3.1 Preliminaries on moduli of cubic surfaces

We denote by

the four-dimensional moduli space of smooth (complex) cubic surfaces, where $\mathbb {P} H^0(\mathbb {P}^3,\mathcal {O}_{\mathbb {P}^3}(3))^\circ $ denotes the locus of smooth cubic surfaces embedded in $\mathbb {P}^3$ . We denote by

the GIT compactification, and by

(3.1) $$ \begin{align} \pi :{\mathcal {M}}^{\operatorname{K}}\longrightarrow {\mathcal {M}}^{\operatorname{GIT}} \end{align} $$

the Kirwan resolution of ${\mathcal {M}}^{\operatorname {GIT}}$ . The GIT stability for cubic surfaces has been completely described (see, e.g., [Reference MumfordMuk, §7.2(b)]). For a cubic surface $S\subseteq \mathbb {P}^3$ :

• • S is stable if and only if it has at worst $A_1$ singularities,

• • S is semi-stable if and only if it is stable, or has at worst $A_2$ singularities, and does not contain the axes of the $A_2$ singularities,

• • S is strictly polystable if and only if it is projectively equivalent to the so-called $3A_2$ cubic surface

  
  which has exactly three singular points, each of which is an $A_2$ singularity.

It is a classical result (see [Reference du Plessis and WallDvG+, (2.4)]) that the GIT compactification

$$ \begin{align*}{\mathcal {M}}^{\operatorname{GIT}}\cong \mathbb {P}(1,2,3,4,5) \end{align*} $$

is a weighted projective space. We will denote by

$$ \begin{align*}D_{A_1}\subseteq {\mathcal {M}}^{\operatorname{GIT}}\end{align*} $$

the so-called discriminant divisor, that is, the closure of the locus of (stable) cubics with an $A_1$ singularity. This divisor is irreducible, as the locus of corresponding cubics in $\mathbb {P} H^0(\mathbb {P}^3,\mathcal {O}_{\mathbb {P}^3}(3))$ is irreducible; the general point is given by the orbit of the locus of cubics of the form $x_0q(x_1,x_2,x_3)+f(x_1,x_2,x_3)$ with $q(x_1,x_2,x_3)$ a smooth quadric, and $f(x_1,x_2,x_3)$ a cubic.

We denote by

$$ \begin{align*}R\subseteq {\mathcal {M}}^{\operatorname{GIT}}\end{align*} $$

the so-called Eckardt divisor, which generically parameterizes smooth cubic surfaces having an Eckardt point—a point that lies on three lines contained in the cubic surface. This divisor is also irreducible, as the corresponding locus in $\mathbb {P} H^0(\mathbb {P}^3,\mathcal {O}_{\mathbb {P}^3}(3))$ is irreducible; the locus of smooth Eckardt cubics is the orbit of the locus of cubics of the form $x_0^2\ell (x_1,x_2,x_3)+f(x_1,x_2,x_3)$ with $\ell (x_1,x_2,x_3)$ a linear form and $f(x_1,x_2,x_3)$ a cubic, with the line and cubic meeting transversally. In the coordinates of the previous equation, the point $(1:0:0:0)$ is then an Eckardt point, and the three lines through the Eckardt point are the ones determined by the Eckardt point and the intersection $\{\ell =f=0\}$ in the hyperplane at infinity. Such an Eckardt cubic in these coordinates has an involution given by $x_0\mapsto -x_0$ . Note that a general smooth cubic surface with an Eckardt point has a unique Eckardt point, and has automorphism group $\mathbb {Z}_2$ , and moreover, the Eckardt divisor R contains the locus of all smooth cubic surfaces that have any nontrivial automorphism (see, e.g., [Reference Dardanelli and van GeemenDD, Table 1 and Fig. 1]).

From the description of stability given above, it follows that ${\mathcal {M}}^{\operatorname {GIT}}$ contains a unique strictly polystable point $\Delta _{3A_2}\in {\mathcal {M}}^{\operatorname {GIT}}$ corresponding to the orbit of the $3A_2$ cubic $S_{3A_2}$ , and consequently the Kirwan resolution ${\mathcal {M}}^{\operatorname {K}}$ of ${\mathcal {M}}^{\operatorname {GIT}}$ is a blowup with center supported at $\Delta _{3A_2}$ (see, e.g., [Reference KirwanKi2], [Reference ZhangZha]); we denote by

$$ \begin{align*}D_{3A_2}\subseteq {\mathcal {M}}^{\operatorname{K}}\end{align*} $$

the exceptional divisor, which is irreducible (as is the case for exceptional divisors in Kirwan blowups, being the quotient of an open subset of the blowup of a smooth irreducible subvariety of a smooth irreducible variety). Note that ${\mathcal {M}}^{\operatorname {K}}$ and $D_{3A_2}$ are, by construction, smooth up to finite quotient singularities. We will denote by $\widetilde {D}_{A_1}\subseteq {\mathcal {M}}^{\operatorname {K}}$ the strict transform of the discriminant divisor $D_{A_1}\subseteq {\mathcal {M}}^{\operatorname {GIT}}$ , and by $\widetilde {R}\subseteq {\mathcal {M}}^{\operatorname {K}}$ the strict transform of the Eckardt divisor $R\subseteq {\mathcal {M}}^{\operatorname {GIT}}$ .

3.2 Geometry of ${\mathcal {M}}^{\operatorname {GIT}}$ as a weighted projective space

In full generality, consider a weighted projective space $\mathbb {P}(q_0,\dots , q_n)=\operatorname {Proj}\mathbb {C}[x_0,\dots ,x_n]=(\mathbb {C}^{n+1}-\{0\})/\mathbb {C}^*$ , where the weight of $x_i$ (resp. the weight of the action of $\mathbb {C}^*$ on $x_i$ ) is $q_i$ for $i=0,\dots ,n$ , where, without loss of generality, we assume that $\gcd (q_0,\dots , q_n)=1$ [Reference DolgachevDol Reference Abramovich, Graber and Vistoli1, Prop. p. 37]. Denoting , where $\mu _{\ell }$ is the multiplicative group of roots of unity of order $\ell $ , and letting the group $G_{\vec q}$ act diagonally on $\mathbb {P}^n$ , we can express the weighted projective space as the quotient

(3.2) $$ \begin{align} \mathbb{P}(q_0,\dots, q_n) = \mathbb{P}^n / G_{\vec q}\,, \end{align} $$

with quotient map

(3.3) $$ \begin{align} h:\mathbb{P}^n \longrightarrow \mathbb{P}(q_0,\dots, q_n)\,. \end{align} $$

This is the map of spaces associated with the identification of the graded ring $\mathbb {C}[x_0,\dots ,x_n]$ as the subring $\mathbb {C}[y_0^{q_1},\dots ,y_n^{q_n}]\subseteq \mathbb {C}[y_0,\dots ,y_n]$ of the standard graded polynomial ring, which can be viewed as the invariant ring for the group $G_{\vec q}$ acting diagonally.

While $G_{\vec q}$ need not be cyclic, the weighted projective space is locally a cyclic quotient. It is covered by the open sets (i.e., $U_i$ is the image of $\{x_i \neq 0\}\subseteq \mathbb {C}^{n+1}-\{0\}$ under the quotient map h), and one has

$$ \begin{align*}U_i \cong \mathbb{C}^n/\mu_{q_i}=\mathbb{C}^n/\langle (\zeta_{q_i}^{q_0}, \dots ,\zeta_{q_i}^{q_{i-1}}, \zeta_{q_i}^{q_{i+1}}, \dots ,\zeta_{q_i}^{q_{n}}) \rangle\,, \end{align*} $$

where $\zeta _{q_i}$ is a primitive root of unity of order $q_i$ . This identification comes from the identification $U_i=\operatorname {Spec}(\mathbb {C}[x_0,\dots ,x_n]_{(x_i)})$ , where $\mathbb {C}[x_0,\dots ,x_n]_{(x_i)}$ is the subring of elements of degree $0$ in the localized ring $\mathbb {C}[x_0,\dots ,x_n]_{x_i}$ , and the identification of the ring $\mathbb {C}[x_0,\dots ,x_n]_{(x_i)}$ with the subring of $\mathbb {C}[z_0,\dots ,z_{i-1},z_{i+1},\dots ,z_n]^{\mu _{q_i}} \subseteq \mathbb {C}[z_0,\dots ,z_{i-1},z_{i+1},\dots ,z_n]$ of invariants for the diagonal action of the cyclic group $\mu _{q_i}=\langle (\zeta _{q_i}^{q_0}, \dots ,\zeta _{q_i}^{q_{i-1}}, \zeta _{q_i}^{q_{i+1}}, \dots ,\zeta _{q_i}^{q_{n}}) \rangle $ .

It is now straightforward to apply the Reid–Shepherd-Barron–Tai (RSBT) criterion to ${\mathcal {M}}^{\operatorname {GIT}}$ , by local computations in these charts.

Proof. We first note that $U_0 \cong \mathbb {C}^4$ . To explain the other charts, we first treat the chart

$$ \begin{align*}U_4= \mathbb{C}^4/\langle g_4 \rangle\,, \end{align*} $$

where

There is only one fixed point of this finite group action, namely the origin. Here, the RSBT criterion tells us that we have to check for all nontrivial powers $g_4^k$ that the inequality

$$ \begin{align*}\lfloor \tfrac{k}{5} \rfloor + \lfloor \tfrac{2k}{5} \rfloor + \lfloor \tfrac{3k}{5} \rfloor + \lfloor \tfrac{4k}{5} \rfloor \geq 1 \end{align*} $$

holds. Altogether we find one singularity in this chart, namely the point $P_4=(0:0:0:0:1)$ , that is, the image of the point $(0,0,0,0,1)\in \mathbb {C}^{n+1}-\{0\}$ under the quotient map h. The singularity at this point is canonical. The other open sets $U_i$ can be treated similarly. The situation for $U_2$ is completely analogous, and we find one further canonical singularity, namely $P_2=(0:0:1:0:0)$ . For $U_3$ , we have to consider

$$ \begin{align*}U_3= \mathbb{C}^4 / \langle g_4 \rangle = \mathbb{C}^4 / \langle i,-1,-i,i \rangle\,. \end{align*} $$

Once again, we find one canonical singular point, namely $P_3=(0:0:0:1:0)$ . Finally, in the chart

$$ \begin{align*}U_1= \mathbb{C}^4 / \langle g_2 \rangle = \mathbb{C}^4 / \langle -1,1,-1,-1 \rangle, \end{align*} $$

we have a one-dimensional fixed locus, namely the line

We also note that $g_4^2=g_2$ and that $P_3 \in L$ . Outside $P_3$ , we have a transversal singularity along L of type $\mathbb {C}^3/\langle (-1,-1,-1) \rangle $ .

In the proof of the lemma, we have also verified that

$$ \begin{align*}\operatorname{Sing}{\mathcal {M}}^{\operatorname{GIT}}= \lbrace P_2\rbrace \cup \lbrace P_4\rbrace \cup L\,, \end{align*} $$

as already stated in [Reference du Plessis and WallDvG+, §6.9].

We note that the space ${\mathcal {M}}^{\operatorname {GIT}}=\mathbb {P}(1,2,3,4,5)$ is $\mathbb {Q}$ -factorial (since it only has finite quotient singularities), and is thus $\mathbb {Q}$ -Gorenstein, but it is not Gorenstein. Indeed, the line bundle $\mathcal {O}_{\mathbb {P}^n}(1)$ descends under the cover h (3.3) to a $\mathbb {Q}$ -Cartier divisor on $\mathbb {P}(q_0,\dots ,q_n)$ , which by abuse of notation we denote by $\mathcal {O}_{\mathbb {P}(q_0,\dots ,q_n)}(1)$ . The lowest multiple of $\mathcal {O}_{\mathbb {P}(q_0,\dots ,q_n)}(1)$ which is Cartier is then $\mathcal {O}_{\mathbb {P}(q_0,\dots ,q_n)}\left (\operatorname {lcm}(q_0, \dots ,q_n)\right )$ (e.g., [Reference FultonF, Prop. p. 63] or [Reference Casalaina-Martin, Grushevsky, Hulek and LazaCLS, Exams. 4.1.5 and 4.2.11]). For the canonical bundle, using the fact that the weighted projective space is a toric variety, and that the covering map h is ramified along the toric divisors, one obtains the standard formula (e.g., [Reference DolgachevDol Reference Abramovich, Graber and Vistoli1, Th. 3.3.4])

(3.4) $$ \begin{align} K_{\mathbb{P}(q_0,\dots,q_n)}=\left(-\sum q_i \right)\mathcal{O}_{\mathbb{P}(q_0,\dots,q_n)}(1)\,. \end{align} $$

Thus, in our case,

(3.5) $$ \begin{align} K_{\mathbb{P}(1,2,3,4,5)}=-15\mathcal{O}_{\mathbb{P}(1,2,3,4,5)}(1)\,, \end{align} $$

and its smallest multiple that is Cartier is $4K_{\mathbb {P}(1,2,3,4,5)}$ .

Classical invariant theory for cubic surfaces explicitly identifies the geometric divisors $D_{A_1}$ (the nodal or discriminant) and R (the Eckardt) divisors inside ${\mathcal {M}}^{\operatorname {GIT}}\cong \mathbb {P}(1,2,3,4,5)$ . For further use, we review this. By [Reference Dolgachev, van Geemen and KondoDvG, 1.3 and 6.4], the discriminant $D_{A_1}$ is given by the equation

(3.6) $$ \begin{align} (I_8^2-2^6 I_{16})^2= 2^{14}(I_{32} + 2^{-3}I_8I_{24})\,, \end{align} $$

where $I_8,I_{16}, I_{24}, I_{32}, I_{40},I_{100}$ are the standard generators of the ring of invariants of the action of $\operatorname {SL}(4,\mathbb {Z})$ on the space of cubics, and the subscripts denote their degrees. As $I_{100}^2$ is a polynomial in the other invariants listed, these degrees show that we are working with $\mathbb {P}(8,16,24,32,40)\cong \mathbb {P}(1,2,3,4,5)$ . Moreover, the Eckardt divisor is given by $I_{100}^2=0$ (e.g., [Reference Dolgachev, van Geemen and KondoDvG, §6.5]).

Pulling back the defining equation (3.6) of $D_{A_1}$ to $\mathbb {P}^4$ under the quotient map h given by (3.3), we obtain

(3.7) $$ \begin{align} h^*D_{A_1}=\left\lbrace (y_0^2-2^6 {y_1}^2)^2= 2^{14}(y_3^4 + 2^{-3}y_0y_2^3)\right\rbrace, \end{align} $$

where $y_0, \dots , y_4$ are homogeneous coordinates on $\mathbb {P}^4$ . Furthermore, [Reference du Plessis and WallDvG+] gives the coordinates of the point $\Delta _{3A_2}\in \mathbb {P}(1,2,3,4,5)$ as

$$ \begin{align*}\Delta_{3A_2}=(8:1:0:0:0) \in D_{A_1}\,, \end{align*} $$

which in particular is a smooth point of ${\mathcal {M}}^{\operatorname {GIT}}$ . It is also a smooth point of $D_{A_1}\subseteq \mathbb {P}(1,2,3,4,5)$ , as in the local coordinates on the open chart $U_0\cong \mathbb {C}^4$ , the defining equation (3.6) of the discriminant divisor becomes $(1-2^6z_1)^2 = 2^{14}(z_3 +2^{-3} z_2)$ , and $\Delta _{3A_2}$ corresponds to the point $(1/8^2,0,0,0)$ , so that taking partial derivatives of this equation at the point $\Delta _{3A_2}$ gives smoothness of $D_{A_1}$ at $\Delta _{3A_2}$ .

We can perform a similar analysis for the Eckardt divisor. As mentioned above, the Eckardt divisor R is defined by $I_{100}^2$ , which is an irreducible polynomial in $I_8,\dots ,I_{40}$ . The exact expression due to Salmon for $I_{100}^2$ in terms of $I_8,\dots ,I_{40}$ no longer seems to be easily accessible in the literature. Dardanelli and van Geemen recently rederived it for their paper [Reference Dolgachev, van Geemen and KondoDvG], and provided us with the expression, which we have omitted to save space; for reference, it is now available on S. Casalaina-Martin’s website. From that description, one can easily see that

(3.8) $$ \begin{align} h^*R=\mathcal{O}_{\mathbb{P}^4}(25). \end{align} $$

As already noted, ${\mathcal {M}}^{\operatorname {GIT}}$ has Picard number $1$ . For further reference, we collect here the classes of various Weil divisors on ${\mathcal {M}}^{\operatorname {GIT}}=\mathbb {P}(1,2,3,4,5)$ :

(3.9) $$ \begin{align} \begin{aligned}K_{{\mathcal {M}}^{\operatorname{GIT}}}&=\mathcal {O}_{\mathbb {P}(1,2,3,4,5)}(-15), & \\ D_{A_1}&=\mathcal {O}_{\mathbb {P}(1,2,3,4,5)}(4)& \text{(Discriminant divisor)},\\ R& =\mathcal {O}_{\mathbb {P}(1,2,3,4,5)}(25)& \text{(Eckardt divisor)}. \end{aligned} \end{align} $$

These come from (3.5), (3.7), and (3.8), respectively. In particular, we note that the following relation holds in $\operatorname {Pic}({\mathcal {M}}^{\operatorname {GIT}})_{\mathbb {Q}}$ :

(3.10) $$ \begin{align} K_{\mathbb{P}(1,2,3,4,5)}= - \frac{15}{4} D_{A_1}\,. \end{align} $$

3.3 Local structure of the Kirwan blowup along the exceptional divisor

We recall some relevant computations from [Reference Casalaina-Martin, Grushevsky, Hulek and LazaCMG+1, App. C]. First, to employ the Luna Slice Theorem, we will want to understand the stabilizer of the $3A_2$ cubic surface $S_{3A_2}$ , as well as its action on a Luna slice.

To begin, given a cubic surface $S\subseteq \mathbb {P}^3$ , we denote by $\operatorname {Aut}(S)$ the automorphisms of S (which are automorphisms of S as a subvariety of $\mathbb {P}^3$ , and therefore we naturally have $\operatorname {Aut}(S)\subseteq \operatorname {PGL}(4,\mathbb {C})$ ). From our GIT setup, we are also interested in $\operatorname {Stab}(S)\subseteq \operatorname {SL}(4,\mathbb {C})$ , the stabilizer subgroup, and, since it is sometimes easier to work with, we will also consider the stabilizer $\operatorname {GL}(S)\subseteq \operatorname {GL}(4,\mathbb {C})$ . We recall from [Reference Casalaina-Martin, Grushevsky, Hulek and LazaCMG+1, App. C] that the former two of these stabilizer groups for $S_{3A_2}$ are two-dimensional, but we also want to work out the finite parts explicitly. To this end, we denote

an auxiliary group, and observe that there is an isomorphism $\mathbb {T}^3\cong D$ given by $(\lambda _1,\lambda _2,\lambda _3)\mapsto \operatorname {diag}(\lambda _1^{-1}\lambda _2^{-1}\lambda _3^3,\lambda _1,\lambda _2,\lambda _3)$ . We also want

and note the isomorphism $\mathbb {T}^2\times \mu _4\cong D'$ given by $(\lambda _1,\lambda _2,i^j)\mapsto \operatorname {diag}(\lambda _1^{-1}\lambda _2^{-1}i^{3j},\lambda _1,\lambda _2,i^j)$ . To see that this map is an isomorphism, note that this certainly gives an inclusion $\mathbb {T}^2\times \mu _4\hookrightarrow D'$ , and given $\operatorname {diag}(\lambda _0,\lambda _1,\lambda _2,\lambda _3)\in D'$ , the two equations together give $\lambda _3^4=1$ , so that $\lambda _3=i^j$ for some j. Finally, we denote

and note the isomorphism $\mathbb {T}^2\cong D"$ given by $(\lambda _1,\lambda _2)\mapsto \operatorname {diag}(\lambda _1^{-1}\lambda _2^{-1},\lambda _1,\lambda _2,1)$ .

We use the notation $\mathbb {S}_3$ for the group of matrices obtained from the group of invertible diagonal $3\times 3$ complex matrices by applying all possible permutations of the columns. The determination of the relevant stabilizer groups is parallel to the case of the $3D_4$ cubic threefold, treated in [Reference Casalaina-Martin, Grushevsky, Hulek and LazaCMG+1, Prop. B.6], and proceeds by an explicit computation, which we put in Appendix A.1.

We now describe the action of the stabilizer of the $S_{3A_2}$ cubic on the Luna slice. The description of the Luna slice appears in [Reference ZhangZha, p. 52], but for clarity, particularly for the action of the stabilizer, we include a discussion here as well.

Lemma 3.4 ( $3A_2$ -Luna slice).

A Luna slice for $S_{3A_2}$ , normal to the orbit $\operatorname {SL}(4,\mathbb {C})\cdot [S_{3A_2}] \subseteq \mathbb {P}^{19}$ , is isomorphic to $\mathbb {C}^6$ , spanned by the six monomials

$$ \begin{align*}x_0^3,\ x_1^3,\ x_2^3,\ x_0^2x_3,\ x_1^2x_3, \ x_2^2x_3 \end{align*} $$

in the tangent space $H^0(\mathbb {P}^3,\mathcal {O}_{\mathbb {P}^3}(3))$ .

The Luna slice can be projectively completed to give a $\mathbb {P}^6$ :

$$ \begin{align*}\mathbb{P}^6 = \{\alpha_0x_0^3 +\alpha_1x_1^3 +\alpha_2x_2^3+\alpha_{\widehat 0} x_0^2x_3+\alpha_{\widehat 1} x_1^2x_3 +\alpha_{\widehat 2} x_2^2x_3 +\alpha_{3A_2}(x_0x_1x_2+x_3^3) \}\end{align*} $$

$$ \begin{align*}\subseteq \mathbb{P} H^0(\mathbb{P}^3,\mathcal{O}_{\mathbb{P}^3}(3))=\mathbb{P}^{19}\,. \end{align*} $$

The action of $\operatorname {Stab}(S_{3A_2})$ and $\operatorname {GL}(S_{3A2})$ on the projectively completed Luna slice is given by their inclusion into the groups $\operatorname {SL}(4,\mathbb {C})$ and $GL(4,\mathbb {C})$ , respectively, with the given actions of those groups on $H^0(\mathbb {P}^3,\mathcal {O}_{\mathbb {P}^3}(3))$ from the GIT setup.

The action of $\operatorname {Stab}(S_{3A_2})$ and $\operatorname {GL}(S_{3A2})$ on the Luna slice is the natural induced action. In terms of Lemma 3.3(3) and the description in (3.15), the action of an element $\operatorname {diag}(\lambda _0,\lambda _1,\lambda _2,\lambda _3) $ in D or $D'$ is given by

(3.18) $$ \begin{align} \begin{aligned} (\lambda_0,\lambda_1,\lambda_2,\lambda_3)&\cdot (\alpha_0,\alpha_1,\alpha_2,\alpha_{\widehat 0} ,\alpha_{\widehat 1} ,\alpha_{\widehat 2} )=\\ &=\left(\left(\frac{\lambda_0}{\lambda_3}\right)^3\alpha_0,\left(\frac{\lambda_1}{\lambda_3}\right)^3\alpha_1,\left(\frac{\lambda_2}{\lambda_3}\right)^3\alpha_2, \left(\frac{\lambda_0}{\lambda_3}\right)^2\alpha_{\widehat 0} , \left(\frac{\lambda_1}{\lambda_3}\right)^2\alpha_{\widehat 1} , \left(\frac{\lambda_2}{\lambda_3}\right)^2\alpha_{\widehat 2} \right)\,, \end{aligned} \end{align} $$

and the action of $\sigma \in S_3\subseteq \operatorname {GL}(S_{3A_2})$ is given by

(3.19) $$ \begin{align} \sigma\cdot (\alpha_0,\alpha_1,\alpha_2,\alpha_{\widehat 0} ,\alpha_{\widehat 1} ,\alpha_{\widehat 2} )= (\alpha_{\sigma(0)},\alpha_{\sigma(1)},\alpha_{\sigma(2)},\alpha_{\widehat {\sigma(0)}},\alpha_{\widehat{\sigma(1)}},\alpha_{\widehat{\sigma(2)}})\,. \end{align} $$

Proof. For $S_{3A_2}=\lbrace F_{3A_2}=0\rbrace $ , the matrix $DF_{3A_2}$ , whose entries span the tangent space to the orbit of the $3A_2$ cubic, is given by (see [Reference Casalaina-Martin, Grushevsky, Hulek and LazaCMG+1, §4.2.3] for similar computations for the $3D_4$ cubic threefold)

$$ \begin{align*}DF_{3A_2}=\begin{pmatrix} x_0x_1x_2&x_1^2x_2&x_1x_2^2&x_1x_2x_3\\ x_0^2x_2&x_0x_1x_2&x_0x_2^2&x_0x_2x_3\\ x_0^2x_1&x_0x_1^2&x_0x_1x_2&x_0x_1x_3\\ 3x_0x_3^2&3x_1x_3^2&3x_2x_3^2&3x_3^3\\ \end{pmatrix}\,. \end{align*} $$

Since all entries of this matrix are monomial, the only possible linear relations are pairwise equalities, up to a constant factor. One sees that the only monomial that repeats more than once is $x_0x_1x_2$ , and thus all linear relations satisfied by the entries of $DF_{3A_2}$ are

$$ \begin{align*}(DF_{3A_2})_{00}=(DF_{3A_2})_{11}=(DF_{3A_2})_{22}\,. \end{align*} $$

This means that the normal space to the orbit ( $\dim \mathbb {P}^{19}-\dim \text { orbit } = 19-(16-3)=6$ ) is spanned by the six monomials

$$ \begin{align*}x_0^3,\ x_1^3,\ x_2^3,\ x_0^2x_3,\ x_1^2x_3, \ x_2^2x_3\,. \end{align*} $$

The given action of the stabilizer on the Luna slice can be seen, for instance, from taking the Luna slice to be the affine space

$$ \begin{align*}\alpha_0x_0^3 +\alpha_1x_1^3 +\alpha_2x_2^3+\alpha_{\widehat 0} x_0^2x_3+\alpha_{\widehat 1} x_1^2x_3 +\alpha_{\widehat 2} x_2^2x_3 +(x_0x_1x_2+x_3^3)\subseteq H^0(\mathbb{P}^3,\mathcal{O}_{\mathbb{P}^3}(3))\,, \end{align*} $$

then using the fact that $\lambda _0\lambda _1\lambda _2=\lambda _3^3$ , and dividing through the natural action by $\lambda _3^3$ , to fix the cubic form $x_0x_1x_2+x_3^3$ in the affine space.

We now turn to the Eckardt divisor whose generic point parameterizes smooth cubics with a nontrivial automorphism, and determine the multiplicity with which it contains $S_{3A_2}$ . The proof is by an elaborate lengthy explicit computation using the explicit form of the action on the Luna slice, and is given in Appendix A.2.

One can see from the above that for the action of $\operatorname {Stab}(S_{3A_2})$ on the normal space, the stabilizer of a general line will be trivial. In fact, when we blow up $(\mathbb {P}^{19})^{ss}$ along the orbit $\operatorname {SL}(4,\mathbb {C}) \cdot [S_{3A_2}]$ in the Kirwan blowup process, then in the Luna slice, we are blowing up the origin in $\mathbb {C}^6$ , with exceptional divisor $\mathbb {P}^5$ . The lemma above gives the action of the stabilizer on this $\mathbb {P}^5$ .

For future use, we first describe the semi-stable locus for this action of the stabilizer on  $\mathbb {P}^5$ .

Proof. The action on the Luna slice given by (3.18) gives the following action of $\mathbb {T}^2\simeq D"=\operatorname {Aut}(S_{3A_2})^\circ $ on the $\mathbb {C}^6$ with coordinates T, of which the exceptional divisor is the projectivization:

(3.20) $$ \begin{align} \begin{aligned} (T_0,T_1,T_2,T_{\widehat 0},T_{\widehat 1},T_{\widehat 2})&\mapsto (\lambda_0^3T_0,\lambda_1^3 T_1,\lambda_2^3 T_2, \lambda_0^2T_{\widehat 0},\lambda_1^2T_{\widehat 1},\lambda_2^2T_{\widehat 2})\\ &=(\lambda_1^{-3}\lambda_2^{-3} T_0,\lambda_1^3 T_1,\lambda_2^3 T_2, \lambda_1^{-2}\lambda_2^{-2}T_{\widehat 0},\lambda_1^2T_{\widehat 1},\lambda_2^2T_{\widehat 2}) \end{aligned} \end{align} $$

(here we are acting by $\operatorname {diag}(\lambda _0,\lambda _1,\lambda _2,1)$ with $\lambda _0\lambda _1\lambda _2=1$ , and thus expressing $\lambda _0=\lambda _1^{-1}\lambda _2^{-1}$ ). The action is by multiplying each coordinate by a monomial in $\lambda _1,\lambda _2$ , and thus $\mathbb {C}^6$ is decomposed into a direct sum of six one-dimensional torus representations. Plotting the weights of each monomial in $\mathbb {R}^2$ , a point in $\mathbb {C}^6$ is stable if and only if the convex hull of the set of weights corresponding to nonzero coordinates contains the origin in $\mathbb {R}^2$ (e.g., [Reference Allcock and FreitagAll, Lem. 3.10] or [Reference DolgachevDol Reference Alexeev2, Th. 12.2]). The weight diagram consists of two points on each of three rays from the origin. Thus, a convex hull of some subset of these six weights contains the origin if and only if this subset contains at least one weight from each ray. This is to say, a point is stable if and only if at least one of its two coordinates $T_0$ and $T_{\widehat 0}$ is nonzero, and so forth. Thus, the unstable points in $\mathbb {C}^4\subseteq \mathbb {C}^6$ are precisely those given by a pair of equations $T_i=T_{\widehat i}=0$ for some i.

We note that, as is the case for any Kirwan desingularization, there are no strictly semi-stable points on this exceptional divisor. We now describe the finite stabilizers along the exceptional divisor. This is another detailed explicit computation using the stabilizer computed in Lemma 3.3, and we give it in Appendix A.3. We note that the proof actually allows us to determine all possible stabilizers, but we will not need this information.

We conclude the section with the following non-transversality result.

4.1 Preliminaries on the ball quotient model

We will now describe the compactifications of the ball quotient model of the moduli space of cubic surfaces. Before delving into the specifics for cubic surfaces, we first recall the compactification of ball quotients in general, referring to [Reference Ash, Mumford, Rapoport and TaiAMR+] for the general details of the constructions. Let

be an n-dimensional ball. Alternatively, we can realize $\mathcal {B}_n$ as follows. Let $\mathcal {O}$ be the ring of integers of an imaginary quadratic field $\mathbb {Q}(\sqrt {d})$ , and let $\Lambda $ be a free $\mathcal {O}$ -module equipped with a Hermitian form h of signature $(1,n)$ . Then

$$ \begin{align*}\mathcal{B}_n=\left\{ [z] \in \mathbb{P}(\Lambda \otimes \mathbb{C}): h(z)>0 \right\}\,. \end{align*} $$

For an arithmetic subgroup $\Gamma \subseteq \operatorname {SU}(1,n)$ , there is a quotient quasi-projective analytic space $\mathcal {B}_n/\Gamma $ , which by construction has at worst finite quotient singularities. This quotient admits a projective Baily–Borel compactification $(\mathcal {B}_n/\Gamma )^*$ defined as the $\operatorname {Proj}$ of the ring of automorphic forms with respect to $\Gamma $ . Geometrically, the boundary $(\mathcal {B}_n/\Gamma )^*-\mathcal {B}_n/\Gamma =c_{F_1}\sqcup \dots \sqcup c_{F_r}$ consists of a finite number of points, called cusps. These are in $1$ -to- $1$ correspondence with the $\Gamma $ -orbits of isotropic lines in $\Lambda _{\mathbb {Q}(\sqrt {d})}$ . There are no higher-dimensional cusps since the signature is $(1,n)$ , implying that no other isotropic subspaces exist.

In the case of ball quotients, there exists a unique toroidal compactification $\overline {\mathcal {B}_n/\Gamma }$ . Uniqueness follows since all tori involved have rank $1$ . More precisely, after dividing by the unipotent radical of the parabolic subgroup that stabilizes a given cusp, the quotient locally looks like an open set in $D^* \times \mathbb {C}^{n-1} \subseteq \mathbb {C}^* \times \mathbb {C}^{n-1}$ that contains $\{0\}\times \mathbb {C}^{n-1}$ in its closure, where here $D^*$ is the punctured unit disk. The toroidal compactification is then simply obtained by adding the divisor $\{0\}\times \mathbb {C}^{n-1}$ . As a result, the boundary $\overline {\mathcal {B}_n/\Gamma }-\mathcal {B}_n/\Gamma =T_{F_1}\sqcup \dots \sqcup T_{F_r}$ consists of a finite disjoint union of smooth (up to finite quotient singularities) irreducible divisors, each of which is in fact a finite quotient of an abelian variety. The natural map $p:\overline {\mathcal {B}_n/\Gamma }\to (\mathcal {B}_n/\Gamma )^*$ simply contracts each divisor $T_{F_i}$ to the cusp $c_{F_i}$ (see Theorem 4.6 for a detailed discussion of the case relevant in this paper).

With this setup, we return to the case of cubic surfaces, and describe the period map to the ball quotient. Specifically, considering the triple cover of $\mathbb {P}^3$ branched along a cubic surface, one obtains a cubic threefold, and via the period map for cubic threefolds, taking the $\mathbb {Z}_3$ action into account, one obtains a period map to a four-dimensional ball quotient, $\mathcal {M}\rightarrow {\mathcal {B}_4/\Gamma }$ (see [Reference Abramovich, Graber and VistoliACT]). This is an open embedding, and the complement of the image is the Heegner divisor $D_n=\mathcal {D}_n/\Gamma \subseteq {\mathcal {B}_4/\Gamma }$ . It turns out that in this case the rational period map ${\mathcal {M}}^{\operatorname {GIT}}\dashrightarrow {(\mathcal {B}_4/\Gamma )^*}$ to the Baily–Borel compactification extends to an isomorphism, taking the discriminant divisor $D_{A_1}\subseteq {\mathcal {M}}^{\operatorname {GIT}}$ to the (closure of the) Heegner divisor $D_n\subseteq {(\mathcal {B}_4/\Gamma )^*}$ (which is denoted this way for nodal). Under this isomorphism, the unique strictly polystable point $\Delta _{3A_2}\in {\mathcal {M}}^{\operatorname {GIT}}$ corresponding to the $3A_2$ cubic is identified with the sole cusp $c_{3A_2}=\partial {(\mathcal {B}_4/\Gamma )^*}$ . The natural map $p:{\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$ contracts the irreducible boundary divisor $T_{3A_2}$ to $c_{3A_2}$ . From now on, we will write $D_{A_1}=D_n$ for the discriminant divisor, where we use $D_{A_1}$ when we are thinking of it from the GIT point of view, and $D_n$ when thinking of the ball quotient—to emphasize the context we are in.

In summary, for the case of cubic surfaces, we have a diagram

(4.1)

where f is a birational map that restricts to an isomorphism

(4.2) $$ \begin{align} f:{\mathcal {M}}^{\operatorname{K}}-D_{3A_2} \cong {\overline{\mathcal {B}_4/\Gamma }}- T_{3A_2}\,. \end{align} $$

In [Reference KirwanKi2] and [Reference ZhangZha], the (intersection) Betti numbers of the spaces ${\mathcal {M}}^{\operatorname {GIT}}\cong {(\mathcal {B}_4/\Gamma )^*}$ and ${\mathcal {M}}^{\operatorname {K}}$ were computed. In [Reference Casalaina-Martin, Grushevsky, Hulek and LazaCMG+1, §C.2], the Betti numbers of ${\overline {\mathcal {B}_4/\Gamma }}$ were computed, and they turned out to be the same as for ${\mathcal {M}}^{\operatorname {K}}$ , which served as motivation for our query as to whether these two compactifications are isomorphic, which is the main subject of the current paper.

4.2 The toroidal compactification via marked cubic surfaces

An indispensable tool in the study of cubic surfaces is the group $W(E_6)$ , which is the automorphism group for the configuration of the $27$ lines on a smooth cubic. The moduli space of cubic surfaces $\mathcal {M}$ admits a natural $W(E_6)$ cover $\mathcal {M}_m$ parameterizing marked smooth cubic surfaces, that is, cubics together with a labeling of the $27$ lines. Since the automorphism group of a smooth cubic surface acts faithfully on the primitive cohomology, it follows that $\mathcal {M}_m$ is in fact smooth. Furthermore, Naruki [Reference NarukiN] constructed a smooth normal crossing compactification $\overline {\mathcal {N}}$ of $\mathcal {M}_m$ , which admits various geometric interpretations (see, e.g., [Reference Hacking, Keel and TevelevHKT]). In this section, we use the geometry of the Naruki model $\overline {\mathcal {N}}$ to get a good hold on the space of interest in our paper, ${\overline {\mathcal {B}_4/\Gamma }}$ .

By construction, the marked moduli space $\mathcal {M}_m$ is a Galois cover of $\mathcal {M}$ with Galois group $W(E_6)$ . This cover is compatible with the ball quotient construction of Allcock, Carlson, and Toledo [Reference Abramovich, Graber and VistoliACT]. Specifically, the monodromy group $\Gamma $ for cubic surfaces contains a normal subgroup $\Gamma _m\unlhd \Gamma $ with

(4.3) $$ \begin{align} \Gamma/\Gamma_m\cong W(E_6) \times \{ \pm 1\} \end{align} $$

(see [Reference Abramovich, Graber and VistoliACT, (3.12)]). We observe that $-1 $ acts trivially on the ball $\mathcal {B}_4$ . This leads to a $W(E_6)$ cover ${\mathcal {B}_4/\Gamma }_m\rightarrow {\mathcal {B}_4/\Gamma }$ . Furthermore, this cover extends to the Baily–Borel compactifications (cf. [Reference Abramovich, Graber and VistoliACT, Th. 3.17]), and then also to the toroidal compactifications—essentially because in the ball quotient case, the toroidal compactification is canonical, and thus there is an automatic extension. In summary, the following holds:

Proposition 4.1. With notation as above, we have the following diagram:

(4.4)

where all the spaces in the top row admit a $W(E_6)$ action, and the morphisms are $W(E_6)$ -equivariant, whereas the spaces in the bottom row are the quotients with respect to this $W(E_6)$ action.

It was shown recently that the marked toroidal and Naruki compactifications coincide.

Theorem 4.4 [Reference Gallardo, Kerr and SchafflerGKS].

The Naruki compactification $\overline {\mathcal {N}}$ is isomorphic to the toroidal compactification ${\overline {\mathcal {B}_4/\Gamma _m}}$ . More precisely, there is a $W(E_6)$ -equivariant commutative diagram

The Naruki compactification has a well-understood structure, mostly due to Naruki [Reference NarukiN], with some further later clarifications by other authors.

With these preliminaries, we can extract a series of immediate consequences on the boundary of $\mathcal {M}\subseteq {\overline {\mathcal {B}_4/\Gamma }}$ . First, Proposition 4.1 and Theorem 4.6(0) together show that this is a normal crossing compactification in a stack sense.

For further reference, we also record the structure of the toroidal boundary divisor.

4.3 Proof of Theorem 1.1

At this point, we are able to establish one of our main results, namely that the period map does not extend to an isomorphism between the Kirwan compactification and the toroidal compactification of the ball quotient model. We have seen by Proposition 3.9 that the nodal and boundary divisors do not meet transversally, even generically, in the Kirwan model, whereas an immediate consequence of the above discussion is that they do so in the toroidal model.

As a corollary of Theorem 1.1, we have the following result, which contrasts with Corollary 4.8 for the toroidal compactification, and strengthens Proposition 3.9 (which was itself used in the proof of Theorem 1.1) for the Kirwan compactification.

5.1 Divisors and relations in ${(\mathcal {B}_4/\Gamma )^*}$

The ball quotients come equipped with a natural $\mathbb {Q}$ -line bundle $\lambda $ , the so-called Hodge line bundle, that gives the polarization for the Baily–Borel compactification (and pulls back to a big and nef bundle on the toroidal compactification). Since ${(\mathcal {B}_4/\Gamma )^*}\cong {\mathcal {M}}^{\operatorname {GIT}}\cong \mathbb {P}(1,2,3,4,5)$ has Picard rank $1$ , the divisors $\lambda $ , $ D_n,$ and R must all be proportional. It turns out that one can express the classes of these divisors in terms of $\lambda $ as a consequence of the work of Borcherds. Specifically, the following holds.

Taking into account the ramification orders $6$ for the discriminant divisor, which corresponds to the sum of all short mirrors, and $2$ for the Eckardt divisor, which corresponds to the sum of all long mirrors (see esp. [Reference Abramovich, Graber and VistoliACT, §11]), we obtain the following equalities in $\operatorname {Pic}_{\mathbb {Q}}({(\mathcal {B}_4/\Gamma )^*})$ :

(5.1) $$ \begin{align} 4\lambda=\frac{1}{6} D_n;\qquad 75\lambda=\frac{1}{2} R\,. \end{align} $$

Similarly, in the marked case, where the map $\mathcal {B}_4\rightarrow {(\mathcal {B}_4/\Gamma _m)^*}$ is only ramified along the nodal locus with index $3$ (e.g., [Reference Abramovich, Graber and VistoliACT, Th. 7.26]), the following holds in $\operatorname {Pic}_{\mathbb {Q}}(({\mathcal {B}_4/\Gamma }_m)^*)$ :

$$ \begin{align*} 4\lambda_m=\frac{1}{3} D_{n,m};\qquad 75\lambda_m= R_m\,. \end{align*} $$

Under the identification ${(\mathcal {B}_4/\Gamma )^*}\cong \mathbb {P}(1,2,3,4,5)$ , we have seen (3.9) that $D_n=\mathcal {O}_{\mathbb {P}(1,2,3,4,5)}(4)$ and $R=\mathcal {O}_{\mathbb {P}(1,2,3,4,5)}(25)$ (which gives $R=\frac {25}{4} D_n$ , agreeing with (5.1)), either of which combined with (5.1) give the following relation between the natural polarizations on ${(\mathcal {B}_4/\Gamma )^*}$ and $\mathbb {P}(1,2,3,4,5)$ :

(5.2) $$ \begin{align} \mathcal{O}_{\mathbb{P}(1,2,3,4,5)}(1)=6\lambda\,. \end{align} $$

We now turn to the canonical divisor. By the general theory of ball quotients (e.g., [Reference Mumford, Fogarty and KirwanMum, Prop. 3.4]), this is given by a multiple of the Hodge line bundle $\lambda $ , adjusted by a contribution due to the ramification (see, e.g., [Reference AllcockAle Reference Abramovich, Graber and Vistoli1, Th. 3.4]). The relevant multiple of $\lambda $ is “ $\dim +1$ .” We also note that $\lambda $ is given by the canonical automorphy factor given by the Jacobian, and that modular forms of weight k are exactly the sections of $\lambda ^{\otimes k}$ . The ramification divisor in our case is the union of the discriminant divisor $D_n$ and the Eckardt divisor R, which have branch orders $6$ and $2$ , respectively. Thus, we obtain

(5.3) $$ \begin{align} K_{{(\mathcal {B}_4/\Gamma )^*}}=5\lambda - \frac{5}{6} D_n - \frac{1}{2} R =-90\lambda \,. \end{align} $$

By (5.2), we see that $K_{{(\mathcal {B}_4/\Gamma )^*}}=\mathcal {O}_{\mathbb {P}(1,2,3,4,5)}(-15)$ , which agrees with the canonical bundle of $\mathbb {P}(1,2,3,4,5)$ (e.g., (3.9)). In the marked case, as the map $\mathcal {B}_4\rightarrow {(\mathcal {B}_4/\Gamma _m)^*}$ is only ramified along the nodal locus with index $3$ , we thus obtain

(5.4) $$ \begin{align} K_{{(\mathcal {B}_4/\Gamma _m)^*}}=5\lambda - \frac{2}{3} D_{n,m}\,. \end{align} $$

The same general considerations give the formulas for the canonical bundles of the toroidal compactifications.

5.2 Computations of discrepancies

In order to enable us to compute the top self-intersection of the canonical class, in this section, we compute how divisors on the Baily–Borel compactifications compare to divisors on the toroidal compactifications.

Corollary 5.3. In the notation above, the canonical class is given by

(5.5) $$ \begin{align} K_{{\overline{\mathcal {B}_4/\Gamma _m}}}=p_m^*K_{{(\mathcal {B}_4/\Gamma _m)^*}}+T_{3A_2,m}\,. \end{align} $$

To descend to the unmarked case, we need to understand the intersection of the boundary divisor $T_{3A_2}$ with the two ramification divisors $\widetilde D_{n,m}$ and $\widetilde R_{m}$ .

Proof. The same argument as in Corollary 5.3 gives the first item. As discussed in Theorem 4.6, the divisors $\widetilde D_{n,m}$ and ${T_{3A_2,m}}$ intersect transversely, and [Reference NarukiN, Th. 11.2’] describes this intersection precisely. For a fixed irreducible component $T_0$ of $\widetilde T_{3A_2}$ , which we have seen is isomorphic to $(\mathbb {P}^1)^{\times 3}$ , the intersection of $\widetilde D_{n,m}$ with it consists of nine irreducible components of type $\{\textrm {pt}\}\times \mathbb {P}^1\times \mathbb {P}^1\subseteq \mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^1$ , and thus of class $\mathcal {O}_{(\mathbb {P}^1)^{\times 3}}(1,0,0)$ (up to permuting the coordinates). The claim (2) thus follows.

The stabilizer $S_{3A_2,m}\subseteq W(E_6)$ of $T_0$ was identified in Theorem 4.6. Taking $U_m$ to be a suitable invariant neighborhood of $T_0$ , locally near $T_0$ , the map ${\overline {\mathcal {B}_4/\Gamma _m}}\rightarrow {\overline {\mathcal {B}_4/\Gamma }}$ is simply the quotient $U_m\to U_m/S_{3A_2,m}\cong U$ , with U a neighborhood of the toroidal boundary $T_{3A_2}\subseteq {\overline {\mathcal {B}_4/\Gamma }}$ . Since $S_{3A_2,m}$ contains a normal subgroup $(S_3)^{\times 3}$ , we can take the intermediate quotient $U'=U_m/(S_3)^{\times 3}$ and obtain a diagram

compatible with Theorem 4.6(3) and Corollary 4.9. Let $T'\subseteq U'$ be the quotient $T_0/((S_3)^{\times 3})\cong (\mathbb {P}(2,3))^{\times 3}\cong (\mathbb {P}^1)^{\times 3}$ .

To describe $\widetilde R_{m}\cap T_0$ , first recall that $U_m\to U$ is ramified along the Eckardt and nodal loci, with order $2$ along each. The factorization $U_m\xrightarrow {\alpha } U'\xrightarrow {\beta } U$ has the property that $\alpha $ is ramified along the nodal locus, whereas $\beta $ is ramified along the Eckardt locus. Indeed, in the language of [Reference Abramovich, Graber and VistoliACT], the subgroup $(S_3)^{\times 3}\subseteq S_ {3A_2,m}$ is generated by short roots (corresponding to the nine nodal components meeting $T_0$ ), and the residual $S_3=S_ {3A_2,m}/(S_3)^{\times 3}$ is generated by classes of long roots (corresponding to the Eckardt locus, and geometrically to cubics with an extra involution). In this description, it is clear that the restriction of the Eckardt locus to $T'$ is simply the sum of the three small diagonals (each of type $\mathcal {O}_{(\mathbb {P}^1)^{\times 3}}(1,1,0)$ , up to permutation), and thus of class $\mathcal {O}_{(\mathbb {P}^1)^{\times 3}}(2,2,2)$ . The pullback via $\alpha _{\mid T_0}$ of this class will be of type $\mathcal {O}_{(\mathbb {P}^1)^{\times 3}}(12,12,12)$ . Since $\alpha $ is not ramified along the Eckardt locus, this will be also the class of the reduced divisor $\widetilde {R}_{m}|_{T_0}$ .

Remark 5.5. The last two claims in Proposition 5.4 can be obtained also by a purely arithmetic argument. As alluded to in the proofs above, they follow by counting the short and long roots incident to a fixed cusp in ${(\mathcal {B}_4/\Gamma _m)^*}$ . To explain this, we consider the vector space $\mathbb {F}_3^5$ , equipped with the standard orthogonal form of signature $(4,1)$ . It is well known that the orthogonal group can then be identified as

$$ \begin{align*} \operatorname{O}(\mathbb{F}_3^5) \cong W(E_6) \times \{ \pm 1 \}. \end{align*} $$

An element $[w] \in \mathbb {P}(\mathbb {F}_3^5)$ is called isotropic, short, or long, depending on whether the norm of w equals $0,1$ , or $2$ (this does not depend on the chosen representative in $\mathbb {F}_3^5$ ). By [Reference AlexeevAF, §2] (also [Reference Abramovich, Graber and VistoliACT]), these elements enumerate the cusps, the components of the discriminant divisor $\widetilde D_{n,m}$ , and of the Eckardt divisor $\widetilde R_{m}$ , respectively. We further know from [Reference AlexeevAF, Prop. 2.1] that there are $40$ isotropic, $36$ short, and $45$ long elements in $ \mathbb {P}(\mathbb {F}_3^5)$ , and that the Weyl group $W(E_6)$ acts transitively on each of these three sets.

Counting the number of components of the discriminant divisor $\widetilde D_{n,m}$ and of the Eckardt divisor $\widetilde R_{m}$ intersecting an irreducible component $T_0$ of $T_{3A_2,m}$ as above is then an easy enumeration. Indeed, the choice of the cusp, and of $T_0$ , means fixing an isotropic element $h\in \mathbb {P}(\mathbb {F}_3^5)$ , and a straightforward count shows that there are $9$ short and $18$ long vectors orthogonal to h, which counts the number of irreducible components of $\widetilde {D}_{n,m}$ and $\widetilde {R}_m$ that intersect $T_0$ .

To describe the intersection of these components of $\widetilde D_{n,m}$ and $\widetilde R_{m}$ with $T_0$ , note that since the stabilizer subgroup of h in $W(E_6)$ acts transitively on the set of all sort (and also on the set of all long) vectors orthogonal to h, by symmetry it is enough to understand the intersection with $T_0$ of only one component of $\widetilde D_{n,m}$ and one component of $\widetilde R_{m}$ with $T_0$ , which can be seen, for example, from the standard local coordinates near a boundary component. All we need is then to understand the involutions which account for the degree $2$ ramification along $\widetilde D_n$ and $\widetilde R$ . In the case of the nodal divisor, this is, up to symmetry, an involution on a factor $\mathbb {P}^1$ which fixes some point $p\in \mathbb {P}^1$ , and then the restriction of the component of $\widetilde D_{n,m}$ to $T_0$ , fixed under this involution, is $p \times (\mathbb {P}^1)^{\times 2}$ , which gives the divisor class $ \mathcal {O}_{(\mathbb {P}^1)^{\times 3}}(1,0,0)$ . Adding up all nine components of $\widetilde D_{n,m}$ that meet $T_0$ , we obtain $ \mathcal {O}_{(\mathbb {P}^1)^{\times 3}}(3,3,3)$ . In the case of the Eckardt divisor, the involution is given by interchanging two of the factors of $(\mathbb {P}^1)^{\times 3}$ , while fixing the third factor. In the case of interchanging the first two factors, the fixed locus is then equal to $\Delta _{3A_2,12} \times \mathbb {P}^1$ , and has class $ \mathcal {O}_{(\mathbb {P}^1)^{\times 3}}(1,1,0)$ . Summing over all $18$ components of $\widetilde R_m$ which meet $T_0$ , we obtain $\mathcal {O}_{(\mathbb {P}^1)^{\times 3}}(12,12,12)$ .

We can now compare the discrepancies in the pullbacks of the discriminant and Eckardt divisors for the moduli of marked cubic surfaces.

Remark 5.7. It is interesting to note that the formulas above are compatible with those of Proposition 5.2 that were obtained by general considerations. Specifically, using (5.4) and Corollaries 5.3 and 5.6, we get

$$ \begin{align*} K_{{\overline{\mathcal {B}_4/\Gamma _m}}}&=p_m^*K_{{(\mathcal {B}_4/\Gamma _m)^*}}+T_{3A_2,m}\\&= 5\lambda -\frac{2}{3} (\widetilde{D}_{n,m}+3T_{3A_2,m})+T_{3A_2,m}\\&= 5\lambda -\frac{2}{3} \widetilde{D}_{n,m}-T_{3A_2,m}\,, \end{align*} $$

agreeing indeed with Proposition 5.2(2).

The main result of this section is the computation of the discrepancy of $p^*:{\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$ in the unmarked case, as this coefficient will be crucial for computing the top self-intersection number of $K_{{\overline {\mathcal {B}_4/\Gamma }}}$ .

Proposition 5.8. The canonical bundle of the toroidal compactification of the ball quotient model of the moduli space of cubic surfaces is given by the formula:

$$ \begin{align*} K_{{\overline{\mathcal {B}_4/\Gamma }}}= p^*K_{{(\mathcal {B}_4/\Gamma )^*}}+16T_{3A_2}\,. \end{align*} $$

Proof. Since in (4.4) we computed the discrepancy for the finite cover ${\overline {\mathcal {B}_4/\Gamma _m}}\rightarrow {(\mathcal {B}_4/\Gamma _m)^*}$ , a standard computation (see, e.g., [Reference Kollár and MoriKo Reference Alexeev2, 2.3]) gives

$$ \begin{align*}K_{{\overline{\mathcal {B}_4/\Gamma }}}= p^*K_{{(\mathcal {B}_4/\Gamma )^*}}+aT_{3A_2} \end{align*} $$

for

$$ \begin{align*} a= \frac{1}{r(T_{3A_2,m})}\left((1+\mu_{\widetilde D_{n,m}}+\mu_{\widetilde R_m})+1)\right)-1\,, \end{align*} $$

where $r(T_{3A_2,m})=1$ is the ramification index for the cover ${\overline {\mathcal {B}_4/\Gamma _m}}\rightarrow {\overline {\mathcal {B}_4/\Gamma }}$ along the toroidal boundary divisor (Theorem 4.6(4)), and $\mu _{\widetilde D_{n,m}}=3$ and $\mu _{\widetilde R_m}=12 $ are defined so that $p_m^* D_{n,m} = \widetilde D_{n,m}+\mu _{\widetilde D_{n,m}}T_{3A_2,m}$ and $p_m^* R_m = \widetilde R_m + \mu _{\widetilde R_m} T_{3A_2,m}$ (Corollary 5.6). We conclude that $a= ((1+3+12) +1)-1=16$ as claimed.

Remark 5.9. For completeness, we note that the analog of Corollary 5.6 in the unmarked case is

$$ \begin{align*} p^*D_n&= \widetilde D_n+6T_{3A_2}\,, \end{align*} $$

$$ \begin{align*} p^*R&= \widetilde R+24T_{3A_2}\,. \end{align*} $$

Similarly to Remark 5.7, these formulas are compatible with Propositions 5.8 and 5.2(1) and (5.3), giving a double check of our computations.

5.3 Self-intersection numbers for the toroidal compactification

Using the fact that the Baily–Borel compactification is a weighted projective space ${(\mathcal {B}_4/\Gamma )^*}\cong {\mathcal {M}}^{\operatorname {GIT}}\cong \mathbb {P}(1,2,3,4,5)$ , and (5.2), we conclude the following.

Corollary 5.10. On ${(\mathcal {B}_4/\Gamma )^*}$ , the following holds:

$$ \begin{align*}\left(K_{{(\mathcal {B}_4/\Gamma )^*}}\right)^4=\frac{(-15)^4}{5!}=\frac{15^3}{2^3}=\frac{3,375}{8}\end{align*} $$

and $\lambda ^4=\frac {1}{5!\cdot 6^4}=\frac {1}{2^73^55}=\frac {1}{155,520}$ .

Using the description of toroidal boundary given by Theorem 4.6, we obtain the following.

Lemma 5.11. The self-intersection numbers of the toroidal boundary divisors are

(5.6) $$ \begin{align} (T_{3A_2,m})^4&=-240\quad (\mbox{on }{\overline{\mathcal {B}_4/\Gamma _m}}), \end{align} $$

(5.7) $$ \begin{align} (T_{3A_2})^4&= -\frac{1}{6^3}\quad (\mbox{on }{\overline{\mathcal {B}_4/\Gamma }}). \end{align} $$

Finally, we can compute the top degree self-intersection of the canonical class on the toroidal compactification.

Theorem 5.12. The top self-intersection number of the canonical class on ${\overline {\mathcal {B}_4/\Gamma }}$ is

(5.8) $$ \begin{align} (K_{{\overline{\mathcal {B}_4/\Gamma }}})^4=\frac{25,589}{2^33^3}=\frac{25,589}{216}\,. \end{align} $$

Proof. From Proposition 5.8, we get

$$ \begin{align*}(K_{{\overline{\mathcal {B}_4/\Gamma }}})^4=(K_{{(\mathcal {B}_4/\Gamma )^*}})^4+16^4(T_{3A_2})^4\,.\end{align*} $$

Substituting the numbers from Corollary 5.10 and Lemma 5.11, the conclusion follows.

6.1 Setup

We start with the general setup of a GIT triple $(X,L,G)$ , where X is a scheme of finite type (we will continue to work over $\mathbb {C}$ to simplify notation, but the argument works over any algebraically closed field of characteristic zero, and, with minor adjustments, in positive characteristic, as well), L is an ample line bundle on X, and G is a connected reductive linear algebraic group acting on X, with L being G-linearized. We will also make the following assumptions that put us into the setup for Kirwan’s work:

• • X is a smooth quasi-projective variety.

• • $X^s \ne \emptyset $ , that is, the stable locus is nonempty.

Note that from the second condition, and say the Luna Slice Theorem, it follows that there are a Zariski dense open subvariety $U\subseteq X^s$ and a finite group $G_X$ such that for all $x\in U$ the stabilizer $G_x\subseteq G$ is isomorphic (although not necessarily equal) to $G_X$ ; that is, $G_X$ is the stabilizer of some general point of X.

We denote by

the GIT quotient.

6.2 Canonical classes for GIT quotients

Since $X^{ss}$ is normal, so is Y (e.g, [Reference DolgachevDol Reference Alexeev2, Prop. 3.1, p. 45]), and so canonical classes $K_{X^{ss}}$ and $K_Y$ are defined. Note that while $K_{X^{ss}}$ is Cartier, recall that there are elementary examples of quotients of smooth varieties by reductive groups that are not $\mathbb {Q}$ -Gorenstein (e.g., [Reference Braun, Greb, Langlois and MoragaBGL+, Exam.7.1]); in other words, $K_Y$ need not be $\mathbb {Q}$ -Cartier.

6.2.1 The stable locus

Now, let be the stable locus, and to fix notation, we have the map

$$ \begin{align*}q^s: X^{s}\longrightarrow Y^s= X^s/G\,. \end{align*} $$

Note that since X is smooth, the Luna Slice Theorem implies that $Y^s$ is étale locally the quotient of a smooth variety U by some finite group $G_i$ . In particular, $Y^s$ is $\mathbb {Q}$ -factorial, and there is a well-defined pullback $q^{s*}$ for $\mathbb {Q}$ -Weil divisor classes.

In this situation, we have the following Riemann–Hurwitz lemma.

Lemma 6.1 (Riemann–Hurwitz for the stable locus).

Let $R^s$ be the divisorial locus in $X^s$ that has a stabilizer strictly containing $G_X$ (i.e., the union of codimension $1$ irreducible components of the locus of points in $X^s$ where the stabilizer is not isomorphic to $G_X$ ), let $R^s=\bigcup R_i^s$ be its decomposition into irreducible components, and let $G_{R_i}$ be the stabilizer of a general point of $R_i^s$ . Then

$$ \begin{align*} K_{X^s}&= q^{s*}K_{Y^s}+\sum (|G_{R_i}|/|G_X| -1)R^s_i\,. \end{align*} $$

Proof. It suffices to check étale locally. The Luna Slice Theorem implies that, up to a smooth factor, the quotient $q^s:X^s\to Y^s$ is étale locally equivalent to the quotient $U\to U/G_i$ for a smooth scheme U and some finite group $G_i$ . Computing the canonical bundles is then a standard computation for the ramified cover $U\to U/G_i$ (see, e.g., [Reference Kollár and MoriKo Reference Alexeev2, pp. 63–64]).

6.2.2 Strictly semi-stable locus of codimension at least $2$

The computations for the stable locus carry over immediately to the general case, as long as the strictly semi-stable locus is of codimension at least $2$ . From now on, we will thus assume that $X^{ss}-X^{s} \subseteq X^{ss}$ and $Y-Y^s\subseteq Y$ are codimension at least $2$ . Under this assumption, one can define a pullback $q^*$ on $\mathbb {Q}$ -Weil divisor classes by restricting to the stable locus $Y^s$ (which, as noted above, is $\mathbb {Q}$ -factorial), pulling back to $X^s$ , and then extending over the boundary, which is assumed to be of codimension at least $2$ . This immediately yields the following.

Corollary 6.2 (Riemann–Hurwitz).

Assume that $X^{ss}-X^{s} \subseteq X^{ss}$ and $Y-Y^s\subseteq Y$ have codimension at least $2$ . Then the same conclusion as in Lemma 6.1 holds:

$$ \begin{align*} K_{X^{ss}}&= q^{*}K_{Y}+\sum (|G_{R_i}|/|G_X| -1)R_i\,, \end{align*} $$

where $R_i$ is the closure of $R_i^s$ in $X^{ss}$ .

Remark 6.3. The codimension at least $2$ hypothesis above does rule out some standard GIT constructions. For instance, Corollary 6.2 is not applicable in the case of the GIT moduli space of cubic curves ${\mathcal {M}}^{\operatorname {GIT}}_{\operatorname {curve}}$ , as the locus of strictly semi-stable cubic curves is the locus of $A_1$ cubic curves, which is a codimension $1$ locus in the semi-stable locus in the Hilbert scheme $\mathbb {P}^{9}=\mathbb {P} H^0(\mathbb {P}^2,\mathcal {O}_{\mathbb {P}^2}(3))$ . Of course, ${\mathcal {M}}^{\operatorname {GIT}}_{\operatorname {curve}}$ is simply equal to $\mathbb {P}^1$ , so this particular situation is trivial.

6.3 Canonical bundle of the Kirwan blowup

Here, we consider a step in the Kirwan blowup process:

We refer the reader to Kirwan’s various papers on the topic for the details on Kirwan blowups, or to our paper [Reference Casalaina-Martin, Grushevsky, Hulek and LazaCMG+1] for a summary. For the convenience of the reader, recall that this includes the data of $\operatorname {Stab}^\circ \le G$ , a maximal dimensional connected component of a stabilizer, and the associated locus

which is a smooth closed subvariety of $X^{ss}$ . Note that in Kirwan’s papers and in [Reference Casalaina-Martin, Grushevsky, Hulek and LazaCMG+1], $\operatorname {Stab}^\circ $ is denoted by “R”; this would conflict with our notation here, that R is the Eckardt (ramification) divisor, and so we use the (possibly) more transparent $\operatorname {Stab}^\circ $ in the current paper.

Lemma 6.4. Assume that $X^{ss}-X^{s} \subseteq X^{ss}$ and $Y-Y^s\subseteq Y$ are codimension at least $2$ . Let $x\in Z_{\operatorname {Stab}^\circ }^{ss}$ be a general point, let $\mathcal {N}_x$ be the fiber of the normal bundle to $G\cdot Z_{\operatorname {Stab}^\circ }^{ss}$ in $X^{ss}$ at x, let $\ell _x\subseteq \mathcal {N}_x$ be a general line through the origin, and let $c=\operatorname {codim}_X (G\cdot Z_{\operatorname {Stab}^\circ }^{ss})$ . Denote by $\widetilde R_i$ the strict transform of the ramification divisor $R_i$ in $X^{ss}$ , denote by $G_{F}\subseteq G_x$ the stabilizer of the general line $\ell _x$ , and denote by $\mu _i$ the coefficient defined by $\tilde \pi ^* R_i=\widetilde R_i+\mu _i F$ . In terms of these invariants, the canonical bundles admit the following expressions:

(6.1) $$ \begin{align} K_{\widetilde X^{ss}}&= \tilde q^{*}K_{\widetilde Y}+\sum (|G_{ R_i}|/|G_X| -1)\widetilde R_i + (|G_{F}|/|G_X| -1)F = \tilde \pi^* K_{X^{ss}} + (c-1) F\,, \end{align} $$

(6.2) $$ \begin{align}\hspace{-12pt} K_{\widetilde Y}& = \pi ^*K_Y+\left(\frac{c+\sum (|G_{ R_i}|/|G_X| -1)\mu_i }{|G_F|/|G_X|}-1\right)E, \ \ \ \text{ if } Y \text{ is } \mathbb{Q}\text{-Gorenstein.} \end{align} $$

Proof. The first equality in (6.1) just follows from Corollary 6.2, using that F is the projectivized normal bundle to the orbit $G\cdot Z^{ss}_{\operatorname {Stab}^\circ }\subseteq X^{ss}$ , and the stabilizer of a generic point of F is therefore the stabilizer of the projectivized normal space at a generic point. The second equality in (6.1) is just the formula for the canonical bundle of the blowup of a smooth variety along a smooth subvariety.

For (6.2), we use (6.1). Indeed, substituting the expressions $K_{\widetilde Y}= \pi ^*K_Y +a(E,Y,0)E$ and $K_{X^{ss}}= q^* K_Y+ \sum (|G_{ R_I}|/|G_X| -1)R_i$ , we obtain

$$ \begin{align*} K_{\widetilde X^{ss}}&= \tilde q^{*}\pi ^*K_{Y}+ a(E,Y,0)\left( |G_F|/|G_X| \right) F+\sum (|G_{ R_i}|/|G_X| -1)\widetilde R_i + (|G_{F}|/|G_X| -1)F\\ &= \tilde q^{*} \pi ^*K_{Y}+ \sum (|G_{ R_i}|/|G_X| -1)\widetilde R_i + \left((a(E,Y,0)+1)|G_{F}|/|G_X| -1\right)F,\\ K_{\widetilde X^{ss}}& = \tilde \pi^* q^*K_{Y} + \sum (|G_{ R_i}|/|G_X| -1)\widetilde R_i +\sum (|G_{ R_i}|/|G_X| -1)\mu_i F+ (c-1) F \\ &= \tilde \pi^* q^*K_{Y} + \sum (|G_{ R_i}|/|G_X| -1)\widetilde R_i + \left(c-1 +\sum (|G_{ R_i}|/|G_X| -1)\mu_i \right)F\,. \end{align*} $$

Solving for $a(E,Y,0)$ gives the result.

Remark 6.7 (Kirwan blowups with boundary).

Still under the assumption that $X^{ss}-X^{s} \subseteq X^{ss}$ and $Y-Y^s\subseteq Y$ are codimension at least $2$ , if we consider the case of a boundary $(Y,\Delta _Y)$ and assume that $K_Y+\Delta _Y$ is $\mathbb {Q}$ -Cartier, then setting , and letting $\Delta _{\widetilde Y}$ be the strict transform of $\Delta _Y$ in $\widetilde Y$ , we have

(6.3) $$ \begin{align} K_{\widetilde Y}+\Delta_{\widetilde Y}& = \pi ^*(K_Y+\Delta_Y)+\left(\frac{a(F,X^{ss},\Delta_{X^{ss}})+1+\sum (|G_{ R_i}|/|G_X| -1)\mu_i }{|G_F|/|G_X|}-1\right)E\,. \end{align} $$

Indeed, setting $\Delta _{\widetilde X^{ss}}$ to be the strict transform of $\Delta _{X^{ss}}$ in $\widetilde X^{ss}$ , we have $\tilde q^*\Delta _{\widetilde Y}= \Delta _{\widetilde X^{ss}}$ ; in our situation, both the strict transform and pullback are defined by restricting to the locus where the morphisms are either isomorphisms or étale, and then taking closures, and so the strict transform and pullback commute. Then the same analysis as above using

$$ \begin{align*}K_{\widetilde Y}+\Delta_{\widetilde Y}= \pi ^*(K_Y+\Delta_Y) +a(E,Y,\Delta_Y)E,\end{align*} $$

$$ \begin{align*}K_{\widetilde X^{ss}}+\Delta_{\widetilde X^{ss}}=\tilde \pi^*(K_{X^{ss}}+\Delta_{X^{ss}}) +a(F,X^{ss},\Delta_{X^{ss}})F,\end{align*} $$

$$ \begin{align*}K_{X^{ss}}+\Delta_{X^{ss}}= q^*(K_Y+\Delta_Y)+ \sum (|G_{ R_i}|/|G_X| -1)R_i\end{align*} $$

gives (6.3). Note that from (6.3), it follows that if $(X^{ss}, \Delta _{X^{ss}} = q^*\Delta _Y)$ is klt, then so is $(Y,\Delta _Y)$ ; we emphasize that we started by assuming $K_Y+\Delta _Y$ was $\mathbb {Q}$ -Cartier.

6.4 Computation of $K_{{\mathcal {M}}^{\operatorname {K}}}$

We now specialize the general discussion of the previous section to the particular case of the moduli of cubic surfaces. We want to apply Corollary 6.2 to compute $K_{{\mathcal {M}}^{\operatorname {GIT}}}$ , and then $K_{{\mathcal {M}}^{\operatorname {K}}}$ . To do this for ${\mathcal {M}}^{\operatorname {GIT}}$ , recall that the locus of unstable points in $\mathbb {P}^{19}$ has codimension $\ge 2$ , and the locus of strictly semi-stable points has codimension $\ge 2$ in the semi-stable locus, both in $(\mathbb {P}^{19})^{ss}$ as well as in ${\mathcal {M}}^{\operatorname {GIT}}$ . Similarly, for ${\mathcal {M}}^{\operatorname {K}}$ , the locus of unstable points in the blowup of $(\mathbb {P}^{19})^{ss}$ has codimension $\ge 2$ , and the locus of strictly semi-stable points has codimension $\ge 2$ within the semi-stable locus, both in the blowup of $(\mathbb {P}^{19})^{ss}$ as well as in ${\mathcal {M}}^{\operatorname {K}}$ (see [Reference Casalaina-Martin, Grushevsky, Hulek and LazaCMG+1]). We are thus in the setup of the previous section.

Recall from §3 that the locus of cubics in ${\mathcal {M}}^{\operatorname {GIT}}$ with nontrivial stabilizers is the irreducible Eckardt divisor R, and the cubic surface parameterized by a general point of R has automorphism group $\mathbb {Z}_2$ . Thus, the Riemann–Hurwitz formula (Corollary 6.2) in this case gives

$$ \begin{align*}K_{(\mathbb {P}^{19})^{ss}} = q^* K_{{\mathcal {M}}^{\operatorname{GIT}}} + R\,. \end{align*} $$

The class of the Eckardt divisor is known (see Remark 3.2).

$R=\mathcal {O}_{\mathbb {P}^{19}}(100)$ , where $\mathcal {O}_{\mathbb {P}^{19}}(1)$ is the restriction to $(\mathbb {P}^{19})^{ss}$ of the hyperplane section in $\mathbb {P}^{19}$ . Thus, we have

$$ \begin{align*}-\mathcal {O}_{\mathbb {P}^{19}}(20) = q^*K_{{\mathcal {M}}^{\operatorname{GIT}}} + \mathcal {O}_{\mathbb {P}^{19}}(100)\,, \end{align*} $$

giving

$$ \begin{align*}q^*K_{{\mathcal {M}}^{\operatorname{GIT}}} = -\mathcal {O}_{\mathbb {P}^{19}}(120)\,. \end{align*} $$

Recall also (e.g., Remark 3.2) that the discriminant has class $\mathcal {O}_{\mathbb {P}^{19}}(32)$ in our situation. In other words, as the discriminant in $(\mathbb {P}^{19})^{ss}$ descends to give the Weil divisor $D_{A_1}$ on ${\mathcal {M}}^{\operatorname {GIT}}$ , the divisor $\mathcal {O}_{\mathbb {P}^{19}}(1)$ descends to ${\mathcal {M}}^{\operatorname {GIT}}$ (as a $\mathbb {Q}$ -divisor) to give the $\mathbb {Q}$ -Cartier divisor $\frac {1}{32}D_{A_1}$ . This finally gives

(6.4) $$ \begin{align} K_{{\mathcal {M}}^{\operatorname{GIT}}} = -\frac{120}{32}D_{A_1} = -\frac{15}{4}D_{A_1}\,, \end{align} $$

agreeing with the computation in (3.10).

We now compute $K_{{\mathcal {M}}^{\operatorname {K}}}$ using the same general machinery. Recall that the Kirwan desingularization for the case of cubic surfaces is obtained by a single blowup, supported at the point $\Delta _{3A_2}\in {\mathcal {M}}^{\operatorname {GIT}}$ corresponding to the orbit of the $3A_2$ cubic surface $S_{3A_2}$ .

Corollary 6.8.

$$ \begin{align*} K_{{\mathcal {M}}^{\operatorname{K}}} = \pi ^* K_{{\mathcal {M}}^{\operatorname{GIT}}} + 20 D_{3A_2}\,. \end{align*} $$

6.5 Intersection numbers for divisors on DM stacks

On DM stacks, the top self-intersection numbers of canonical classes are rational numbers. We need the following statement regarding the denominators that may appear, which is essentially [Reference AlexeevAGV, Prop. 2.1.1].

Proposition 6.10. Let $\mathcal {X}$ be a smooth proper DM stack over a field K of characteristic $0$ with coarse moduli space $\Phi :\mathcal {X}\to X$ of dimension d. For each geometric point ${p:\operatorname {Spec} \overline K \to \mathcal {X}}$ , denote by $e_p$ the exponent of the automorphism group of p, and denote by e the least common multiple of the numbers $e_p$ for all geometric points of $\mathcal {X}$ . For any divisor classes $D_1,\dots ,D_d\in \operatorname {CH}_{d-1}(X)$ , we have

$$ \begin{align*}D_1\cdots D_d \in \frac{1}{e^d}\mathbb{Z}\,. \end{align*} $$

Proof. We first recall that for line bundles $M_1,\dots ,M_d$ on X, with divisor classes $[M_1], \dots , [M_d]\in \operatorname {CH}_{d-1}(X)$ , we have by definition

Using the fact that X is $\mathbb {Q}$ -factorial, this allows us to define the intersection number $D_1\cdots D_d$ as a rational number for any divisor classes $D_1,\dots ,D_d\in \operatorname {CH}_{d-1}(X)$ .

To prove the bound on the denominators, we argue as follows. For any divisor $D\in \operatorname {CH}_{d-1}(X)$ , the pullback $\Phi ^*D$ is the divisor class associated with some line bundle $\mathcal {L}$ on $\mathcal {X}$ (given an étale presentation $P:U\to \mathcal {X}$ , the pullback $P^* \Phi ^*D$ is the class of a line bundle on U, which descends to $\mathcal {X}$ ). The fact that $\mathcal {L}^{\otimes e}$ descends to a line bundle M on X is [Reference AlexeevAGV, Lem. 2.1.2]. Thus, we have $(eD_1) \cdots (e D_d) \in \mathbb {Z}$ , completing the proof.