb C$ under the isomorphism above. $D_i$’s are disjoint to each other and the union $D=\amalg \, D_i$ is $\mathfrak S_n$-invariant. Therefore, the blowing-up of $\widetilde{W}^{(n)\prime }$ along D is $\mathfrak S_n$-equivariant. As it is locally isomorphic to $\tilde{f} ^{(n-1)}$, we get an $\mathfrak S_n$-equivariant small resolution of $\widetilde{X}^{(n)\prime }$ by induction on n. The centers of the blowing-ups are strict transform of torus invariant (non-$\mathbb Q$-Cartier) divisors on $\widetilde{X}^{(n)\prime }$, so the resolution is a toric morphism. $\square$

Remark 2.1.1

The toric variety $\widetilde{Z}^{(n)\prime }$ also appeared in [23] as the local model of ‘augmented relative Hilbert scheme’.

2.2.    From this proposition, one immediately sees that the relative self-product $\widetilde{X}^{(n)}$ admits an $\mathfrak S_n$-equivariant small projective toric resolution

$$\begin{aligned} \tilde{\mu }^{(n)}=(\tilde{\mu }^{(n)\prime }\times {{\mathrm{id}}}): \widetilde{Z}^{(n)} = \widetilde{Z}^{(n)\prime }\times \mathbb C^n \rightarrow \widetilde{X}^{(n)}=\widetilde{X}^{(n)\prime }\times \mathbb C^n. \end{aligned}$$

Now we let $Z^{(n)}=\widetilde{Z}^{(n)}/\mathfrak S_n =(\widetilde{Z}^{(n)\prime } \times \mathbb C^n)/\mathfrak S_n$. Then, we get a small projective birational morphism

$$\begin{aligned} \mu ^{(n)}: Z^{(n)}\rightarrow X^{(n)} \end{aligned}$$

and an induced family

$$\begin{aligned} \rho ^{(n)}=\pi ^{(n)} \circ \mu ^{(n)} : Z^{(n)}\rightarrow B. \end{aligned}$$

We want to study the singular locus of $Z^{(n)}$. For that purpose, we need a description of the toric birational morphism $\widetilde{\mu }^{(n)\prime }: \widetilde{Z}^{(n)\prime }\rightarrow \widetilde{X}^{(n)\prime }$ in terms of fan.

2.3.   The blowing-up $\tilde{f}^{(n)}$ appeared above corresponds to the star subdivision (see [5], §11.1 for the definition) $\Theta ^{(n)}$ of the cone $\sigma ^{(n)}$ with respect to the ray spanned by ${}^t(1,0,\ldots , 0)$. The fact that the ray is a one dimensional face of $\sigma ^{(n)}$ corresponds to the smallness of $\tilde{f}^{(n)}$. The cone $\sigma ^{(n)}$ is spanned by the column vectors of the $(n+1)\times 2^n$ matrix $C^{(n)}$. One can check that the resolution $\tilde{\mu }^{(n)\prime }: \widetilde{Z}^{(n)\prime }\rightarrow \widetilde{X}^{(n)\prime }$ is given by the consecutive star subdivisions of $\sigma ^{(n)}$ with respect to first $(2^n-1)$ column vectors (in this order).

Let $\Delta ^{(n)}$ be the resulted fan in $N\otimes \mathbb R$ and $\widetilde{Z}^{(n)\prime }$ is the toric variety $X(\Delta ^{(n)})$. By the proof of Proposition 2.1, one sees that $\mathfrak S_n$ acts transitively on the set of open subsets $\{\widetilde{W}^{(n)\prime }\cap (\mathbb C^{2n}\times U_i)\}_{i=1}^n$. This means that $\mathfrak S_n$ acts transitively on the set of maximal cones $\{\theta _i\}_{i=1}^n$ of the fan $\Theta ^{(n)}$ corresponding to $\widetilde{W}^{(n)\prime }$. Actually, $\mathfrak S_n$ acts on it via the permutation of the index set $\{1,2,\ldots , n\}$. Each cone $\theta _i$ can be identified with $\sigma ^{(n-1)}$ and its stabilizer subgroup ${{\mathrm{Stab}}}(\theta _i)\subset \mathfrak S_n$ is nothing but the subgroup of permutations that leave i invariant, which is naturally isomorphic to $\mathfrak S_{n-1}$. An inductive argument infers that the set of maximal cones of the fan $\Delta ^{(n)}$ consists of n! cones and $\mathfrak S_{n}$ acts on the set transitively. By the construction, $\Delta ^{(n)}$ contains a cone $\delta ^{(n)}$ that is generated by the column vectors of

$$\begin{aligned} \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 1 &{}\quad \cdots &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad \cdots &{}\quad 1\\ &{}\quad &{}\quad &{}\quad \ddots &{}\quad \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 \end{pmatrix} . \end{aligned}$$

(5)

2.4.    Let $\bar{\Delta }^{(n)}$ be the Coxeter fan of $A_{n-1}$-root system, namely the fan whose maximal cones are Weyl chambers of $A_{n-1}$-root system on the weight lattice $\overline{N}=\mathbb Z^{n-1}$. Here, we adopt somewhat non-standard realization of $A_{n-1}$-root system. Regardless of inner product, we set the non-zero vectors in $\overline{N}=\mathbb Z^{n-1}$ whose entries are 0 or 1 the positive primitive weight vectors, and the negative primitive weight vectors are the negation of the positive primitive weight vectors. Note that this determines an $\mathfrak S_n$-action on the lattice $\overline{N}=\mathbb Z^{n-1}$, namely for $k<n-1$

$$\begin{aligned} (k\ \ k+1)&= \left( \begin{array}{c|c@{\quad }c|c} I_{k-1} &{} &{} &{} \\ \hline &{} 0 &{} 1 &{} \\ &{} 1 &{} 0 &{} \\ \hline &{} &{} &{} I_{n-k-2} \end{array}\right) \; \text{ and } \; \\ (n-1\ \ n)= & {} \begin{pmatrix} 1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad -1 \\ 0 &{}\quad 1 &{}\quad \cdots &{}\quad 0 &{}\quad -1 \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 &{}\quad -1\\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad -1 \end{pmatrix}. \end{aligned}$$

Now we consider the projective toric variety $X(\bar{\Delta }^{(n)})$. It has been studied by several authors [7, 17, 22] in connection with combinatorics theory. Speaking in a geometric language, $X(\bar{\Delta }^{(n)})$ is the canonical elimination of indeterminacy of the standard Cremona transformation of degree $n-1$ in $\mathbb P^{n-1}$ ([6], Example 7.2.5). More precisely, we have a sequence of projective birational morphisms

$$\begin{aligned} g: X(\bar{\Delta }^{(n)})=X_1\overset{g_1}{\longrightarrow } X\longrightarrow \cdots \longrightarrow X_{n-2}\overset{g_{n-2}}{\longrightarrow } X_{n-1}=\mathbb P^{n-1}, \end{aligned}$$

where $g_i$ is the blowing-up of the strict transform of the union of the linear subspaces defined by vanishing of $i+1$ projective coordinates ([7], Lemma 5.1). We take a fan $\Phi$ in $\overline{N}=\mathbb Z^{n-1}$ spanned by

$$\begin{aligned} v_1=\begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix},\; v_2=\begin{pmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix},\; \dots ,\; v_{n-1}=\begin{pmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix}, \; v_n=\begin{pmatrix} -1 \\ -1 \\ \vdots \\ -1 \end{pmatrix}. \end{aligned}$$

The fan $\bar{\Delta }^{(n)}$ is a subdivision of $\Phi$ and the associated toric morphism $X(\bar{\Delta }^{(n)})\rightarrow X(\Phi )=\mathbb P^{n-1}$ is nothing but the above-mentioned birational morphism g.

Let $D_{pos}$ (resp. $D_{neg}$) be a torus invariant divisor on a toric variety $X(\bar{\Delta }^{(n)})$ corresponding to the sum of positive (resp. negative) primitive weight vectors. We take a homogeneous coordinate $[x_1:\dots : x_n]$ on $\mathbb P^{n-1}$ such that the prime divisor corresponding to $v_j$ is the hyperplane $(x_j=0)$ for $1\le j\le n$. Then, the $\mathfrak S_n$-action on the toric variety $X(\Phi )$ coincides with the natural permutation of coordinates on $\mathbb P^{n-1}$:

$$\begin{aligned} s\cdot [x_1:\dots :x_n] = [x_{s(1)}:\dots :x_{s(n)}]\quad (s\in \mathfrak S_n). \end{aligned}$$

Under this choice of coordinate, we have $D_{neg}=g^* {{\mathrm{div}}}(x_n)$. Let $\Phi '$ be another $\mathbb P^{n-1}$-fan on $\overline{N}$ spanned by $-v_1,\; -v_2,\; \dots ,\; -v_n$. $\bar{\Delta }^{(n)}$ is again a subdivision of $\Phi '$ and let $h:X(\bar{\Delta }^{(n)})\rightarrow X(\Phi ')=\mathbb P^{n-1}$ be the associated birational morphism. Then, the composition is nothing but the standard Cremona transformation

$$\begin{aligned}{}[x_1:\dots :x_n]\mapsto \left[ \frac{1}{x_1} : \dots : \frac{1}{x_n}\right] , \end{aligned}$$

and we have $D_{pos}=h^*{{\mathrm{div}}}(x_n)$. In the below, we sometimes denote the toric variety $X(\bar{\Delta }^{(n)})$ by $X(A_{n-1})$ for easy recognition.

Proposition 2.5

The toric variety $\widetilde{Z}^{(n) \prime }=X(\Delta ^{(n)})$ is isomorphic to the total space of a rank 2 vector bundle

$$\begin{aligned} \mathscr {O}(-D_{pos})\oplus \mathscr {O}(-D_{neg}) \end{aligned}$$

over $X(\bar{\Delta }^{(n)})=X(A_{n-1})$. Moreover, the projection

$$\begin{aligned} \eta ^{(n)}: X(\Delta ^{(n)}) \rightarrow X(\bar{\Delta }^{(n)}) \end{aligned}$$

is $\mathfrak S_n$-equivariant.

Proof

Let $Q: N\rightarrow N$ be an automorphism defined by left multiplication of the matrix

$$\begin{aligned} Q= \begin{pmatrix} 1 &{}\quad 1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad -1 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad -1 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad \cdots &{}\quad 0 &{}\quad -1 \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 &{}\quad -1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 1 \end{pmatrix}. \end{aligned}$$

Then, we see that

(6)

In particular, $Q\delta ^{(n)}$ is generated by column vectors of

Let $p:N=\mathbb Z^{n+1} \rightarrow \overline{N}=\mathbb Z^{n-1}$ be the projection to middle $(n-1)$-factors. From the matrix representation (3), it is easy to see that the composition

$$\begin{aligned} \Pi =p\circ Q:N\rightarrow \overline{N} \end{aligned}$$

is $\mathfrak S_n$-equivariant. As the maximal cones in $\Delta ^{(n)}$ are $\mathfrak S_n$-translates of $\delta ^{(n)}$ and $\Pi$ is $\mathfrak S_n$-equivariant, we know that $\Pi$ is compatible with fans $\Delta ^{(n)}$ and $\bar{\Delta }^{(n)}$, and induces a toric morphism $\eta ^{(n)}: X(\Delta ^{(n)})\rightarrow X(\bar{\Delta }^{(n)})$. Moreover, one sees from (6) and [5] Proposition 7.3.1 that $X(\Delta ^{(n)})$ is the total space of the direct sum of line bundles $\mathscr {O}(-D_{pos})\oplus \mathscr {O}(-D_{neg})$.

$\square$

2.6.   The coordinate $[x_1:\dots :x_n]$ on $\mathbb P^{n-1}$ gives a convenient system of local coordinate on $X(A_{n-1})$ as in the following way. Let $\bar{\delta }^{(n)}$ be the positive Weyl chamber generated by the column vectors of

$$\begin{aligned} \begin{pmatrix} 1 &{}\quad 1 &{}\quad \cdots &{}\quad 1 \\ 0 &{}\quad 1 &{}\quad \cdots &{}\quad 1 \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 \end{pmatrix}. \end{aligned}$$

This is a smooth cone and the corresponding affine open subset U, which is isomorphic to $\mathbb C^{n-1}$, has a toric coordinate

$$\begin{aligned} \left( \frac{x_1}{x_2},\frac{x_2}{x_3},\ldots ,\frac{x_{n-1}}{x_n}\right) . \end{aligned}$$

A maximal cone in the $A_{n-1}$-Coxeter fan $\bar{\Delta }^{(n)}$ is written as $s\cdot \bar{\delta }^{(n)}$ for some $s\in \mathfrak S_n$. It is immediate to see that the corresponding affine open subset $U_s$ has a toric coordinate

$$\begin{aligned} \left( \frac{x_{s(1)}}{x_{s(2)}},\frac{x_{s(2)}}{x_{s(3)}},\ldots ,\frac{x_{s(n-1)}}{x_{s(n)}}\right) . \end{aligned}$$

2.7.   Let $\lambda =(\lambda _1,\ldots , \lambda _r)$ be a partition of n, i.e., let $\lambda$ satisfy $\lambda _1\geqslant \cdots \geqslant \lambda _r>0$ and $\lambda _1+\cdots +\lambda _r=n$. We set $l_j=\sum \nolimits _{i=1}^j \lambda _i$ for $1\leqslant j\leqslant r$ (and $l_0=0$ for convenience). Let $c_j\; (1\leqslant j\leqslant r)$ be a cyclic permutation

$$\begin{aligned} c_j=(l_{j-1}+1\; \ldots \; l_j) \end{aligned}$$

of length $\lambda _j$ and we take a standard element

$$\begin{aligned} \begin{aligned} s=s_{\lambda }&=(1 \; \ldots \; l_1)(l_1+1\; \ldots \; l_2)\cdots (l_{r-1}+1\; \ldots \; l_r)\\&=c_1c_2 \ldots c_r \end{aligned} \end{aligned}$$

(7)

of $\mathfrak S_n$ in the conjugacy class determined by $\lambda$.

Let $\overline{N}^{\langle s\rangle }$ be the sublattice of s-fixed vectors. If we denote the standard basis of $\overline{N}=\mathbb Z^{n-1}$ by $e_1,\dots , e_{n-1}$, $\overline{N}^{\langle s\rangle }$ is generated by

$$\begin{aligned} e_1+\cdots +e_{l_1},\; e_{l_1+1}+\cdots +e_{l_2},\; \dots , \; e_{l_{r-2}+1}+\cdots +e_{l_{r-1}}, \end{aligned}$$

so that $\overline{N}^{\langle s \rangle }\cong \mathbb Z^{r-1}$. Every s-fixed point on $X(A_{n-1})$ is contained in an open subset

$$\begin{aligned} X_{\overline{N}}(\bar{\Delta }^{(n)}\cap (N^{\langle s\rangle }\otimes \mathbb R))\cong (\mathbb C^*)^{n-r} \times X(A_{r-1}), \end{aligned}$$

which $\mathfrak S_n$-equivalently birationally dominates $(\mathbb C^*)^{n-r}\times \mathbb P^{r-1}$. Let

$$\begin{aligned} t_{ji}=\frac{x_{l_{j-1}+i+1}}{x_{l_{j-1}+i}},\; \mathbf t_j=(t_{j1},\ldots , t_{j,\lambda _j-1}), \; \text{ and } \; y_k = x_{l_{k-1}+1}. \end{aligned}$$

Then,

$$\begin{aligned} \left( (\mathbf t_1,\ldots , \mathbf t_r),[y_1:\ldots :y_r]\right) = \left( (t_{11},\ldots , t_{1,\lambda _1-1}\, ;\, \ldots \, ;\, t_{r1},\ldots , t_{r,\lambda _r-1}),[y_1:\ldots :y_r]\right) \end{aligned}$$

is a coordinate on $(\mathbb C^*)^{n-r}\times \mathbb P^{r-1}$ and the action of $c_j$ is given by

$$\begin{aligned}&\left( (\mathbf t_1 ; \, \ldots \, ; \;\; t_{j1},\ldots , t_{j,\lambda _j-1} \;\; ;\, \ldots \,; \; \mathbf t_r ),[y_1:\cdots :y_j:\cdots :y_r]\right) \\&\quad \mapsto \left( \left( \mathbf t_1 ; \ldots ; \; t_{j2},\ldots , t_{j,\lambda _j-1}, \frac{1}{t_{j1}\cdots t_{j,\lambda _j-1}} \; ; \ldots \,;\; \mathbf t_r\right) , [y_1:\cdots :t_{j1}y_j:\cdots :y_r]\right) . \end{aligned}$$

In particular, if a point on $(\mathbb C^*)^{n-r} \times X(A_{r-1})$ is fixed by s, we necessarily have

$$\begin{aligned} t_{j1}=\cdots =t_{j,\lambda _j-1}=\alpha _j \end{aligned}$$

for all $1\leqslant j\leqslant r$, where $\alpha _j$ is a $\lambda _j$th root of unity. Let us fix such a point

$$\begin{aligned} \varvec{\alpha }=(\alpha _1,\ldots , \alpha _1;\ldots ;\alpha _r,\ldots , \alpha _r) \in (\mathbb C^*)^{n-r} \end{aligned}$$

and $p\in \mathfrak S_r$. Let $U_p\cong \mathbb C^{r-1}$ be the affine open subset of $X(A_{r-1})$ corresponding to p. At a point $(\varvec{\alpha },\xi )\in (\mathbb C^*)^{n-r}\times U_p$, the acton of $c_j$ is given by

$$\begin{aligned}&\left( \varvec{\alpha }, \left( \frac{y_{p(1)}}{y_{p(2)}},\dots ,\frac{y_{p(r-1)}}{y_{p(r)}}\right) \right) \nonumber \\&\quad \mapsto \left( \varvec{\alpha }, \left( \frac{y_{p(1)}}{y_{p(2)}},\dots , \frac{1}{\alpha _j}\cdot \frac{y_{p(p^{-1}(j)-1)}}{y_j}, \alpha _j \cdot \frac{y_j}{y_{p(p^{-1}(j)+1)}}, \dots , \frac{y_{p(r-1)}}{y_{p(r)}}\right) \right) , \end{aligned}$$

(8)

thus the action of $s=c_1\ldots c_r$ is of the form

$$\begin{aligned} \left( \varvec{\alpha }, \left( \frac{y_{p(1)}}{y_{p(2)}},\dots ,\frac{y_{p(r-1)}}{y_{p(r)}}\right) \right) \mapsto \left( \varvec{\alpha }, \left( \frac{\alpha _{p(1)}}{\alpha _{p(2)}}\cdot \frac{y_{p(1)}}{y_{p(2)}},\dots , \frac{\alpha _{p(r-1)}}{\alpha _{p(r)}}\cdot \frac{y_{p(r-1)}}{y_{p(r)}}\right) \right) . \end{aligned}$$

(9)

In particular, for the torus invariant point $\xi _p$, the origin of $U_p\cong \mathbb C^{r-1}$, $(\varvec{\alpha }, \xi _p)$ is always an s-fixed point.

2.8.    Now we consider the action of s on the fiber $\mathbb C^2$ of $\mathscr {O}(-D_{pos})\oplus \mathscr {O}(-D_{neg})$ over a fixed point. In §2.4, we saw that $D_{neg}=g^*{{\mathrm{div}}}(x_n)$. Noting that $n=l_r$ and $s(n)=l_{r-1}+1$, we have

$$\begin{aligned} (s^{-1})^* D_{neg}=g^*{{\mathrm{div}}}(x_{l_{r-1}+1}) = D_{neg}+{{\mathrm{div}}}\left( \frac{x_{l_{r-1}+1}}{x_n}\right) , \end{aligned}$$

hence an isomorphism of invertible sheaves

Let us take an s-fixed point $(\varvec{\alpha },\xi )\in (\mathbb C^*)^{n-r}\times U_p\; (p\in \mathfrak S_r)$ and set $\nu =p^{-1}(r)$. Then, it is easy to see that

$$\begin{aligned} \mathscr {O}_{(\mathbb C^*)^{n-r}\times U_p}(-D_{neg}) =\frac{y_{p(\nu +1)}}{y_{p(\nu )}} \dots \frac{y_{p(r)}}{y_{p(r-1)}} \mathscr {O}_{(\mathbb C^*)^{n-r}\times U_p}. \end{aligned}$$

Therefore, for any $\xi \in U_p$, the action of s on the fiber $\mathscr {O}(-D_{neg})\otimes \kappa (\varvec{\alpha }, \xi )$ is given by the composition

namely by a multiplication of $\alpha _{p(r)}$. One also sees by the same argument that the action of s on the fiber $\mathscr {O}(-D_{pos})\otimes \kappa (\varvec{\alpha }, \xi )$ is given by the multiplication of $\alpha _{p(1)}^{-1}$.

Lemma 2.9

Let $s\in \mathfrak S_n$ as in (7) and take a fixed point $q=(q_1,q_2)\in \widetilde{Z}^{(n)}=X(\Delta ^{(n)})\times \mathbb C^n$. Assume $\eta ^{(n)}(q_1)=(\varvec{\alpha }, \xi )\in (\mathbb C^*)^{n-r}\times U_p\subset X(\bar{\Delta }^{(n)})$. Then, the eigenvalues of the Jacobian matrix $J_q(s)$ of s at q is

$$\begin{aligned} (\underbrace{ \zeta _1,\dots , \zeta _1 ^{\lambda _1-1}; \; \dots \; ; \zeta _r, \dots ,\zeta _r ^{\lambda _r-1} }_{{\mathrm{(i)}}}; \ \ \ \underbrace{ \frac{\alpha _{p(1)}}{\alpha _{p(2)}}, \dots , \frac{\alpha _{p(r-1)}}{\alpha _{p(r)}} }_{{\mathrm{(ii)}}}; \\ \underbrace{ \alpha _{p(1)}^{-1}, \alpha _{p(r)} }_{{\mathrm{(iii)}}}; \ \ \ \underbrace{ 1, \zeta _1,\dots , \zeta _1 ^{\lambda _1-1}; \; \dots \; ; 1, \zeta _r, \dots ,\zeta _r ^{\lambda _r-1} }_{{\mathrm{(iv)}}} ). \end{aligned}$$

In particular, we always have $\det \left( J_q(s)\right) =1$.

Proof

The part (i) comes from the action of s on $(\mathbb C^*)^{n-r}$. More precisely, as we saw in §2.7, s acts on $(\mathbb C^*)^{n-r}$ by

$$\begin{aligned} (\dots ; \; t_{j1},\dots , t_{j,\lambda _j-1} \; ; \dots ) \mapsto \left( \dots ; \; t_{j2},\dots , t_{j,\lambda _j-1}, \frac{1}{t_{j1}\dots t_{j.\lambda _j-1}} \; ; \dots \right) , \end{aligned}$$

therefore the corresponding Jacobian matrix at $t_{ji}=\alpha _j\; (1\leqslant j\leqslant r,\, 1\leqslant i\leqslant \lambda _j-1)$ is a block matrix with components of the form

$$\begin{aligned} \begin{pmatrix} 0 &{}\quad 1 &{}\quad 0 &{}\quad \cdots &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 \\ -1 &{}\quad -1 &{}\quad -1 &{}\quad \cdots &{}\quad -1 \end{pmatrix}, \end{aligned}$$

whose characteristic polynomial is $1+t+\dots +t^{\lambda _j-1}$. This gives the part (i). The part (ii) comes from the action (9) of s on $U_p$, and the part (iii) is the action on the fiber of $\mathscr {O}(-D_{pos})\oplus \mathscr {O}(-D_{neg})$ (§2.8). The part (iv) is the contribution from the permutation representation of $\mathfrak S_n$ on the second factor $\mathbb C^n$ in $\widetilde{Z}^{(n)}$. $\square$

Theorem 2.10

The quotient $Z^{(n)}=\widetilde{Z}^{(n)}/\mathfrak S_n$ has only Gorenstein canonical quotient singularities and the singular fiber $(\rho ^{(n)})^{-1}(0)\subset Z^{(n)}$ of the induced family $\rho ^{(n)}: Z^{(n)}\rightarrow B$ is a divisor with V-normal crossings ([21], Definition (1.16)).

Proof

V-normal crossingness automatically follows from the fact that the morphism $\widetilde{Z}^{(n)} = X(\Delta ^{(n)})\times \mathbb C^n\rightarrow B$ is toric and each $s\in \mathfrak S_n$ acts on $\widetilde{Z}^{(n)}$ by a toric morphism. Therefore, it is enough to show that for every point $q\in \widetilde{Z}^{(n)}$, the stabilizer subgroup ${{\mathrm{Stab}}}_{\mathfrak S_n}(q)$ is contained in $SL(T_q\widetilde{Z}^{(n)})$. This is equivalent to say that for any $s\in \mathfrak S_n$ and s-fixed point $q\in \widetilde{Z}^{(n)}$, the determinant of the Jacobian matrix $J_q(s)$ for the action of s at q is 1, which is nothing but the last assertion of the lemma above. $\square$

Remark 2.10.1

It follows from the theorem that $X^{(n)}$ also has only Gorenstein canonical singularities (but not $\mathbb Q$-factorial, unlike $Z^{(n)}$). Since $\mu ^{(n)}:Z^{(n)}\rightarrow X^{(n)}$ is small, and therefore $K_{Z^{(n)}}=\mu ^{(n)\, *}K_{X^{(n)}}$, it is clear that $X^{(n)}$ has only canonical singularities. The following argument to show that $X^{(n)}$ is Gorenstein is suggested by the referee; Let D be an effective divisor such that $-D$ is $\mu ^{(n)}$-ample ([12], Lemma 6.28). Then, for a sufficiently small positive rational numver $\varepsilon$, the pair $(Z^{(n)}, \varepsilon D)$ is klt and $\mu ^{(n)}$ is a contraction of $(K_{Z^{(n)}}+\varepsilon D)$-negative extremal curves. Cone theorem ([12], Theorem 3.25) implies that there exists a Cartier divisor B on $X^{(n)}$ such that $K_{Z^{(n)}}=\mu ^{(n)\, *}B$ since $(K_{Z^{(n)}}\cdot C)=0$ for every curve C that is contracted by $\mu ^{(n)}$. This shows that $K_{X^{(n)}}\sim _{\mathbb Q} B$ and therefore $K_{X^{(n)}}$ is Cartier.

4 Stringy E-polynomial

3.1. The stringy E-function is a cohomological invariant defined for varieties with only log terminal singularities. We review the formula, which can be seen as a definition in our purpose, of the stringy E-function for (global) quotient variety. For the foundation of the theory of stringy E-functions, we refer [1, 24].

Let X be a variety. By general theory of mixed Hodge structures, the compact support cohomology $H_c^k(X,\mathbb Q)$ carries a canonical mixed Hodge structure, and hence the Hodge number $h^{p,q}(H^k_c(X))$ is defined. We define the E-polynomial $E(X)\in \mathbb Z[u,v]$ of X by

$$\begin{aligned} E(X)=\sum _{p,q,k} (-1)^k\ h^{p,q}(H^k_c(X))\ u^pv^q. \end{aligned}$$

As customary, we denote the E-polynomial of the affine line $\mathbb A^1$ by $\mathbb L$:

$$\begin{aligned} \mathbb L= E(\mathbb A^1)=uv. \end{aligned}$$

If X is a toric variety, X is stratified into tori of various dimensions in Zariski topology, therefore, E(X) can be written as a polynomial in $\mathbb L$.

Let M be a non-singular algebraic variety of dimension n and G a finite group acting on M. We denote a set of complete representatives of the conjugacy classes of G by ${{\mathrm{Conj}}}(G)$. Let $F_g$ be the locus of g-fixed points on M for $g\in {{\mathrm{Conj}}}(G)$. For each point $q\in F_g$, the Jacobian matrix $J_q(g)$ is diagonalizable. We list its eigenvalues as

$$\begin{aligned} \left( e^{2\pi i \theta _1}, \dots , e^{2\pi i \theta _n}\right) \quad (0\leqslant \theta _j< 1), \end{aligned}$$

where $\theta _j$ is a rational number whose denominator is a divisor of the order of g. We define the age (or shift number) of g at q by

$$\begin{aligned} {{\mathrm{age}}}(g; q) = \sum _{j=1}^n \theta _j. \end{aligned}$$

The age gives a locally constant function

$$\begin{aligned} {{\mathrm{age}}}: F_g\rightarrow \mathbb Q. \end{aligned}$$

For $\nu \in \mathbb Q$ , we define $F_{g,\nu }={{\mathrm{age}}}^{-1} (\nu )$. The age of g is an integer if and only of $J_q(g)\in SL(T_qM)$.

From now on, let us assume $J_q(g)\in SL(T_qM)$ for all $g\in G$ and $q\in M$, namely we assume M / G is Gorenstein canonical. We also assume that the quotient map $M\rightarrow M/G$ has no ramification divisor, i.e., ${{\mathrm{codim}}}F_g>1$ for any $g\in G$. Then, the stringy E-polynomial of the quotient variety M / G is given by

$$\begin{aligned} E_{st}(M/G) = E(M/G) + \!\!\! \sum _{\begin{array}{c} {{\mathrm{id}}}\ne g\in {{\mathrm{Conj}}}(G) \\ \nu \in \mathbb Z \end{array}} \!\!\! E\left( F_{g,\nu }/Z(g)\right) \cdot \mathbb L^{\nu }, \end{aligned}$$

(10)

where Z(g) is the centralizer of $g\in G$. The right hand side is called the orbifold E-function of M / G (see [1], Definition 6.3), which is known to be the same as the stringy E-function that, in turn, is defined in a quite different way ([1] Theorem 7.5). The summands other than E(M / G) are called twisted sectors, while we call E(M / G) the untwisted sector.

3.2.   Now, let us move on to the case $M=\widetilde{Z}^{(n)}$ and $G=\mathfrak S_n$. The untwisted sector $E(\widetilde{Z}^{(n)}/\mathfrak S_n)=E(Z^{(n)})$ can be calculated by the character formula for the cohomology group $H^*(X(A_{n-1}),\mathbb Q)$ due to Procesi, Dolgachev–Lunts, and Stembridge.

Lemma 3.3

$E(X(A_{n-1})/\mathfrak S_n)=(1+\mathbb L)^{n-1}$.

Proof

This is essentially Theorem 3.1 of [22], which states that the $\mathfrak S_n$-invariant part of the cohomology ring $H^*(X(A_{n-1}))^{\mathfrak S_n}$ has a basis $\left\{ y_J \; |\; J\subset \{ \text{ simple } \text{ roots } \} \right\}$ indexed by all the subset of the set of simple roots. Moreover the degree of $y_J$ is $2\cdot |J|$. Therefore, $H^{2k}(X(A_{n-1})) ^{\mathfrak S_n}$ is of dimension $\left( {\begin{array}{c}n-1\\ k\end{array}}\right)$. Since $X(A_{n-1})$ is a smooth toric variety, the whole $H^{2k}$ has Hodge type (k, k) ([5] Theorem 12.5.3), and we obtain

$$\begin{aligned} E(X(A_{n-1}))/\mathfrak S_n)=\sum _{k=0} ^{n-1} \left( {\begin{array}{c}n-1\\ k\end{array}}\right) (uv)^k = (1+\mathbb L)^{n-1}. \end{aligned}$$

$\square$

Proposition 3.4

$E(Z^{(n)})=\mathbb L^{n+2}(1+\mathbb L)^{n-1}$.

Proof

We recall that $\widetilde{Z}^{(n)} = X(\Delta ^{(n)})\times \mathbb C^n$ and $X(\Delta ^{(n)})$ is the total space of a rank 2 vector bundle over $X(A_{n-1})$. Poincaré duality implies

$$\begin{aligned} H^{2k+4}_c(X(\Delta ^{(n)})) \cong H^{2k} (X(A_{n-1})) \otimes H^4_c(\mathbb C^2) \end{aligned}$$

and therefore

$$\begin{aligned} H^{2k+2n+4}_c(\widetilde{Z}^{(n)}) \cong H^{2k} (X(A_{n-1})) \otimes H^4_c(\mathbb C^2)\otimes H^{2n}_c(\mathbb C^n). \end{aligned}$$

The one-dimensional space $H^{2m}_c(\mathbb C^m)$ is spanned by the fundamental class so that a finite group action always leaves it invariant. Thus we get

$$\begin{aligned} E(\widetilde{Z}^{(n)}/\mathfrak S_n) = E(X(A_{n-1})) \cdot \mathbb L^{n+2} = \mathbb L^{n+2}(1+\mathbb L)^{n-1} \end{aligned}$$

by the previous lemma. $\square$

3.5.   For a subset $M\subset \{1,\ldots , r\}$, we define $\mathfrak S M$ to be the subgroup consisting of elements in $\mathfrak S_r$ that leave each element of $\{1,\ldots , r\}\backslash M$ invariant. Let

$$\begin{aligned} \varphi : \{1,\ldots , r\}\rightarrow T \end{aligned}$$

be a map to a set T and $\{\beta _1,\ldots , \beta _k\}$ the set of its values. Taking the level sets $M(\varphi ) _j = \varphi ^{-1}(\beta _j)$ $(1\leqslant j\leqslant k)$, we get a partition $M(\varphi )=\{M(\varphi )_i\}$,

$$\begin{aligned} \{1,\ldots , r\}=\coprod _{j=1}^k M(\varphi )_j\, , \end{aligned}$$

and

$$\begin{aligned} m(\varphi ) = (|M(\varphi )_1|,\ldots , |M(\varphi )_k|) \end{aligned}$$

is a (not necessarily non-increasing) partition of r. We will call $M(\varphi )$ (or $m(\varphi )$) the multiplicity partition of $\varphi$. We define

$$\begin{aligned} \mathfrak S_{M(\varphi )}=\mathfrak SM(\varphi )_1 \times \dots \times \mathfrak SM(\varphi )_k \subset \mathfrak S_r. \end{aligned}$$

This is a Young subgroup of $\mathfrak S_r$.

3.6.   Let $\lambda =(\lambda _1,\dots , \lambda _r)$ be a length r partition of n. It defines a non-increasing map $\lambda : \{1,\dots ,r\}\rightarrow \mathbb Z$ by $j\mapsto \lambda _j$, and therefore we have the associated multiplicity partition $M(\lambda )$ and the Young subgroup $\mathfrak S_{M(\lambda )}$. If $m(\lambda ) = (m_1,\dots , m_k)$, we have

$$\begin{aligned} M(\lambda )_j = \left\{ \sum _{i=1}^j m_{i-1}+1,\dots , \sum _{i=1}^j m_i\right\} , \end{aligned}$$

where $m_0=0$ by convention. For each partition $\lambda =(\lambda _1,\ldots , \lambda _r)$, we define

$$\begin{aligned} \Theta _{\lambda } =\left\{ \theta =(\theta _1,\ldots , \theta _r) \; | \; \theta _j = \frac{a_j}{\lambda _j},\; a_j\in \mathbb Z,\; 0\leqslant a_j < \lambda _j\right\} \end{aligned}$$

and call an element $\theta \in \Theta _{\lambda }$ an angle type associated with $\lambda$. We say that an angle type $\theta =(\theta _1, \ldots ,\theta _r) \in \Theta _{\lambda }$ is standard if

$$\begin{aligned} \theta _{\sum _{i=1}^j m_{i-1}+1} \geqslant \dots \geqslant \theta _{\sum _{i=1}^j m_i} \end{aligned}$$

for all $1\leqslant j\leqslant k$. An angle type $\theta$ also determines a map

$$\begin{aligned} \theta : \{1,\ldots ,r\}\rightarrow \mathbb Q\cap [0,1) \end{aligned}$$

and we have the associated multiplicity partition $M(\theta )$ and the Young subgroup $\mathfrak S_{M(\theta )}$.

3.7.   Each angle type $\theta \in \Theta _{\lambda }$ determines a point $\varvec{\alpha }(\theta )\in (\mathbb C^*)^{n-r}$ by

$$\begin{aligned} \varvec{\alpha }(\theta ) = (\underbrace{e^{2\pi i \theta _1}, \dots ,e^{2\pi i \theta _1}}_{{(\lambda _1-1)-\mathrm{times}}}\ ;\ \ldots \ ;\ \underbrace{e^{2\pi i \theta _r}, \ldots ,e^{2\pi i \theta _r}}_{{(\lambda _r-1)-\mathrm{times}}} ). \end{aligned}$$

Now let $s=s_{\lambda }\in \mathfrak S_n$ be the standard element in the conjugacy class determined by $\lambda$, as in (7). Let $\overline{F}_s$ be the set of s-fixed points on $X(A_{n-1})$. As we saw in §2.7, a point in $\overline{F}_s$ is of the form $(\varvec{\alpha }(\theta ),\xi )\in (\mathbb C^*)^{n-r}\times X(A_{r-1}) \subset X(A_{n-1})$ for some angle type $\theta \in \Theta _{\lambda }$. Therefore, if we define

$$\begin{aligned} \overline{F}_s ^{\theta } =\overline{F}_s \cap \left( \{\varvec{\alpha }(\theta )\}\times X(A_{r-1})\right) , \end{aligned}$$

then we have $\overline{F}_s=\coprod \nolimits _{\theta \in \Theta _{\lambda }} \overline{F}_s^{\theta }$. We naturally identify $\overline{F}_s^{\theta }$ with the corresponding closed subset of $X(A_{r-1})$.

The centralizer Z(s) for $s\in \mathfrak S_n$ is generated by the cyclic permutations $c_1,\ldots , c_r$ in the notation of (7) and the permutations of the cycles of the same length among $\{c_1,\ldots ,c_r\}$ (see [20], Proposition 1.1.1). The subgroup H(s) generated by $c_1,\ldots , c_r$ in $\mathfrak S_n$ is a normal subgroup of Z(s) and we have $Z(s)/H(s)\cong \mathfrak S_{M(\lambda )}$.

Lemma 3.8

Let $V\subset \overline{F}_s$ be a union of connected components that is invariant under the action of the centralizer subgroup Z(s). Then, Z(s) acts on the cohomology group $H^*(V)$ via a natural action of $\mathfrak S_{M(\lambda )}$, namely $H^*(V/Z(s))\cong H^*(V/\mathfrak S_{M(\lambda )})$.

Proof

The permutations of the cycles of the same length in $\{c_1,\ldots , c_r\}$ act on $\mathbb P^{r-1}$ through corresponding permutations of homogeneous coordinates $[y_1:\dots :y_r]$, and accordingly they naturally act on $X(A_{r-1})$. The action is identified with the natural action of $\mathfrak S_{M(\lambda )}\subset \mathfrak S_r$. Therefore, it is sufficient to show that the subgroup H(s) acts on $H^*(V)$ trivially. H(s) leaves $\alpha (\theta )$ invariant and it acts on the torus invariant closed subset $\overline{F}_s^{\theta }$ as a finite subgroup of the open dense torus associated to $N^{\langle s\rangle }$ of $X(A_{r-1})$. H(s) also leaves $H^*(\overline{F}_{s}^{\theta }\cap V)$ invariant, since the cohomology group $H^*(\overline{F}_{s}^{\theta }\cap V)$ is generated by the fundamental cycles of torus invariant closed subvarieties of a smooth projective toric variety $\overline{F}_{s}^{\theta }\cap V$ (see [5], Lemma 12.5.1 and Theorem 12.5.3). $\square$

3.9 Let $\varphi :\{1, \ldots , r\}\rightarrow T$ be a map. We define the adjacency function

$$\begin{aligned} {{\mathrm{adj}}}(\varphi ):\{1, \ldots , r\}\rightarrow \mathbb Z \end{aligned}$$

of $\varphi$ inductively by

$$\begin{aligned} {{\mathrm{adj}}}(\varphi )(1)=1,\quad {{\mathrm{adj}}}(\varphi )(j)= {\left\{ \begin{array}{ll} {{\mathrm{adj}}}(\varphi )(j-1) &{} \text{ if } \quad \varphi (j)=\varphi (j-1) \\ {{\mathrm{adj}}}(\varphi )(j-1)+1 &{} \text{ if } \quad \varphi (j)\ne \varphi (j-1) \end{array}\right. } . \end{aligned}$$

Let $\lambda$ be a length r partition of n, $\theta \in \Theta _{\lambda }$ an angle type, and $p\in \mathfrak S_r$. We define a function

$$\begin{aligned} \theta \star p : \{1,\ldots , r\} \rightarrow \mathbb Z \end{aligned}$$

by $\theta \star p={{\mathrm{adj}}}(\theta \circ p)\circ p^{-1}$ and let $M(\theta \star p)$ be the corresponding level set partition of $\{1,\ldots , r\}$. The partition defines a Young subgroup $\mathfrak S_{M(\theta \star p)}\subset \mathfrak S_r$. As $M(\theta \star p)$ is a refinement of the partition $M(\theta )$, $\mathfrak S_{M(\theta \star p)}$ is a subgroup of $\mathfrak S_{M(\theta )}$. Let $\bar{p} =\mathfrak S_{M(\theta \star p)} p$ be the right coset in $\mathfrak S_r$ and $P(\theta )=\{\bar{p} \; |\; p\in \mathfrak S_r\}$. Then, it is easy to see that $P(\theta )$ is a partition of $\mathfrak S_r$. We define

$$\begin{aligned} \tau _{\bar{p}} = \bigcap _{q\in \bar{p}} q \bar{\delta }^{(r)}, \end{aligned}$$

where $\bar{\delta }^{(r)}$ is the positive Weyl chamber of the $A_{r-1}$-root system as in §2.6.

Proposition 3.10

The set of connected components of $\overline{F}_s^{\theta }$ agrees with the set of orbit closures $\{V(\tau _{\bar{p}})\; |\; \bar{p}\in P(\theta )\}$. Moreover, if $m(\theta \star p) = (r_1,\ldots , r_k)$, then $V(\tau _{\bar{p}}) \cong X(A_{r_1-1})\times \cdots \times X(A_{r_k-1})$.

Proof

A codimension one face of $p\bar{\delta }^{(r)}$ is cut out by a hyperplane of invariant vectors under a transposition $(p(j)\;\; p(j+1))$ for some $1\leqslant j<r$. The face corresponds to an affine line with coordinate $y_{p(j)}/y_{p(j+1)}$. Taking the action (9) into account, this affine line consists of s-fixed point if and only if $\theta _{p(j)}=\theta _{p(j+1)}$. On the other hand, it is easy to verify that

$$\begin{aligned} \mathfrak S_{M(\theta \star p)}=\left\langle (p(j)\;\; p(j+1))\; | \; \theta _{p(j)}=\theta _{p(j+1)}\right\rangle \subset \mathfrak S_r \end{aligned}$$

and if $q\in \bar{p}$, then $\mathfrak S_{M(\theta \star q)}=\mathfrak S_{M(\theta \star p)}$ as subgroups of $\mathfrak S_r$. By construction, the orbit $O(\tau _{\bar{p}})$ is a torus with the coordinates

$$\begin{aligned} \left\{ \frac{y_{p(j)}}{y_{p(j+1)}}\; \bigg | \; \theta _{p(j)}=\theta _{p(j+1)}\right\} . \end{aligned}$$

In particular, every point in $O(\tau _{\bar{p}})$, and therefore of $V(\tau _{\bar{p}})$, is fixed by s, namely $V(\tau _{\bar{p}})$ is a connected component of $\overline{F}_s^{\theta }$.

On the other hand, since s acts on $X(A_{r-1})$ via a cyclic subgroup of the torus, every connected component V of $\overline{F}_s^{\theta }$ is a torus invariant closed subset of $X(A_{r-1})$. In particular, V contains a torus invariant point $\xi _p$ corresponding to a maximal cone $p\bar{\delta }^{(r)}$ for some $p\in \mathfrak S_r$. The above argument shows that $V(\tau _{\bar{p}})$ is the connected component of $\overline{F}_s^{\theta }$ containing $\xi _p$. Hence we know that $V=V(\tau _{\bar{p}})$.

The image fan on $\overline{N}^{\langle s\rangle }{/}\left( \langle (\tau _{\bar{p}})\rangle _{\mathbb R}\cap \overline{N}^{\langle s \rangle }\right)$ corresponding to $V(\tau _{\bar{p}})$ is the Coxeter complex of the root system corresponding to the Young subgroup $\mathfrak S_{M(\theta \star p)}\cong \mathfrak S_{r_1}\times \cdots \times \mathfrak S_{r_k}$, where $m(\theta \star p)=(r_1,\ldots , r_k)$. It immediately follows that $V(\tau _{\bar{p}})\cong X(A_{r_1-1})\times \cdots \times X(A_{r_k-1})$. $\square$

3.11.    We denote the connected component $V(\tau _{\bar{p}})\subset \overline{F}_s^{\theta }$ by $\overline{F}_s^{\theta ,\bar{p}}$. Let $F_s$ be the closed subset of s-fixed points on $X(\Delta ^{(n)})$ and $F_s^{\theta ,\bar{p}}\subset X(\Delta ^{(n)})$ the union of connected components of $F_s$ that map to $\overline{F}_s^{\theta , \bar{p}}$. If we define $\phi (\theta , \bar{p})$ to be the number of 0 in the set $\{\theta _{p(1)},\, \theta _{p(r)}\}$, the calculation in §2.8 implies that $F_s^{\theta ,\bar{p}}\rightarrow \overline{F}_s^{\theta ,\bar{p}}$ is a vector bundle of rank $\phi (\theta , \bar{p})$. In particular $F_s^{\theta ,\bar{p}}$ is connected. We know that the s-fixed point locus of $\widetilde{Z}^{(n)}$ is a disjoint union of $F_s^{\theta ,\bar{p}}\times (\mathbb C^n)^{\langle s\rangle }\subset X(\Delta ^{(n)})\times \mathbb C^n = \widetilde{Z}^{(n)}$.

Lemma 3.12

The value of the age function at a point in $F_s^{\theta ,\bar{p}} \times (\mathbb C^n)^{\langle s\rangle }$ is given by

$$\begin{aligned} a(s;\theta , \bar{p}) = n -r + \sum _{j=1}^{r-1} \left\{ \theta _{p(j+1)} - \theta _{p(j)} \right\} + \{1-\theta _{p(1)}\}+ \theta _{p(r)}, \end{aligned}$$

(11)

where $\{t\}=t-\lfloor t \rfloor$ is the fractional part of t.

Proof

This is just a consequence of Lemma 2.9, noting that the contribution from the parts (i) and (iv) sum up to

$$\begin{aligned} \sum _{j=1}^r\sum _{i=1}^{\lambda _j-1} 2\cdot \frac{i}{\lambda _j} =\sum _{j=1}^r (\lambda _j-1) = n -r . \end{aligned}$$

$\square$

3.13 Let $\Theta _{\lambda } ^{st}$ be the set of standard angle types. $\mathfrak S_{M(\lambda )} \subset \mathfrak S_r$ acts on the set of angle types $\Theta _{\lambda }$ by permutation of the factors and each orbit contains a unique standard element $\theta$. Therefore, there is an identification $\Theta _{\lambda } / \mathfrak S_{M(\lambda )}\cong \Theta _{\lambda }^{st}$. The stabilizer subgroup of $\theta$ is $\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta )}$. Therefore, we get

$$\begin{aligned} \overline{F}_s / \mathfrak S_{M(\lambda )} = \left( \coprod _{\theta \in \Theta _{\lambda }} \overline{F}_s^{\theta } \right) / \mathfrak S_{M(\lambda )} \cong \coprod _{\theta \in \Theta _{\lambda }^{st}} \overline{F}_s^{\theta } / (\mathfrak S_{M(\lambda )}\cap \mathfrak S _{M(\theta )}). \end{aligned}$$

Let $\theta \in \Theta _{\lambda }^{st}$. Then, Proposition 3.10 says that we have a decomposition into connected components

$$\begin{aligned} \overline{F}_{s}^{\theta } = \coprod _{\bar{p}\in P(\theta )} \overline{F}_s^{\theta ,\bar{p}}, \end{aligned}$$

where $P(\theta )=\{\bar{p} = \mathfrak S_{M(\theta \star p)} p\; | \; p\in \mathfrak S_r\}$, and the isomorphism class of $\overline{F}_s^{\theta ,\bar{p}}=V(\tau _{\bar{p}})$ is determined only by the multiplicity partition $m(\theta \star p)$. $\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta )}$ naturally acts on $P(\theta )$. Let $\overline{P}(\theta )$ be a complete system of representative for the quotient set $P(\theta )/(\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta )})$. Then, we get

$$\begin{aligned} \overline{F}_s^{\theta } /(\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta )}) \cong \coprod _{\bar{p}\in \overline{P}(\theta )} \overline{F}_s^{\theta ,\bar{p}} /(\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta \star p)}). \end{aligned}$$

Combining what we have obtained above, finally we get the following

Theorem 3.14

We keep the notation above. The stringy E-polynomial of $Z^{(n)}$ is given by the following formula:

$$\begin{aligned} E_{st}(Z^{(n)})= & {} \mathbb L^{n+2}(1+\mathbb L)^{n-1} \nonumber \\&+ \sum _{\begin{array}{c} \lambda \vdash n \\ \lambda \ne (1^n) \end{array}}\sum _{\theta \in \Theta _{\lambda }^{st}} \sum _{\bar{p} \in \overline{P}(\theta )} E(\overline{F} _{s_{\lambda }} ^{\theta ,\overline{p}}/(\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta \star p)})) \cdot \mathbb L^{\phi (\theta , \bar{p})+r(\lambda )+a(s; \theta ,\bar{p})},\nonumber \\ \end{aligned}$$

(12)

where $s_\lambda$ is the standard permutation associated with $\lambda$ as in (7) and $r(\lambda )$ stands for the length of the partition $\lambda$.

3.15.   We remark that one can actually calculate the term $E(\overline{F} _{s_{\lambda }} ^{\theta ,\overline{p}}/(\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta \star p)}))$ in the formula (12) as follows.

We saw in Proposition 3.10 that

$$\begin{aligned} \overline{F}_s^{\theta , \bar{p}}\cong X(A_{r_1-1})\times \cdots \times X(A_{r_k-1}) \end{aligned}$$

if $m(\theta \star p)=(r_1,\ldots , r_k)$. Since $\mathfrak S_{M(\lambda )} \cap \mathfrak S_{M(\theta \star p)}$ is a Young subgroup of

$$\begin{aligned} \mathfrak S_{M(\theta \star p)}\cong \mathfrak S_{r_1}\times \cdots \times \mathfrak S_{r_k}, \end{aligned}$$

there is a partition $\mu _j\vdash r_j$ for each $1\leqslant j\leqslant k$ such that

$$\begin{aligned} \mathfrak S_{M(\lambda )} \cap \mathfrak S_{M(\theta \star p)} \cong \mathfrak S_{\mu _1}\times \cdots \times \mathfrak S_{\mu _k}, \end{aligned}$$

where $\mathfrak S_{\mu _j}\subset \mathfrak S_{r_j}$ is the Young subgroup associated with the partition $\mu _j\vdash r_j$. Therefore, we have

$$\begin{aligned} E(\overline{F} _{s_{\lambda }} ^{\theta ,\overline{p}}/(\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta \star p)})) = E(X(A_{r_1-1})/\mathfrak S_{\mu _1})\times \cdots \times E(X(A_{r_k-1})/\mathfrak S_{\mu _k}). \end{aligned}$$

On the other hand, $E(X(A_{r-1})/\mathfrak S_{\mu })$ for $\mu \vdash r$ can be calculated by the character formula of Procesi, Dolgachev–Lunts, and Stembridge: if we define $\chi _l(A_{n-1})$ to be the character of the $\mathfrak S_n$-representation $H^{2l}(X(A_{n-1}),\mathbb Q))$ and $\chi [A_{n-1},q]=\sum \nolimits _{l=0}^{n-1} \chi _l(A_{n-1})q^l$ the generating function, then we know that

$$\begin{aligned} 1+\sum _{n\geqslant 1} \chi [A_{n-1},q]t^n= & {} \frac{1+\sum \nolimits _{m\geqslant 1} h_m t^m}{1-\sum \nolimits _{m\geqslant 2}(q+\cdots +q ^{m-1}) h_mt^m} \\= & {} 1+ h_1t +h_2(1+q)t^2+(h_3+(h_1h_2+h_3)q+h_3q^2)t^3\\&+ \,(h_4+(h_2^2+h_1h_3+h_4)(q+q^2)+h_4q^3)t^4+\cdots , \end{aligned}$$

where $h_m$ is the character of the trivial representation of $\mathfrak S_m$, or rather, the complete symmetric function of degree m (see [22], Theorem 6.2, see also [7, 17]). Therefore, if $\mu =(m_1,\ldots , m_k)$, the character inner product gives the formula

$$\begin{aligned} E(X(A_{r-1}/\mathfrak S_{\mu })) = ( h_{\mu }, \chi [A_{r-1},\mathbb L] ), \end{aligned}$$

where $h_{\mu }=h_{m_1}\cdots h_{m_k}$.

3.16.   Let X be a variety with only log terminal singularities. The change of variable formula for motivic integration ([1], Theorem 3.5, or [24], Theorem 66) implies that if $f:Y\rightarrow X$ is a proper birational morphism that is crepant, i.e., $K_Y=f^*K_X$, the stringy E-function is invariant: $E_{st}(Y)=E_{st}(X)$ ([1], Theorem 3.8). In particular, if $f:Y\rightarrow X$ is a crepant resolution, we have $E_{st}(X)=E_{st}(Y)=E(Y)$.

In our situation, as the birational morphism $\mu ^{(n)}:Z^{(n)}\rightarrow X^{(n)}$ is small, it is in particular crepant. Thus we have $E_{st}(X^{(n)})=E_{st}(Z^{(n)})$. If the symmetric product $X^{(n)}$ admits a crepant resolution $Y\rightarrow X^{(n)}$, we have $E(Y)=E_{st}(Z^{(n)})$. Moreover if the induced family $Y\rightarrow B$ is semistable, and if we denote the singular fiber by $Y_0$, Poincaré duality and homotopy invariance implies

$$\begin{aligned} H^p_c (Y)^* \cong H^{2n-p+2}(Y)\cong H^{2n-p+2}(Y_0). \end{aligned}$$

Therefore, the polynomial $E(Y)=E_{st}(Z^{(n)})$ encodes the cohomological information of the singular fiber of a semistable model. In general, $X^{(n)}$ may not admit a crepant resolution; nevertheless the Gorenstein canonical orbifold model $Z^{(n)}$ is already a good substitute for a minimal semistable model of $X^{(n)}$ on the level of cohomology.

Proposition 3.17

The stringy E-polynomial $E_{st}(Z^{(n)})$ for $n=2,3,4,5$ is as follows:

$$\begin{aligned} E_{st}(Z^{(2)})= & {} \mathbb L^5+2\mathbb L^4+\mathbb L^3, \\ E_{st}(Z^{(3)})= & {} \mathbb L^7+3\mathbb L^6+5\mathbb L^5+2\mathbb L^4, \\ E_{st}(Z^{(4)})= & {} \mathbb L^9+4\mathbb L^8+11\mathbb L^7+14\mathbb L^6+4\mathbb L^5, \\ E_{st}(Z^{(5)})= & {} \mathbb L^{11}+5\mathbb L^{10}+17\mathbb L^9+35\mathbb L^8+30\mathbb L^7+6\mathbb L^6. \end{aligned}$$

Proof

We demonstrate the case $n=4$. The other cases are similar (but much more complicated in the case $n=5$). Nontrivial partition $\lambda$ of 4 is one of $\lambda = (2,1^2),\, (2^2),\, (3,1),\, (4)$. We calculate the contributions to twisted sectors case by case. We represent $p\in \mathfrak S_r$ by a sequence $p=[p(1), p(2), \ldots , p(r)]$ to make it short.

$\underline{\hbox {Case }\lambda = (2,1^2)}$. The length of the partition is $r=3$. The set of standard angle type in this case is $\Theta _{(2,1^2)}^{st} = \{ (0^3),\, (1/2,0^2)\}$. We also note that $\mathfrak S_{M(\lambda )}=\mathfrak S\{2,3\}$.

Case $\theta = (0^3)$: In this case, we have $\mathfrak S_{M(\theta )}=\mathfrak S_{M(\theta \star p)}=\mathfrak S_3$ for all $p\in \mathfrak S_3$. It immediately follows that $P(\theta )=\{\bar{1}\}$, $\overline{F}_{(1\; 2)}^{(0^3)} = X(A_2)$, and $\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta \star 1)}=\mathfrak S\{2,3\}$. We need to know $E(X(A_2)/\mathfrak S\{2,3\})$, which is calculated by the argument in §3.15 as follows:

$$\begin{aligned} E(X(A_2)/\mathfrak S\{2,3\})= & {} (h_1h_2,\chi [A_2,\mathbb L]) \\= & {} (h_1h_2, h_3+(h_1h_2+h_3)\mathbb L+h_3\mathbb L^2)\\= & {} 1+3\mathbb L+\mathbb L^2. \end{aligned}$$

Noting $\phi =2,\, a=1$, the associated twisted sector is $\mathbb L^8+3\mathbb L^7+\mathbb L^6$.

Case $\theta = (1/2,0^2)$: $\mathfrak S_{M(\theta )}=\mathfrak S\{2,3\}$. $P(\theta )$ consists of

$$\begin{aligned} \overline{[1,2,3]}=\overline{[1,3,2]}, \overline{[2,1,3]}, \overline{[3,1,2]}, \text{ and } \overline{[2,3,1]}=\overline{[3,2,1]}. \end{aligned}$$

However, $\overline{[2,1,3]}$ and $\overline{[3,1,2]}$ is in the same oribt under the action of $\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta )}=\mathfrak S\{2,3\}$, therefore we have $\overline{P}(\theta ) = \{ \bar{1},\, \overline{(1\; 2)},\, \overline{(1\; 2\; 3)}\}$. It is straightforward to get the following table:

p	$\mathfrak S_{M(\theta \star p)}$	$\theta \circ p$	$\phi$	a	Twisted sector
1	$\mathfrak S\{2,3\}$	(1 / 2, 0, 0)	1	2	$(1+\mathbb L)\mathbb L^6$
$(1\; 2)$	$\{1\}$	(0, 1 / 2, 0)	2	2	$\mathbb L^7$
$(1\; 2\; 3)$	$\mathfrak S\{2,3\}$	(0, 0, 1 / 2)	1	2	$(1+\mathbb L)\mathbb L^6$

In total, the contribution is $3\mathbb L^7+2\mathbb L^6$.

$\underline{\hbox {Case }\lambda = (2^2)}$. We have $r=2,\, \mathfrak S_{M(\lambda )}=\mathfrak S_2$, and

$$\begin{aligned} \Theta _{(2^2)}^{st}=\{(0,0),\, (1/2,0),\, (1/2,1/2)\}. \end{aligned}$$

Case $\theta = (0^2)$: $\mathfrak S_{M(\theta )}=\mathfrak S_{M(\theta \circ p)}=\mathfrak S_2$ for all $p\in \mathfrak S_2$, and $P(\theta )=\{\bar{1}\}$ as before. $\overline{F}_{(1\; 2)(3\; 4)}^{(0^2)} = X(A_1)$ and $\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta )}=\mathfrak S_2$, $\phi =2$, and $a=2$, so that the associated twisted sector is $E(X(A_1)/\mathfrak S_2)\mathbb L^6 = \mathbb L^7+\mathbb L^6$.

Case $\theta = (1/2,0)$: Since $\mathfrak S_{M(\theta )}=\{1\}$, $P(\theta )=\{\bar{1},\, \overline{(1\; 2)}\}$ and $\overline{F}_{(1\; 2)(3\; 4)}^{(0^2)}$ consists of two points. As we have $\phi =1,\, a=3$ in both cases, the associated twisted sector is $2\mathbb L^6$.

Case $\theta = (1/2,1/2)$: $\mathfrak S_{M(\theta )}=\mathfrak S_{M(\theta \star p)}=\mathfrak S_2$, and $P(\theta )=\{\bar{1}\}$. As before, we know $\overline{F}_{(1\; 2)(3\; 4)}^{(1/2,1/2)} = X(A_1)$ and $\mathfrak S_{M(\lambda )}\cap \mathfrak S_{M(\theta )}=\mathfrak S_2$. Since $\phi =0, a=3$ the associated twisted sector is $E(X(A_1)/\mathfrak S_2)\mathbb L^5 = \mathbb L^6+\mathbb L^5$.

$\underline{\hbox {Case}\,\, \lambda = (3,1)}$. In this case, $r=2,\, \mathfrak S_{M(\lambda )}=\{1\}$, $\Theta _{(3,1)}^{st} = \{ (0^2),\, (1/3,0),\, (2/3,0)\}$.

Case $\theta = (0^2)$: As before, $\mathfrak S_{M(\theta )}=\mathfrak S_{M(\theta \star p)}=\mathfrak S_2$ and $P(\theta ) =\{1\}$. Therefore, $\overline{F}_{(1\; 2\; 3)}^{(0^2)}=X(A_1)$. As $\phi =2, a=2$, the twisted sector is $\mathbb L^7+\mathbb L^6$.

Case $\theta = (1/3,0)$: $\mathfrak S_{M(\theta )}=1$, $P(\theta ) = \{\bar{1},\, \overline{(1\; 2)}\}$ and $\overline{F}_{(1\; 2\; 3)}^{(1/3,0)}$ consists of two points. As $\phi =1, a=3$ in this case, the twisted sector is $2\mathbb L^6$.

Case $\theta = (2/3,0)$: This case is completely the same as in the case $\theta =(1/3,0)$. The twisted sector is $2\mathbb L^6$.

$\underline{\hbox {Case }\lambda = (4)}$. In this case $r=1$ and we always have $\mathfrak S_{M(\lambda )} =\mathfrak S_{M(\theta )}=\{1\}$. Therefore $P(\theta )=\{\bar{1}\}$ and $\overline{F}^{\theta }_{(1\; 2\; 3\; 4)}$ is just one point set. As we have

$\theta$	$\phi$	a	Twisted sector
(0)	2	3	$\mathbb L^6$
(1 / 4), (2 / 4), (3 / 4)	0	4	$\mathbb L^5$

the twisted sector in total is $\mathbb L^6+3\mathbb L^5$.

Summing up everything, finally we get

$$\begin{aligned} E_{st}(X^{(4)})= & {} \mathbb L^6 (1+\mathbb L)^3 \\&+\left( \mathbb L^8+3\mathbb L^7+\mathbb L^6\right) +\left( 3\mathbb L^7+2 \mathbb L^6\right) +\left( \mathbb L^7+ \mathbb L^6\right) +2\mathbb L^6 \\&+ \left( \mathbb L^6+ \mathbb L^5\right) +\left( \mathbb L^7+\mathbb L^6\right) +2\mathbb L^6+2\mathbb L^6 + \left( \mathbb L^6+ 3\mathbb L^5\right) \\= & {} \mathbb L^9+4\mathbb L^8+11\mathbb L^7+14\mathbb L^6+4\mathbb L^5. \end{aligned}$$

$\square$

Remark 3.17.1

In comparison with the case of ${{\mathrm{Hilb}}}^n(S)$ for smooth algebraic surface S (see e.g. [16]), it is interesting to look for a formula of the generating function

$$\begin{aligned} \sum _{n\geqslant 0} E_{st}(Z^{(n)})\cdot t^n . \end{aligned}$$

Unfortunately, due to combinatorial complication in Theorem 3.14, the author does not yet have a good answer to this question at the time of writing.

5 Minimal model

In this section, we will discuss more birational modifications of $Z^{(n)}$, in particular minimal models of $Z^{(n)}$.

Theorem 4.1

There exists a projective birational morphism $\nu ^{(n)}:Y^{(n)} \rightarrow Z^{(n)}$ satisfying the following conditions:

• (i) $Y^{(n)}$ has only Gorenstein terminal quotient singularities.

• (ii) $\nu ^{(n)}$ is a crepant divisorial contraction, namely $K_{Y^{(n)}}=\nu ^{(n)\, *} K_{Z^{(n)}}$ and the exceptional set of $\nu ^{(n)}$ is a divisor.

• (iii) Let $\psi ^{(n)}:Y^{(n)}\rightarrow B$ be the composition $\rho ^{(n)}\circ \nu ^{(n)}$. Then, its general fiber is ${{\mathrm{Hilb}}}^n(\mathbb C^*\times \mathbb C)$ and the singular fiber is a divisor with V-normal crossings.

Although the existence of a minimal model of $Z^{(n)}$ is a consequence of the general theory of minimal model program (MMP) of higher dimensional algebraic varieties [3], here we stick to an explicit construction of a minimal model $Y^{(n)}$ so that we have a good control on the singularities of the total space and the singular fiber of the resulted minimal model. The claims (i) and (iii) will not follow from a straightforward application of MMP.

4.2.   From the description of §1.6, the natural morphism $\tilde{\rho }^{(n)\prime }:\widetilde{Z}^{(n)\prime }=X(\Delta ^{(n)})\rightarrow B=\mathbb C$ is a toric morphism associated with the lattice homomorphism

$$\begin{aligned} g= (1\; 1\; 0\; \dots \; 0) : N=\mathbb Z^{n+1} \rightarrow \mathbb Z. \end{aligned}$$

If we take a basis of N consisting of the column vectors of Q in the proof of Proposition 2.5, g is represented by a matrix $(1\; 0\; \dots \; 0\; 1)$. It immediately follows that the primitive generator for each ray in the fan $\Delta ^{(n)}$, namely the column vectors of (6), has multiplicity one with respect to g. This means that the fiber of $\tilde{\rho }^{(n)\prime }$ over the origin is the union of all the torus invariant divisors of $X(\Delta ^{(n)})$. It is easy to see from the description (3) the $\mathfrak S_n$-action on N that its restriction to ${{\mathrm{Ker}}}(g)$ is the permutation representation. Therefore, the restriction of $\tilde{\rho }^{(n)\prime }$ to the torus $N\otimes \mathbb C^*\rightarrow \mathbb C^*$ is a trivial family of the permutation action on $(\mathbb C^*)^n$, and the $\mathfrak S_n$-quotient of $N\otimes \mathbb C^*\times \mathbb C^n\rightarrow \mathbb C^*$ is a trivial family of ${{\mathrm{Sym}}}^n(\mathbb C^*\times \mathbb C)$. It follows that $Z^{(n) \circ }_n=\rho ^{(n)\; -1} (\mathbb C^*)$ has a crepant divisorial resolution

$$\begin{aligned} \psi ^{(n) \circ }: Y^{(n) \circ }\rightarrow Z^{(n) \circ } \end{aligned}$$

that is a family of Hilbert–Chow morphism ${{\mathrm{Hilb}}}^n(\mathbb C^*\times \mathbb C)\rightarrow {{\mathrm{Sym}}}^n(\mathbb C^*\times \mathbb C)$. We prove that $\psi ^{(n)\circ }$ extends to a crepant birational morphism $\psi ^{(n)}:Y^{(n)}\rightarrow Z^{(n)}$.

Lemma 4.3

Let $s\in \mathfrak S_n$. The connected component $F^{\theta , \bar{p}}_s$ (see §3.11) of the s-fixed point locus in $X(\Delta ^{(n)})$ has intersection with the open dense torus $N\otimes \mathbb C^*$ if and only if $\theta$ is a zero-sequence $(0^r)$ where r is the length of the partition $\lambda$ associated with s.

Proof

Let us assume that $F^{\theta , \bar{p}}_s$ has a point in common with $N\otimes \mathbb C^*$. Then, we necessarily have $\phi (\theta ,\bar{p})=2$, that is, $\theta _p{(1)}=\theta _{p(r)}=0$. A point $\left( \varvec{\alpha }, \left( \frac{y_{p(1)}}{y_{p(2)}},\ldots ,\frac{y_{p(r-1)}}{y_{p(r)}}\right) \right) \in X(A_{n-1})$ is in the open dense torus if and only if $\frac{y_{p(j)}}{y_{p(j+1)}}\ne 0$ for all j. Taking the action map (9) into account, it follows that $\theta _j- \theta _{j+1}\in \mathbb Z$. Therefore, we must have $\theta =(0^r)$. The converse also follows from the action map (9) and the definition of $F^{\theta , \bar{p}}_s$. $\square$

4.4.    Let $F^0$ be the union of the fixed point loci of trivial angle type $F_s^{(0^{r})}$ for all $s\in \mathfrak S_n$ (here r is the length of the associated partition $\lambda$ to $s\in \mathfrak S_n$). Since $F_s^{(0^r)}$ is the total space of a rank 2 vector bundle over $\overline{F}_s^{(0^{r-1})}\cong X(A_{r-1})$, we know $\dim F_s^{(0^{r})}=r+1$. In particular $F^0$ is a divisor in $X(\Delta ^{(n)})$.

Lemma 4.5

Let $q=(q_1,q_2)\in \widetilde{Z}^{(n)}$. We define

$$\begin{aligned} {{\mathrm{Stab}}}^0(q) = \{s\in {{\mathrm{Stab}}}(q)\; |\; q_1\in F_s^{(0^r)}\}. \end{aligned}$$

Then,

• (i) ${{\mathrm{Stab}}}^0(q)$ is a Young subgroup of $\mathfrak S_n$.

• (ii) ${{\mathrm{Stab}}}^0(q)$ is a normal subgroup of ${{\mathrm{Stab}}}(q)$.

Proof

(i) Let $\bar{q}_1$ be an image of $q_1$ under the composition

$$\begin{aligned} X(\Delta ^{(n)}) \rightarrow X(A_{n-1})\rightarrow \mathbb P^{n-1}. \end{aligned}$$

From the description in §2.7, one sees that $q_1$ is an s-fixed point of trivial angle type if and only if its coordinate satisfies the relation $\displaystyle \frac{x_{s(i)}}{x_i} = 1$ for all i such that $s(i)\ne i$. If $q_1$ is also t-fixed point of trivial angle typle, we have $\displaystyle \frac{x_{t\circ s(i)}}{x_i} = \frac{x_{t(s(i))}}{x_{s(i)}}\frac{x_{s(i)}}{x_i}=1$, and hence ${{\mathrm{Stab}}}^0(q)$ is a subgroup of ${{\mathrm{Stab}}}(q)$. Also by the characterization above, one sees that if $s\in {{\mathrm{Stab}}}^0(q)$ has a cycle decomposition

$$\begin{aligned} s = (i_1 \,\cdots \, i_{l_1})(i_{l_1+1}\, \cdots \, i_{l_2})\cdots (i_{l_{r-1}+1}\, \cdots \, i_{l_r}), \end{aligned}$$

all the elements in the subgroup

$$\begin{aligned} \mathfrak S\{i_1 \cdots i_{l_1}\} \times \mathfrak S\{i_{l_1+1} \cdots i_{l_2}\}\times \dots \times \mathfrak S\{i_{l_{r-1}+1} \cdots i_{l_r}\} \end{aligned}$$

is also contained in ${{\mathrm{Stab}}}^0(q)$. It implies that ${{\mathrm{Stab}}}^0(q)\subset \mathfrak S_n$ is a Young subgroup.

(ii) Assume that $s\in {{\mathrm{Stab}}}(q)\subset \mathfrak S_n$ has the partition type $\lambda$ of length r. Then, by Lemma 2.9, $s\in {{\mathrm{Stab}}}^0(q)$ if and only if the multiplicity of 1 in the eigenvalues of the action of s on $T_q\widetilde{Z}^{(n)}$ is $2r+1$. As the eigenvalues are constant in a conjugacy class, we know that ${{\mathrm{Stab}}}^0(q)$ is a normal subgroup of ${{\mathrm{Stab}}}(q)$. $\square$

4.6.    Let $q=(q_1,q_1)\in \widetilde{Z}^{(n)}=X(\Delta ^{(n)})\times \mathbb C^n$ and assume that $q_1\in F^0$. Then, by Lemma 2.9, the action of an element $s\in {{\mathrm{Stab}}}^0(q)$ on the tangent space $T_q\widetilde{Z}^{(n)}$ has eigenvalues

$$\begin{aligned}&(\zeta _1,\dots , \zeta _1 ^{\lambda _1-1}; \; \dots \; ; \zeta _r, \dots ,\zeta _r ^{\lambda _r-1};\; \underbrace{1,\dots , 1}_{r+1}\; ; \\&\qquad \qquad 1, \zeta _1,\dots , \zeta _1 ^{\lambda _1-1}; \; \dots \; ; 1, \zeta _r, \dots ,\zeta _r ^{\lambda _r-1} \; )\qquad \quad \end{aligned}$$

if s has partition type $\lambda =(\lambda _1,\ldots , \lambda _r)$. In particular, the character of the representation ${{\mathrm{Stab}}}^0(q)\rightarrow GL(T_q \widetilde{Z}^{(n)})$ agrees with the character of the permutation representation of the Young subgroup ${{\mathrm{Stab}}}^0(q)\subset \mathfrak S_n$ on $\mathbb C^n\oplus \mathbb C^n\oplus \mathbb C$, the direct sum of two copies of a permutation representation and a trivial representation. Let $U_q$ be a sufficiently small ${{\mathrm{Stab}}}(q)$-invariant open neighborhood of $q\in \widetilde{Z}^{(n)}$. If ${{\mathrm{Stab}}}^0(q)$ is of the form $\mathfrak S_{\mu }$ for a partition $\mu = (\mu _1,\ldots , \mu _k)$, $V_q=U_q/{{\mathrm{Stab}}}^0(q)$ is locally isomorphic to a product of a neighborhood of a general cycle in ${{\mathrm{Sym}}}^n(\mathbb C^2)$ of the form

$$\begin{aligned} \sum _{i=1}^k \mu _i a_i\quad (a_i\in \mathbb C^2), \end{aligned}$$

and a complex line $\mathbb C$. Therefore, it admits a crepant resolution $\widetilde{V}_q\subset {{\mathrm{Hilb}}}^n(\mathbb C^2)\times \mathbb C$. By a theorem of Haiman ([10], Theorem 5.1), $\widetilde{V}_q$ can be regarded as an open subset of $\mathfrak S_n\text{- }{{\mathrm{Hilb}}}(\mathbb C^{2n+1})$, so the quotient group $G_q={{\mathrm{Stab}}}(q)/{{\mathrm{Stab}}}^0(q)$ naturally acts on $\widetilde{V}_q$ and the quotient $\widetilde{V}_q/G_q$ gives a partial resolution

$$\begin{aligned} \psi ^{(n)}_{2,q} : \widetilde{V}_q/G_q \rightarrow \overline{U}_q \end{aligned}$$

of the image $\overline{U}_q$ of $U_q$ in $Z^{(n)}=\widetilde{Z}^{(n)}$. Since $\overline{U}_q\subset Z^{(n)}$ has only canonical singularities, $\psi ^{(n)}_{2,q}$ is again a crepant birational morphism. As the partial resolution $\psi ^{(n)}_{2,q}$ naturally glue with the Hilbert-Chow morpihsm $\psi ^{(n) \circ }: Y^{(n) \circ }\rightarrow Z^{(n) \circ }$ at the image of q in $Z^{(n)}$ for every $q=(q_1,q_2)$ with $q_1\in F^0$, we get an extended crepant partial resolution $\psi ^{(n)}:Y^{(n)}\rightarrow Z^{(n)}$.

4.7.   To finish the proof of Theorem 4.1, we check that $\psi ^{(n)}$ constructed above satisfies the conditions (i) and (iii).

Let $D_1,\ldots ,D_k\subset X(\Delta ^{(n)})$ be torus invariant prime divisors. At a point $q\in F^0$, they are defined by $y_j=0$ (not all but for some j’s) in the notation of §2.7. Lemma 2.9 implies that the intersection $D_1\cap \dots \cap D_k$ is (if not empty) transversal to the action of $\mathfrak S_n$. Therefore, the strict transform $D'_i$ of the image of $D_i$ in $\widetilde{V}_q$ is a smooth divisor and intersecting transversally along the exceptional divisor. As the quotient group $G_q={{\mathrm{Stab}}}(q)/{{\mathrm{Stab}}}^0(q)$ acts on the coordinate $(y_1,\ldots , y_r)$ via a torus $(\mathbb C^*)^r$, the normal crossing divisor $\sum D'_i$ is preserved by the action of $G_q$. Therefore, the singular fiber of $Y^{(n)}\rightarrow B$ is a divisor with V-normal crossings. This proves the condition (iii).

The condition (i) is a consequences of the characterization of terminal quotient singularity (see, [13] Theorem 2.3). Let $\tilde{\Gamma }$ be a unique smooth irreducible divisor on $\widetilde{Z}^{(n)}$ dominating $F^{(0^{n-1})}_{(1\; 2)}\subset X(\Delta ^{(n)})$ and $\Gamma$ the image of $\tilde{\Gamma }$ in $Z^{(n)}$. Lemma 3.12 implies that the age function always satisfies $a(s;\theta , \bar{p})\geqslant n-r$ if s has the partition type of length r (regardless of a choice of primitive root of unity). In particular, if $a(s;\theta , \bar{p})=1$, we necessarily have $r=n-1$, therefore the associated partition should be $\lambda =(2,1^{n-1})$. Moreover, in that case, one need to have $\theta _1=\dots =\theta _r$ and $\theta _r=0$, namely $\theta =(0^{n-1})$. This implies that an exceptional divisor of discrepancy 0 over $Z^{(n)}$ necessarily dominates $\Gamma$ (see [19] Remark (3.2)). On the other hand, as $\psi ^{(n)}$ is a crepant resolution of the singularity of $Z^{(n)}$ at the generic point of $\Gamma$, $Y^{(n)}$ has no crepant exceptional divisor over it. This implies (i) and completes the proof of Theorem 4.1.

4.8.   The recent construction of degeneration of Hilbert schemes by Gulbrandsen et al. [9] seems to be strongly related to the problem. In this paragraph, we use the notation of [9] freely. We can show that, for the expanded degeneration $X[n]\rightarrow \mathbb A^{n+1}$, there is a natural isomorphism between the GIT quotient of the stable locus of the relative symmetric product ${{\mathrm{Sym}}}^n(X[n]/\mathbb A^{n+1})^s$ by $G[n]=(\mathbb C^*)^n$ and our $Z^{(n)}$:

$$\begin{aligned} \varepsilon ^{(n)}:{{\mathrm{Sym}}}^n(X[n]/\mathbb A^{n+1})^s /\!\!/G[n] \overset{\sim }{\longrightarrow } Z^{(n)}. \end{aligned}$$

Therefore, the relative Hilbert–Chow morphism

$$\begin{aligned} {{\mathrm{Hilb}}}^n(X[n]/\mathbb A^{n+1})^s\rightarrow {{\mathrm{Sym}}}^n(X[n]/\mathbb A^{n+1})^s \end{aligned}$$

gives a birational morphism

$$\begin{aligned} \psi ^{(n),GHH}: I^n(X/\mathbb A^1)={{\mathrm{Hilb}}}^n(X[n]/\mathbb A^{n+1})^s /\!\!/G[n] \rightarrow Z^{(n)} \end{aligned}$$

over the base $B=\mathbb A^1$. As the authors claim in [9] that $I^n(X/\mathbb A^1)$ has only (abelian) quotient singularities and has trivial canonical bundle, it will immediately follow that $\psi ^{(n),GHH}$ is an extension of $\psi ^{(n)\circ }$ satisfying the conditions (i)–(iii) of Theorem 4.1. We will discuss the construction of $\varepsilon ^{(n)}$ and the comparison of $\psi ^{(n),GHH}$ and our $\psi ^{(n)}$ in a forthcoming article [15].

4.9.   The theorem asserts that $Y^{(n)}$ is a relatively minimal model of the symmetric product $X^{(n)}$ over B, as $K_{Y^{(n)}}$ is numerically trivial over $X^{(n)}$ and $K_{X^{(n)}}\equiv 0$. The general theory of MMP also suggests that there may be other minimal models of the symmetric product. Actually, if $n=2$ or 3, we can prove that the relative Hilbert scheme ${{\mathrm{Hilb}}}^n (S_2/B)$ is irreducible and admits a small resolution

$$\begin{aligned} H^{(n)}\rightarrow {{\mathrm{Hilb}}}^n (S_2/B)\rightarrow X^{(n)} \end{aligned}$$

such that the natural map $H^{(n)}\rightarrow B$ is semistable (see [14], Theorem 4.3, for the case $n=2$). Moreover, for $n=2$, we can explicitly write down the flop as follows.

The singular fiber of $p_2:S_2\rightarrow B$ consists of two components $S_{2,1}=(x_1=0)$ and $S_{2,2}=(x_2=0)$. Let $C= S_{2,1}\cap S_{2,2}\cong \mathbb A^1$ be the double line. The fiber of the relative Hilbert scheme ${{\mathrm{Hilb}}}^2(S_2/B)\rightarrow B$ consists of three components:

$$\begin{aligned} {{\mathrm{Hilb}}}^2(S_{2,1}),\; {{\mathrm{Hilb}}}^2(S_{2,2}),\; \text{ and } \text{ a } \text{ component } \overline{H}_{12} \text{ birational } \text{ to } S_0\times S_1. \end{aligned}$$

A non-trivial fiber occurs over a cycle $\gamma =2p\in X^{(2)}$. If the support of $\gamma$ lies in the smooth locus of $p_2$, the fiber of ${{\mathrm{Hilb}}}^2(S_2/B)\rightarrow X^{(2)}$ is $\mathbb P^1$. Let us denote the associated cycle by $l_1$. If the support is in the double curve C, the fiber is $\mathbb P^2$ and we denote the class of a line in this $\mathbb P^2$ by $l_2$.

The small resolution $h: H^{(2)}\rightarrow {{\mathrm{Hilb}}}^2(S_2/B)$ is given by a blowing-up along the (non-Cariter) divisor ${{\mathrm{Hilb}}}^2(S_{2,1})$.

Claim

The singular fiber of $H^{(2)}\rightarrow B$ consists of

$$\begin{aligned} H_{ii} = Bl _{{{\mathrm{Hilb}}}^2(C)}({{\mathrm{Hilb}}}^2(S_{2,i}))\;(i=1,2),\quad \text{ and } \quad H_{12}= Bl _{\Delta _C}(S_{2,1}\times S_{2,2}), \end{aligned}$$

where $\Delta _C$ is the diagonal of C in $C\times C\subset S_{2,1}\times S_{2,2}$.

Proof

Let D be the diagonal of $S_2\times S_2$ and W the strict transform of $(p_2\times p_2)^{-1}\Delta _B$ in $Bl_{D}(S_2\times S_2)$, where $\Delta _B\cong B$ is the diagonal of $B\times B$. Then, ${{\mathrm{Hilb}}}^2(S_2/B)$ is nothing but the quotient $W/\mathfrak S_2$. The fiber of $W\rightarrow \Delta _B\cong B$ over the origin $0\in B$ consists of four components

$$\begin{aligned} W_{ii} = Bl_{\Delta _{S_{2,i}}} (S_{2,i}\times S_{2,i}), \quad W_{ij} = Bl_{\Delta _C} (S_{2,i}\times S_{2,j}) \end{aligned}$$

with $i,j\in \{1,2\},\, i\ne j$. If we take $\widetilde{W}=Bl _{W_{11}} W=Bl _{W_{22}} W$, we have an isomorphism $H^{(2)}\cong \widetilde{W}/\mathfrak S_2$. Let us denote by $\widetilde{W}_{ii},\widetilde{W}_{ij}$ the strict transforms of $W_{ii}, W_{ij}$, respectively. It immediately follows that the fiber of $H^{(2)}\rightarrow B$ over $0\in B$ consists of

$$\begin{aligned} \begin{aligned} H_{ii}&\cong \widetilde{W}_{ii}/\mathfrak S_2=Bl _{{{\mathrm{Hilb}}}^2(C)}({{\mathrm{Hilb}}}^2(S_{2,i})) \quad \text{ and }\\ H_{12}&\cong \widetilde{W}_{12}\cong W_{12}=Bl_{\Delta _C} (S_{2,1}\times S_{2,2}), \end{aligned} \end{aligned}$$

noting that $W_{12}$ is smooth and $W_{12}\cap W_{11}$ is a Cartier divisor on $W_{12}$. $\square$

The exceptional divisor E of $Bl_{\Delta _C}(S_{2,1}\times S_{2,2})$ is isomorphic to $\mathbb P^2\times \Delta _C$ and the class of a line on the fiber $\mathbb P^2$ is $l_2$. Here, we remark that the morphism h restricted to the strict transform $C\times C\subset H_{12}$ of $C\times C\subset S_{2,1}\times S_{2,2}$ is the canonical morphism

$$\begin{aligned} C\times C\rightarrow {{\mathrm{Sym}}}^2(C), \end{aligned}$$

while $h_{|H_{12}}$ is birational. It in particular implies that the component $\overline{H}_{12}$ is non-normal.¹ One also sees from the description given in the proof of the claim above that $H_{11}\cap H_{22}$ is $\mathbb P^1$-bundle over ${{\mathrm{Hilb}}}^2(C)={{\mathrm{Sym}}}^2(C)$:

$$\begin{aligned} H_{11}\cap H_{22} = \mathbb P(N_{{{\mathrm{Hilb}}}^2(C)/{{\mathrm{Hilb}}}^2(S_{2,1})}), \end{aligned}$$

and $H_{11}\cap H_{22}\cap H_{12}$ is isomorphic to $C\times C$ that is embedded in $H_{11}\cap H_{22}$ as a double section over ${{\mathrm{Sym}}}^2(C)$. All the exceptional fibers of h are isomorphic to $\mathbb P^1$, whose numerical class we denote by $l_3$. The fiber $\mathbb P^1$ of $H_{11}\cap H_{22}\rightarrow {{\mathrm{Sym}}}^2(C)$ is numerically equivalent to $l_3$.

Now, the relative cone of curves $NE(H^{(2)}/X^{(2)})$ is spanned by $l_1,\, l_2,$ and $l_3$. An easy calculation shows that

$$\begin{aligned} H_{12}\cdot l_1=0,\quad H_{12}\cdot l_2=-2,\quad \text{ and }\quad H_{12}\cdot l_3=2. \end{aligned}$$

As the canonical bundle of $H^{(2)}$ is trivial by [14], Theorem 4.3, $(H^{(n)},\varepsilon H_{12})$ is klt for a sufficiently small positive rational number $\varepsilon$, and Cone Theorem guarantees that there is an extremal contraction of $l_2$, which is a small contraction that contracts E. One sees that $Y^{(2)}$ is nothing but its flop. Actually, the flop produces family of $\mathbb P^1$ over $\Delta _C$ passing through a $\frac{1}{2}(1,1,1,1)$-singularity coming from the fixed point locus with the angle type $\theta =(1/2)$. This is a locally trivial family of toric flop that is called “Francia flop” in [11] (Example 5.1 and Definition 4.1).