Compactifications of moduli of del Pezzo surfaces via line arrangement and K-stability

We know that the Fermat cubic surface is K-stable (cf. [Tia87]). By the openness of the K-stability (cf. Theorem 2.5), a general cubic surface is K-stable. Denote by $\overline{M}^{K}_{3}$ by the K-moduli space of cubic del Pezzo surfaces, which parameterizes K-stable smooth cubic surfaces and their K-polystable limits. Let $X\in\overline{M}^{K}_{3}$ be the limit of a 1-parameter family of $\{X_{t}\}_{t\in T\setminus\{0\}}$ K-semistable smooth cubics.

As the K-semistable surfaces have klt singularities (cf. [Oda13]), in particular have quotient singularities, we let $(x\in X)\simeq(0\in\mathbb{A}^{2}/G_{x})$ be a singular point (if it exists). Then by Theorem 2.7, we have that

which implies that $|G_{x}|\leq 3$. If $|G_{x}|=2$, then $x$ is an $A_{1}$-singularity. If $|G_{x}|=3$, then $x$ is either an $A_{2}$-singularity or a $\frac{1}{3}(1,1)$-singularity. The latter case is ruled out by [KSB88, Proposition 3.10]. We thus conclude that $X$ has at worst $A_{1}$- or $A_{2}$-singularities. It follows from [Fuj90, Section 2] that $X$ can be embedded by $|-K_{X}|$ into $\mathbb{P}^{3}$ as a cubic surface.

Since for hypersurfaces the K-stability implies GIT-stability (cf. [OSS16, Theorem 3.4] or [PT09, Theorem 2]), then we get an open immersion $\mathcal{M}^{K}_{3}\hookrightarrow\mathcal{M}^{\operatorname{GIT}}_{3}$ of moduli stacks, which descends to a birational and injective morphism $\Phi:\overline{M}^{K}_{3}\rightarrow\overline{M}^{\operatorname{GIT}}_{3}$ between good moduli spaces. Notice that the $\overline{M}^{\operatorname{GIT}}_{3}$ is normal by the properties of GIT quotients. As both of these two moduli spaces are proper, then $\Phi$ is a finite map, and thus $\Phi$ is an isomorphism by Zariski Main Theorem.

∎

It is proven in [GKS21, Proposition 5.8] that if $X$ the log canonical threshold is larger than $\frac{1}{9}$ if $X$ has at worst $A_{1}$-singularities.

Now let us deal with the surfaces with $A_{2}$-singularities. We first give an explicit construction of the surface $X_{0}=\{x_{0}^{3}=x_{1}x_{2}x_{3}\}$. Let $(x:y:z)$ be the homogeneous coordinates of $\mathbb{P}^{2}$, then blow up the tangent vector given by $\{y=0\}$ at $(1:0:0)$, the tangent vector given by $\{z=0\}$ at $(0:1:0)$, and the tangent vector given by $\{x=0\}$ at $(0:0:1)$. Denote by $E_{i}$ and $F_{i}$ the exceptional divisors on the blow-up with self-intersection $-2$ and $-1$ respectively, and $H_{i}$ the $(-2)$-curves in the class

respectively, where $i=1,2,3$. Denote this surface by $\widetilde{X}_{0}$. Finally contract the three pairs of $(-2)$-curves to get $X_{0}$. Notice that there are four sections of $\omega^{*}_{\widetilde{X}_{0}}$ coming from $xyz,y^{2}z,z^{2}x,x^{2}y\in H^{0}(\mathbb{P}^{2},\omega^{*}_{\mathbb{P}^{2}})$. They give rise to a morphism from $\widetilde{X}_{0}$ to $\mathbb{P}^{3}$ contracting $(-2)$-curves, with image $X_{0}\subseteq\mathbb{P}^{3}$ given by $x_{0}^{3}=x_{1}x_{2}x_{3}$.

Let $\pi:\widetilde{X}_{0}\rightarrow X_{0}$ be the contraction map. This surface has three $A_{2}$-singularities, and the 27 lines degenerate to the images of $F_{1},F_{2},F_{3}$, denoted by $G_{1}$, $G_{2}$, $G_{3}$, each of which is of multiplicity $9$ (cf. [Tu05, Proposition 4.1(ii)]). Then by computing intersection numbers, we deduce that

and the other two relations for $\pi^{*}G_{2}$ and $\pi^{*}G_{3}$. As a result, we obtain that

for $0<c<\frac{1}{9}$. Thus the pair $(X_{0},c\sum L_{j})$ is log canonical for $0<c<\frac{1}{9}$. The other pairs whose surfaces have $A_{2}$-singularities specially degenerate to this case, so we get the result we desire. ∎

Let $\pi^{\circ}:(\mathscr{X}^{\circ},c\mathscr{D}^{\circ})\rightarrow T\setminus\{0\}$ be a family of K-semistable pairs, where $(\mathscr{X}^{\circ},c\mathscr{D}^{\circ})_{t}$ is isomorphic to a smooth del Pezzo surface with the sum of lines, and $\mathscr{X}\rightarrow T$ be an extension of $\mathscr{X}^{\circ}\rightarrow T\setminus\{0\}$. It suffices to show that there exists a unique extension $\pi:(\mathscr{X},c\mathscr{D})\rightarrow T$ of $\pi^{\circ}$. The uniqueness is apparent to us: as $\mathscr{X}_{0}$ is normal, the filling $\mathscr{D}_{0}$ must be obtained in the following way if it exists. Let $\mathscr{X}^{\operatorname{sm}}$ be the open locus of $\mathscr{X}$ such that $\pi:\mathscr{X}\rightarrow T$ is smooth on $\mathscr{X}^{\operatorname{sm}}$. Taking the scheme-theoretic closure of $\mathscr{D}^{\circ}$, and restricting it to $\mathscr{X}^{\operatorname{sm}}_{0}$, the $\mathscr{D}_{0}$ is the extension (by taking closure) of this restricted divisor on $\mathscr{X}^{\operatorname{sm}}_{0}$ to $\mathscr{X}_{0}$ as a Weil divisor.

Now we only need to display such a filling. Recall that the central fiber $\mathscr{X}_{0}$ has at worst $A_{2}$-singularities. It was displayed in [Cay69] that the lines on general fibers degenerate to lines on $\mathscr{X}_{0}$ with multiplicities, denoted by $\sum L_{i,0}$. It follows from Proposition 3.4 that the pair $(\mathscr{X}_{0},c\sum L_{i,0})$ is log canonical when $c=\frac{1}{9}$. It follows from interpolation that the pair $(\mathscr{X}_{0},c\sum L_{i,0})$ is K-semistable for $0<c\ll 1$, and this gives a desired filling. ∎

We claim that the forgetful map $\varphi:\mathcal{M}^{K}_{3,\varepsilon}\rightarrow\mathcal{M}^{K}_{3}$ is finite. Since $\varphi$ is representable, by Proposition 3.5, it suffices to show that it is quasi-finite. In the proof of Proposition 3.5, we in fact show that for each K-semistable pair $(X,cD)$ in the stack $\mathcal{M}^{K}_{3,\varepsilon}$, the divisor $D$ is the sum of the lines on $X$ counted with multiplicities. For every cubic surface $X$ with at worst $A_{2}$-singularities, there are only finitely many lines on $X$, hence $\varphi$ is quasi-finite. The finite forgetful map $\varphi$ descends to a finite morphism between good moduli spaces $\psi:\overline{M}^{K}_{3,\varepsilon}\rightarrow\overline{M}^{K}_{3}$. As the K-moduli $\overline{M}^{K}_{3}$ is normal, and the morphism $\psi$ is birational and finite, it follows from the Zariski’s Main Theorem that $\psi$ is an isomorphism. ∎

This follows immediately from Proposition 3.4 and Proposition 3.6 that there are no walls for the K-moduli spaces $\overline{M}^{K}_{3,c}$, and they are all isomorphic to the K-moduli of cubic del Pezzo surfaces, the isomorphism being induced by the forgetful map $\varphi$.

∎

As in the proof of Corollary 3.8, we have a family $(\mathcal{X},\mathcal{D})$ over $\mathcal{M}^{K}_{4}$ such that a general fiber is a smooth quartic with the 16 lines on it. By interpolation of K-stability (cf. Theorem 2.4), if we prove that these pairs are log canonical for $0<c\leq\frac{1}{4}$, then $(\mathcal{X},c\mathcal{D})$ has K-semistable fibers for any $0\leq c<1/4$. In particular, the same argument of the second statement of Corollary 3.8 shows that $\overline{M}^{K}_{4,c}\simeq\overline{M}^{K}_{4}$ for any $c\in(0,\frac{1}{4})$.

Now we prove that the each fiber in the family $(\mathcal{X},c\mathcal{D})$ is log canonical for $0\leq c\leq 1/4$. Let $(X,cD=c\sum L_{i})$ be an arbitrary fiber.

1. (i)

   If $X$ is smooth, then the 16 lines are distinct and $\sum L_{i}$ is normal crossing. Thus the pair is log canonical automatically.

2. (ii)

   If $X$ has one $A_{1}$-singularity $p$, then we claim that there exist exactly four lines of multiplicity two and eight lines of multiplicity one, and that these double lines pass through $p$, while the remaining eight lines avoid it.

   First notice that a quintic del Pezzo surface $X$ with exactly one $A_{1}$-singularity is obtained by contracting the $(-2)$-curve on the blow-up of $\mathbb{P}^{2}$ along three distinct points $p_{1},p_{2},p_{3}$ and a tangent vector supported at $p_{4}$. In fact, let $p\in X$ be the singularity, and $\widetilde{X}\rightarrow X$ the blow-up of $X$ at $p$ with an exceptional divisor, which is a $(-2)$-curve. By [HW81, Theorem 3.4], the surface $\widetilde{X}$ is obtained from blowing up $\mathbb{P}^{2}$, thus there must be a $(-1)$-curve $C$ intersecting $E$. Contracting first $C$, then the proper transform of $E$, one get a smooth del Pezzo surface of degree $6$, which is a blow-up of $\mathbb{P}^{2}$ at three distinct points.

   Consider a one-parameter family of projective planes $\mathbb{P}^{2}\times\mathbb{A}^{1}\rightarrow\mathbb{A}^{1}$, and let $l_{1},...,l_{5}$ be five sections of $\mathbb{A}^{1}$ such that over $0\neq t\in\mathbb{A}^{1}$, $l_{1},...,l_{5}$ does not intersect, while over $0$, only $l_{4}$ and $l_{5}$ intersect transversely. Blowing up $\mathbb{P}^{2}\times\mathbb{A}^{1}$ along $l_{1}\cup\cdots\cup l_{5}$ with reduced scheme structure, one get a degeneration $\mathfrak{X}\rightarrow\mathbb{A}^{1}$ of smooth quartic del Pezzo surfaces to a singular one with exactly one $A_{1}$-singlarity. Denote $p_{i}(t)$ the intersection of $l_{i}$ with the fiber $\mathbb{P}^{2}_{t}$ over $t\in\mathbb{A}^{1}$. Then under the family $\mathfrak{X}\rightarrow\mathbb{A}^{1}$, the lines $L(p_{i}(t),p_{4}(t))$ and $L(p_{i}(t),p_{5}(t))$ (for i=1,2,3) on general fibers $\mathfrak{X}_{t}$ degenerate to the same line $L(p_{i}(0),p_{4}(0))=L(p_{i}(0),p_{5}(0))$, and the exceptional divisors over $p_{4}(t)$ and $p_{5}(t)$ also degenerate to the same line, which is the exceptional divisor over $p_{4}(0)=p_{5}(0)$. The other eight lines over general fibers degenerate to distinct lines on $\mathfrak{X}_{0}$. Finally observe that $-K_{\mathfrak{X}/\mathbb{A}^{1}}$ gives rise to an embedding of $\mathfrak{X}$ into $\mathbb{P}^{4}\times\mathbb{A}^{1}$ over $\mathbb{A}^{1}$. Thus this degeneration indeed occurs in the Hilbert scheme.

   Let $E$ be the exceptional divisor of the blow-up $\pi:\widetilde{X}\rightarrow X$. Then $E+\pi^{*}\sum L_{i}$ has simple normal crossing support. Moreover, we have that

   $$A_{(X,c\sum L_{i})}(E)=1-4c>0$$

   for any $0<c<\frac{1}{4}$. As the multiplicity of the proper transform of $L_{i}$ is at most two, then the pair is log canonical.

3. (iii)

   If $X$ has two $A_{1}$-singularities $p$ and $q$, then there exist one line of multiplicity four, four lines of multiplicity two and four lines of multiplicity one. This follows from the same argument as in (i), and in fact the line with multiplicity four passes through both $p$ and $q$, the line with multiplicity two passes through either $p$ or $q$, and the line with multiplicity one avoids both $p$ and $q$. For each singularity, there are exactly eight lines (counted with multiplicities) passing through it. Let $E$ be the exceptional divisor of the blow-up at $p$. Then we have that

   $$A_{(X,c\sum L_{i})}(E)=1-4c>0$$

   for any $0<c<\frac{1}{4}$. As the multiplicity of the proper transform of $L_{i}$ is at most four, then the pair is log canonical.

   The proof of cases (iii) and (iv) are completely the same, we will omit part of the details.

4. (iv)

   Suppose that $X$ has three $A_{1}$-singularities. Then $X$ is obtained as follows: blow up $\mathbb{P}^{2}$ at a general tangent vector and at three curvilinear points on a general line, where two of them collide, then take the ample model. There exist two lines of multiplicity four, and four lines of multiplicity two. For each singularity, there are exactly eight lines (counted with multiplicities) passing through it. For the same reason as in (ii), the pair is log canonical.

5. (v)

   If $X$ has four $A_{1}$-singularities, then it is obtained as follows: blow up $\mathbb{P}^{2}$ at a general point $P$ and at two tangent vectors whose supporting lines pass through $P$, then take the ample model. There are four lines of multiplicity four. For each singularity, there are exactly eight lines (counted with multiplicities) passing through it. For the same reason as in (ii), the pair is log canonical.

∎

First observe that the $(\mathbb{P}^{2},C_{2}+C^{\prime}_{2})$ is the degeneration (in the Hilbert scheme) of pairs $(\mathbb{P}^{2},C_{2}(t)+C^{\prime}_{2}(t))_{t\in T}$ where the boundary divisors consist of two smooth conics with a tacnode and two nodes. It follows from [CS03, Section 3.4 case 2] that the arrangement of the 28 bitangent lines of $C_{2}+C^{\prime}_{2}$ is $\sum L_{i}=6L_{p}+6L_{q}+16L_{pq}$, where $L_{p}$ and $L_{q}$ are tangent lines of the conics at $p$ and $q$ respectively, and $L_{pq}$ is the line connecting $p$ and $q$. It follows immediately that the coefficients of $\frac{1}{2}(C_{2}+C^{\prime}_{2})+c\sum L_{i}$ are all smaller than $1$.

Taking the minimal log resolution of $(\mathbb{P}^{2},\frac{1}{2}(C_{2}+C^{\prime}_{2})+c\sum L_{i})$. By symmetry, we only need to look at the log resolution at $q$. Let $E$ and $F$ be the exceptional divisors over $q$ with self-intersection $-2$ and $-1$ respectively. Then one has $A(E)=1-22c$ and $A(F)=1-28c$, which are positive when $0<c<\frac{1}{28}$. ∎

First observe that the $(\mathbb{P}^{2},C_{2}+M_{1}+M_{1}^{\prime})$ is the degeneration (in the Hilbert scheme) of pairs $(\mathbb{P}^{2},C_{2}(t)+M_{1}(t)+M_{1}^{\prime}(t))_{t\in T}$ where the boundary divisors consist of two distinct lines $M_{1}(t)$, $M_{1}^{\prime}(t)$ and a smooth conic $C_{2}(t)$ tangent to $M_{1}(t)$ at a point $p(t)$ and meeting $M_{1}^{\prime}(t)$ transversely. It follows from [CS03, Section 3.4 case 10] that the arrangement of the 28 bitangent lines of $C_{2}+M_{1}+M_{1}^{\prime}$ is $\sum L_{i}=6M_{1}+6M_{1}^{\prime}+16L_{pq}$, where $L_{pq}$ is the line connecting $p$ and $q$. It follows immediately that the coefficients of $\frac{1}{2}(C_{2}+C^{\prime}_{2})+c\sum L_{i}$ are all smaller than $1$.

Taking the minimal log resolution of $(\mathbb{P}^{2},\frac{1}{2}(C_{2}+M_{1}+M_{1}^{\prime})+c\sum L_{i})$. By symmetry, we only need to look at the log resolution at $q$. Let $E$ and $F$ be the exceptional divisors over $q$ with self-intersection $-2$ and $-1$ respectively. Then one has $A(E)=1-22c$ and $A(F)=1-28c$, which are positive when $0<c<\frac{1}{28}$. ∎

The classification of $C_{4}$ is listed in [CS03, Section 3.4]. For each class, we can run the same argument. For simplicity, we only prove the statement for one class where $C_{4}=M_{1}+M_{2}+M_{3}+M_{4}$ is the union of four general lines (forming three pairs of nodes). In this case, we have that $\sum L_{i}=4(M_{1}+\cdots+M_{7})$, where $M_{5},M_{6},M_{7}$ are the lines joining the three pairs of nodes (cf. [CS03, Section 3.4 case 11]). Then the blow-up of the pair $(\mathbb{P}^{2},\frac{1}{2}C_{4}+c\sum L_{i})$ at the six nodes (with exceptional divisors $E_{1},...,E_{6}$) is log smooth. As the boundary divisors $\frac{1}{2}C_{4}+c\sum L_{i}$ have coefficients less than $1$ and $A_{(\mathbb{P}^{2},\frac{1}{2}C_{4}+c\sum L_{i})}(E_{i})=1-12c>0$, then the pair we consider is log canonical. ∎

We know from [CS03, Lemma 3.3.1] that the 28 lines degenerate to 4 bitangent lines of multiplicity 1 not passing through the nodes, 6 tangent lines of multiplicity 2 passing through exactly one node, and 3 lines of multiplicity 4 containing two nodes. Thus for each node $p$, there are 12 lines passing through it, and at most 4 lines tangential to it. One can blow up at $p$ twice to get a log resolution. We have $A(E)=1-16c>0$ and $A(F)\geq\frac{3}{2}-20c>0$ for $0<c<1/28$, where $E$ and $F$ are the two exceptional divisors over $p$ with self-intersection $-2$ and $-1$ respectively. The singularities at other points are milder, thus $(\mathbb{P}^{2},\frac{1}{2}C_{4}+c\sum L_{i})$ is log canonical as desired.

∎

We know from [CS03, Lemma 3.3.1] that the 28 lines degenerate to 4 bitangent lines of multiplicity 1 not passing through the cusps, and 3 lines of multiplicity 9 containing two cusps. Thus for each cusp $p$, there are 18 lines passing through it, and none of them are tangential to it. One can blow up at $p$ three times to get a log resolution. Let $E,F,G$ be the exceptional divisors with self-intersection $-3,-2,-1$ respectively. We have $A(E)=1-18c>0$, $A(F)=\frac{3}{2}-18c>0$, and $A(G)=2-36c>0$ for $0<c<1/28$. The singularities at other points are milder, thus $(\mathbb{P}^{2},\frac{1}{2}C_{4}+c\sum L_{i})$ is log canonical as desired.