Unimodal singularities and boundary divisors in the KSBA moduli of a class of Horikawa surfaces

Smooth minimal surfaces of general type with $K^2=1$, $p_g=2$, and $q=0$ constitute a fundamental example in the geography of algebraic surfaces, and the 28-dimensional moduli space $\mathbf{M}$ of their canonical models admits a modular compactification $\overline{\mathbf{M}}$ via the minimal model program. We describe eight new irreducible boundary divisors in such compactification parametrizing reducible stable surfaces. Additionally, we study the relation with the GIT compactification of $\mathbf{M}$ and the Hodge theory of the degenerate surfaces that the eight divisors parametrize.


Introduction
The interplay between geometric compactifications and Hodge theory is one of the driving forces in moduli theory. Well-studied cases include abelian varieties [Ale02], K3 surfaces [AE21, AEH21, ABE22, AE22, AET23], algebraic curves [CV11], cubic surfaces [GKS21], cubic fourfolds [Laz10], etc. Recently, there has been a great focus on generalizing this interplay in the case of surfaces of general type -the so-called non-classical cases [KU09,GGLR17]. A particular well-posed case for such generalization is given by algebraic surfaces of general type with geometric genus p g = 2, irregularity q = 0, and K 2 = 1. These were described by Enriques [Enr49] and later studied by Horikawa [Hor76]. In the current work we refer to these surfaces, which are sometimes called I-surfaces, simply as Horikawa surfaces. By the work of Gieseker [Gie77], the canonical models of such surfaces form a 28-dimensional quasi-projective coarse moduli space M.
In the current paper, we consider the compactification perspective offered by the minimal model program. By the work of Kollár, Shepherd-Barron, and Alexeev [KSB88,Ale96,Kol23] there exists a projective and modular compactification parametrizing stable surfaces with K 2 = 1 and χ(O) = 3, which contains M. This is also known as the KSBA compactification, and in this specific case has several irreducible components by [FPRR22]. In this work we focus on the main irreducible component M of this compactification, which parametrizes smoothable surfaces. Our first contribution towards the understanding of M is the following theorem that combines Theorems 3.15 and 4.2.
Following the approach in [Wen21], we review the construction of the moduli space of Horikawa surfaces and its GIT compactification.
Definition 2.2. The vector space of coefficients for a polynomial in normal form F 10 (x, y, z) = z 5 + q 4 (x, y)z 3 + q 6 (x, y)z 2 + q 8 (x, y)z + q 10 (x, y) is the vector space Let U be the open subset of points in the affine space Spec(Sym(V ∨ 10 )) ∼ = A 32 parametrizing smooth Horikawa surfaces in normal form. There is a natural GL 2 -action on U given by linear change of coordinate in x and y. By [Wen21, Lemma 4 and Lemma 7], we have that the points of the 28-dimensional quotient M = U/GL 2 are in bijection with the isomorphism classes of Horikawa surfaces considered. We refer to M as the coarse moduli space of Horikawa surfaces. This coincides with the Gieseker moduli space of canonical surfaces S of general type with K 2 S = 1, p g (S) = 2, and q(S) = 0 [Gie77]. The family of Horikawa surfaces is defined over U by the relative equation H ′ = V (w 2 − z 5 − g 4 (x, y)z 3 − g 6 (x, y)z 2 − g 8 (x, y)z − g 10 (x, y)) ⊆ U × P(1, 1, 2, 5), 5 with proper flat morphism H ′ → U given by the restriction of the projection onto the first factor. This family descends to the geometric quotient by GL 2 giving the family of Horikawa surfaces H → M with pairwise non-isomorphic fibers. Definition 2.3. As discussed in [Wen21, Proposition 9], we have a natural projective GIT compactification of M given by M git := A 32 / /GL 2 ∼ = (A 32 /C * )/ /(GL 2 /C * ) ∼ = P(4 5 , 6 7 , 8 9 , 10 11 )/ /SL 2 , where the GIT quotients are with respect to appropriate linearizations (see [Wen21]) and P(4 5 , 6 7 , 8 9 , 10 11 ) denotes a weighted projective space with n m representing n, . . . , n repeated m-times. • Every geometric fiber X s is a stable variety, dim(X s ) = d, and K d Xs = C; • There exists an invertible sheaf L on X such that for every geometric fiber X s , L| Xs ∼ = O Xs (N(K Xs )). The above stack V is called the Viehweg moduli stack. In the notation of [Kol23,§ 8], this moduli functor is SP(0, d, C). Definition 2.6. Consider the Viehweg moduli stack V for d = 2, C = 1, and N large enough (see Definition 4.1). Let V be the corresponding projective coarse moduli space. The family of Horikawa surfaces H → M in Definition 2.2 induces a morphism h : M → V which is injective on C-points. We denote by M the normalization of the closure of the image of h in V, and we will refer to M as the KSBA compactification of the moduli space of Horikawa surfaces. parametrizes classes of degenerations of degree 10 hypersurfaces in P(1, 1, 2, 5) that appear as double covers of P(1, 1, 2). Therefore, we can determine the surface singularities by considering the singularities of the branch curve V (F 10 (x, y, z)) ⊆ P(1, 1, 2). If the curve is away from the singularity of P(1, 1, 2), then the singularities are locally isomorphic to a plane curve singularity. We now describe the plane curve singularities of interest for the current work. Given a singularity T , we can associate two invariants: the Milnor number µ(T ) and the modality m(T ) of the singularity (also called modulus). These invariants measure the complexity of the singularity and provide a criterion to classify them. From the moduli theory perspective, the modality m(T ) is particularly interesting because it is equal to the dimension of stratum in the base space of the versal deformation where µ is constant, minus 1 (see [Arn76,§ 4]). Another relevant fact is that both µ(T ) and m(T ) are upper semicontinuous, see [GLS07, Chapter I, § 2]. The hypersurface singularities with m(T ) = 0 are precisely the ADE singularities. Therefore, the next case of interest are the singularities with m(T ) = 1. By the classification of log canonical two dimensional hypersurface singularities [LR12, Table 1] in combination with [Arn76, § I.1], we obtain that there are eight plane curve singularities of modality 1 that are not log canonical. These are given in Table 2. Here, the parameter a is generic, and as it varies describes non isomorphic plane curves with the given isolated singularities. Table 2: Local models of the eight isolated non-log canonical singularities of modality 1 that can be attained as degeneration in P(1, 1, 2, 5) of Horikawa surfaces. E 12 z 3 + y 7 + ay 5 z Z 11 yz 3 + y 5 + ay 4 z W 12 z 4 + y 5 + ay 3 z 2 E 13 z 3 + y 5 z + ay 8 Z 12 yz 3 + y 4 z + ay 3 z 2 W 13 z 4 + y 4 z + ay 6 E 14 z 3 + y 8 + ay 6 z Z 13 yz 3 + y 6 + ay 5 z We remark that we label the variables in a different way than in [Arn76, § I.1]. Our choice is to be compatible with the discussion that follows. Furthermore, the above notation denotes germs of singularities up to stable equivalence, that is, up to terms of the form x 2 i . Therefore, for instance, we use the notation E 12 for both the plane curve singularity and for the associated surface singularity w 2 = z 3 + y 7 + ay 5 z. Summarizing, we have the following lemma.
Lemma 2.8. Let X → ∆ be a one-dimensional family of smooth Horikawa surface degenerating a double cover of P(1, 1, 2) with an unique isolated singularity of modality 1. Then the singularity is one of the following: E 12 , E 13 , E 14 , Z 11 , Z 12 , Z 13 , W 12 , W 13 .
We conclude with the following diagram in Figure 1, where the arrow A ← B means that the germ of an A singularity degenerates to the germ of a B singularity. Remark 2.9. Within the context of the main theorems in the introduction (e.g. Theorem 1.1), we highlight that the adjacency diagrams in Figure 1 do not mean that the divisors D Σ are contained in each other or that they intersect generically in M. Additionally, in terms of deformation theory, we note that the germs of E 7 and E 8 degenerate to Z 11 , Z 12 , Z 13 and E 12 , E 13 , E 14 respectively. The equations of E 7 and E 8 are x 4 + y 4 + ax 2 y 2 = 0 with a 2 = 4 and x 3 + y 6 + ax 2 y 2 = 0 with 4a 3 + 27 = 0 respectively. However, after stable replacement, these describe codimension one strata of the boundary of M. In general, in a compact moduli space of stable varieties, it is an open problem to understand the reciprocal relations among the boundary strata corresponding to the singularities in an adjacency diagram of germ of singularities.
2.4. Weighted blow ups. A key step in our work is a partial resolution of isolated singularities via weighted blow ups at a point, which, we briefly describe here. See [KSC04, § 6.38] for a reference. Let (a 1 , . . . , a n ) be a sequence of relatively prime positive integers. We have a natural map A n P(a 1 , . . . , a n ) (x 1 , . . . , x n ) → (x a 1 1 , . . . , x an n ). The weighted blow up of A n with local coordinates (x 1 , . . . , x n ) and weights (a 1 , . . . , a n ) at the origin is the closure of the graph of the above rational map. Its exceptional divisor is isomorphic to P(a 1 , . . . , a n ), and the associated ideal is the integral closure of the ideal (x N/a 1 1 , . . . , x N/an n ) for sufficiently divisible N. In particular, if a 1 = . . . = a n = 1, then we recover the simple blow up of A n at the origin.

Stable replacements of generic one-parameter degenerations of Horikawa surfaces
In this section, we construct the KSBA stable replacements for one-parameter degenerations of Horikawa surfaces over DVRs whose central fiber has a singularity of type Σ. We begin by first analyzing a special class of degenerations defined by a C * -action. The general case of a DVR is handled in § 3.10.
Let C[x, y, z, w] denote the homogeneous coordinate ring of P(1, 1, 2, 5) where x and y have degree 1, z has degree 2, and w has degree 5. Let Σ be one of the eight exceptional families of isolated unimodular surface singularities in Lemma 2.8. Let where f (x, y, z) is a homogeneous polynomial of weighted degree 10 with coefficients in C[t]. Let π : S → C denote the projection onto the second factor with fibers S t = π −1 (t) and ∆ be a neighborhood of 0 ∈ C such that, after restriction to ∆, S becomes a one-parameter family of smooth Horikawa surfaces for t = 0 and for t = 0 the fiber S 0 has exactly one isolated singularity of type Σ. After a sequence of birational modifications of the central fiber of π : S → ∆ and possibly base changes, we obtain a new family S ′ → ∆ ′ whose central fiber S ′ 0 now has semi-log canonical singularities and it has ample canonical class, i.e. S ′ 0 is a stable surface. The surface S ′ 0 is called the stable replacement of the central fiber of S → ∆. The isomorphism class of S ′ 0 corresponds to the limit point in the KSBA compactification M of the arc ∆ • := ∆ \ {0} → M induced by the family S • := S \ S 0 → ∆ • . More precisely, for each singularity Σ we describe some explicit families S → ∆ such that the corresponding isomorphism classes of stable surfaces S ′ 0 generically describe a divisor in M. The discussion is organized as follows. First, in § 3.1 we define such families. Then, in § 3.2 we describe the stable replacements S ′ 0 . Afterward, the remaining subsections contain the proofs of the claims in § 3.2. We show that all the isomorphism classes of S ′ 0 give rise to boundary divisors in M later in § 4.
3.1. Definition of the families. Let Σ be one of the eight singularity types and let (p, q), d as in Table 3. The meaning of these constants is the following: if we assign weight (p, q) to (y, z), the lower degree part of the local singularity for Σ becomes homogeneous of degree d, and this information will be used to compute the stable replacement of the central fiber S 0 .
Definition 3.1. Let V 10 be the complex vector space spanned by the monomials x a y b z c satisfying a + b + 2c = 10. If x a y b z c is a monomial in V 10 , then we define its weight with respect to Σ as wt Σ (x a y b z c ) = pb + qc − d.
The remaining nine monomials can be of positive, negative, or zero weight as the singularity type changes: In particular, for each singularity type, there exist precisely two monomials of weight 0 as listed below. Sing.
m 1 (x, y, z) x 3 y 7 x 3 y 5 z x 2 y 8 x 5 y 5 x 4 y 4 z x 4 y 6 x 5 y 5 x 4 y 4 z Proof. This is an immediate computer assisted check.
Definition 3.3. We let U Σ denote the codimension one subspaces of V 10 consisting of elements for which m 1 and m 2 have the same coefficient. Then, the weight function gives a direct sum decomposition U Σ = U Σ,+ ⊕ U Σ,0 ⊕ U Σ,− where: We let π + , π 0 , and π − denote the projections from U Σ to U Σ,+ , U Σ,0 , and U Σ,− respectively. If W is a subspace of U, then we define W reg = W \ π −1 0 (0) and P(W ) reg = {[w] ∈ P(W ) | w ∈ W reg }. We have that P(W ) reg is an affine patch of the projective space P(W ).
Definition 3.5. We now define a C * -action on V 10 by describing how an element t ∈ C * acts on a given monomial x a y b z c ∈ V of weight ω. We define In particular t ⋆ (m 1 + m 2 ) = m 1 + m 2 , and hence this C * -action descends to an action on U Σ .
Remark 3.6. If v ∈ V 10 then t ⋆ v converges to π + (v) + π 0 (v) as t → 0, and hence t ⋆ v ∈ Definition 3.7. If u ∈ (U Σ ) reg we define the associated family of surfaces to be The corresponding family of curves is with projection π : B → C and fiber B t = π −1 (t) ⊆ P(1, 1, 2).
Remark 3.8. The family B depends only on [u] ∈ P(U Σ ) reg . The limit branch curve is obtained by composing V (·) with the morphism P(U Σ ) reg → P(U Σ,+ ⊕U Σ,0 ) reg of Lemma 3.4. The central fiber S 0 is the double cover of P(1, 1, 2) with branch curve B 0 .
Definition 3.9. Let u ∈ U Σ and consider the one-parameter family S(t ⋆ u) → ∆ in Definition 3.7. The central fiber S 0 ⊆ S(t ⋆ u) is given by V (w 2 − (π 0 + π + )(u)) ⊆ P(1, 1, 2, 5). We say that u is Σ-generic provided the following hold: (1) The central fiber S 0 is singular only at [1 : 0 : 0 : 0], where it has a singularity of type Σ; (2) The other fibers of S(t ⋆ u) → ∆ are smooth Horikawa surfaces in P(1, 1, 2, 5). Such conditions are verified for a generic choice of the coefficients of the polynomial u. This can be observed at the level of the branch curve (3.2), for which we need to show it has only one singularity of type Σ at [1 : 0 : 0]. The idea is that for a generic u, the curve V ((π 0 + π + )(u)) has a singularity of type Σ at [1 : 0 : 0], and one can choose a specific u for which the corresponding curve only has a singularity of type Σ at [1 : 0 : 0]. By the upper semicontinuity of Milnor numbers [GLS07, Theorem 2.6], the generic u has the claimed property. We illustrate an analogous argument in the proof of Theorem 7.1.
Remark 3.10. There exists r > 0 such that S t is smooth for 0 < |t| < r, i.e. u is Σ-generic in the sense of the Definition 3.9. In particular, in order to be Σ-generic, the coefficient of z 5 must be non-zero, otherwise S 0 will also pass through the singular point [0 : 0 : 1 : 0] ∈ P(1, 1, 2, 5).
Finally, the following construction will be used in § 3.3.
Example 3.12. Let us choose Σ = W 12 , so that (p, q) = (4, 5) and d = 20. Let us construct an example of the family described together with the C * -action. The weight function is given by wt Σ (x a y b z c ) = 4b + 5c − 20. So, we can consider where the weights of the monomials are 5, 0, 0, −20 respectively. We have that t ⋆ u = z 5 + x 5 y 5 + x 2 z 4 + t 20 x 10 . Therefore, in the central fiber for t = 0, the limiting curve is given by the vanishing of z 5 + x 5 y 5 + x 2 z 4 = 0. Additionally, we have that which is homogeneous of degree 20 in P(1, 4, 5) with coordinate [t : α : β].

3.2.
Stable replacement of the central fiber of the families. Fix one of the eight singularity types Σ and let S = S(t ⋆ u) → ∆ be one of the families in Definition 3.7 with u Σ-generic. Then S → ∆ comes equipped with a fiberwise Z 2 -action. The quotient by this action S → X = P(1, 1, 2) × ∆ has branch divisor B ⊆ X which fiberwisely gives a curve of weighted degree 10.
Let S ′ → ∆ ′ denote the stable replacement of S → ∆. As this is obtained after a combination of birational modifications of the central fiber and possibly after base changes branched at the origin of ∆, then also S ′ → ∆ ′ comes equipped with a fiberwise Z 2 -action away from the central fiber. It is a standard argument that this action extends to the whole S ′ (see for instance [MS21, Lemma 3.14]). Let S ′ → X ′ be the quotient by this action and let B ′ ⊆ X ′ be the branch locus. Then X ′ , 1 2 B ′ → ∆ ′ is also a family of stable pairs by the work of Alexeev-Pardini [AP12]. In particular, the central fiber S ′ 0 ⊆ S ′ is an appropriate double cover of X ′ 0 ⊆ X ′ . From this discussion it follows that the first goal is to compute the stable replacement of the central fiber of X, 1 2 B → ∆. Definition 3.13. Let X ′ → X be the weighted blow up of the central fiber X 0 ⊆ X at the point ξ = [1 : 0 : 0] with weights (p, q) according to the singularity type Σ (see Table 3). Denote by Y ⊆ X ′ the exceptional divisor of the blow up, which is isomorphic to P(1, p, q). Let Z be the strict transform of the central fiber X 0 ⊆ X, and let The first step is then to prove the following theorem.
Theorem 3.14. The central fiber where Y ∼ = P(1, p, q) and Z = Bl which is Cartier, and K X ′ 0 is also Cartier by applying the adjunction formula on the central fiber of the family X ′ 0 ⊆ X ′ (see [Cor92,Proposition 16.4]). Then, proving that X ′ 0 , 1 2 B ′ 0 is semi-log canonical boils down to show that Y, G + 1 2 B ′ | Y and Z, E + 1 2 B ′ | Z are log canonical. This is proved in Propositions 3.21 and 3.22. Finally, we check the ampleness of Z as given in Table 1. The gluing locus Y ∩ Z is isomorphic to P 1 . The components Y and Z have finite cyclic quotient singularities, and these are contained in the gluing locus. Moreover, the topological Euler characteristics of Y and Z are µ Σ + 3 and 36 − µ Σ respectively, where µ Σ is the Milnor number of the singularity Σ.
Proof. By the discussion at the beginning of § 3.2, the stable replacement S ′ 0 is obtained by taking an appropriate double cover of X ′ 0 . In § 3.5 and § 3.7 we describe the double covers Y → Y and Z → Z. These two surfaces are glued along G → G and E → E, which are isomorphic to P 1 as we prove in § 3.6. We prove that Y and Z have finite quotient singularities, which lie along the gluing locus, in Proposition 3.31. The claimed values of Z , and of the topological Euler characteristic of Z are computed in Proposition 3.27 and Corollary 3.30. The statement about the topological Euler characteristic of Y is proved in Corollary 3.26.
Remark 3.16. Consider S = S(t ⋆ u) → ∆ as in Definition 3.7 with u Σ-generic. One may want to describe the stable replacement directly on S and not passing through X, 1 2 B → ∆ as we described so far. This is done as follows. In the cases where d is even, it is sufficient to take the weighted blow up of S at (t, y, z, w) = (0, 0, 0, 0) with weights (1, p, q, d/2) in the affine patch x = 0. If d is odd, then we first have to perform the base change ∆ → ∆ such that s → t 2 obtaining a new family S → ∆, and then blow up S at (t, y, z, w) = (0, 0, 0, 0) with weights (1, 2p, 2q, d) in the affine patch x = 0.
Before we move on with the proofs of the above claims, we recall some preliminaries about weighted projective planes. A reference for the following well-known facts is [Has00, § 5.1]. Let a, b be two positive coprime integers. Consider the weighted projective plane P(1, a, b) with coordinates [x : Then we have the following linear equivalences: Moreover, abD x generates the Picard group of P(1, a, b), and the intersection numbers among D x , D y , D z are given by Remark 3.17. Let C ⊆ P(1, 1, 2) be an irreducible curve of weighted degree 10. As P(1, 1, 2) is Q-factorial and since 2D x generates Pic(P(1, 1, 2)), there exists a rational constant c such that C = cD x . Intersecting both sides with D z , we obtain that c = 10.
3.3. Proof of semi-log canonicity. Let us prove that the pair X ′ 0 , 1 2 B ′ 0 has semi-log canonical singularities. We already explained that Let us first focus on the former pair. We have that Y ∼ = P(1, p, q) has coordinates [t : α : β] and G = V (t). So, along G, Y has a 1 p (1, q) singularity at [0 : 1 : 0] and a 1 q (1, p) singularity at [0 : 0 : 1]. We now describe the curve B ′ | Y . For this we start with the equation of B ⊆ X.
Lemma 3.18. Let W Σ denote the vector space of homogeneous polynomials of degree d in P(1, p, q). Then the morphism is an isomorphism. Moreover, for generic ω ∈ W Σ there exists σ ∈ Aut(P(1, p, q)) such that Proof. In the following table we report for each singularity type the number of monomials of degree d in P(1, p, q). Sing.
17 18 19 15 16 17 16  17 This matches the number of degree 10 monomials x a y b z c with wt Σ (x a y b z c ) ≤ 0, which were listed in Proposition 3.2. So, to show the map in the statement is an isomorphism, it suffices to show it is surjective. An arbitrary degree d monomial in W Σ is in the form t a α b β c with a + pb + qc = d. This is the image of x 10−b−2c y b z c , which we need to show has non-positive weight. But this is true as wt Σ ( For the statement about the existence of σ ∈ Aut(P(1, p, q)), one simply checks case by case constructing an automorphism σ which makes the nonzero coefficients of θ(m 1 ) and θ(m 2 ) equal, which is the required condition to be in θ(U Σ,0 ⊕ U Σ,− ).
In the next lemma we discuss the intersection B ′ | Y ∩ G depending on the singularity Σ.
14 Lemma 3.19. The points in which B ′ | Y intersects with G are summarized in the table below.
The only solution to α q + β p = 0 and t = 0 in P(1, p, q) is either [0 : 1 : −1] or [0 : −1 : 1] according to the table above. By inspection Remark 3.20. Part of the information of Lemma 3.19 can be found in Table 4, where in each cell the triangle on the right represents Y, By Lemma 3.18 and by the genericity assumption on the curve T 0 (u), we have that the curve B ′ | Y is smooth away from G, which contains the singular points of P(1, p, q). So, we only have to check the log canonicity in a neighborhood of G. As illustrated in Lemma 3.19, G and B ′ | Y intersect transversely at a smooth point of Y , or at a singular torus fixed point of the toric variety Y . So, we have that Y, G + 1 2 B ′ | Y is log canonical by combining [Kol13, Theorem 3.32] with [CLS11, Proposition 11.4.24 (a)].
We now focus on the other pair.
Proposition 3.22. The pair Z, E + 1 2 B ′ | Z is log canonical. Proof. By the Σ-genericity assumption, the curve B ′ | Z is smooth away from E. So, we only have to check log canonicity of the pair in a neighborhood of the exceptional divisor E. The blow up Z = Bl (p,q) ξ P(1, 1, 2) may be singular along E at the torus fixed points t 1 , t 2 . These singularities are toric, and dictated by the weights p and q (see the next Remark 3.23). As we discussed in the previous section, depending on the singularity Σ, the curve B ′ | Z may pass through t 1 or t 2 . So we argue again by cases. These are summarized in Remark 3.23. Let us compute the singularities of Z along E. From a toric perspective, the cone in R 2 corresponding to ξ ∈ P(1, 1, 2) is C(ξ) = (1, 0), (0, 1) , where R ≥0 (1, 0) corresponds to a torus fixed fiber of P(1, 1, 2) and hence R ≥0 (0, 1) to a torus fixed section. By [Has00, Remark after Proposition 4.4] we know that C(ξ) is subdivided by the ray R ≥0 (q, p) by the weighted blow up (that is, the blow up of the ideal (y q , z p )). Therefore, let t 1 , t 2 be the torus fixed points of Z along E with associated cones (1, 0), (q, p) and (0, 1), (q, p) respectively. By [CLS11, Proposition 10.1.2], these give rise to cyclic quotient singularities of type 1 p (1, −q) and 1 q (1, −p) respectively.
To compute the intersection B ′ | Y · G, we can use the calculations carried out in the proof of Lemma 3.19 (these are visually summarized in Table 4). For instance, for Σ = E 12 , B ′ | Y · G = 1, hence c = 21. Repeating this for each singularity we obtain the following table: Sing.
E 12 E 13 E 14 Z 11 Z 12 Z 13 W 12 W 13 The inequality c 2 − p − q > 0 is then verified by the above table (see Table 3 for the values of p and q). In particular, so, in the affine patch {x = 0}, we assign weight p to y and q to z. For a divisor D in P(1, 1, 2), let D denote its strict transform. As Z is a toric surface, we have that the divisor In what follows, we compute these four intersection numbers. We Then, if σ : Z → X denotes the weighted blow up at ξ, the following equalities hold: Combining these equalities with We can then compute the following intersection numbers: where in the last equality we used that . The intersection with D x is 1 2 , independently of the singularity type. The intersections with D y , D z , E are also positive, as shown in the table below as the singularity type varies.
Sing. Sing. Z 12 Z 13 W 12 W 13 3.5. The double cover Y → Y . If d is even, we can form the double cover of P(1, p, q) with branch curve B ′ | Y = T 0 (u) as the hypersurface of degree d in P(1, p, q, d/2) given by w 2 = θ((π 0 + π − )(t ⋆ u)). If d is odd, we use the isomorphism P(1, p, q) ∼ = P(1, 2p, 2q) to construct the double cover as a hypersurface in P(1, 2p, 2q, d). This amounts to replacing t by t 2 and doubling the degrees of α and β. More geometrically, when d is odd, one is constructing the double cover of P(1, p, q) branched along T 0 (u) ∪ V (t). We summarize this information in the table below.
Sing. Proof. Let Y → Y be the minimal resolution of singularities of Y , which is a smooth K3 surface. Then, χ top ( Y ) equals 24 minus the number of exceptional P 1 in Y . [IF00, Table 1] reports the ADE singularities that Y has. From this, one obtains the claim after checking case by case. For instance, if Σ = E 12 , then Y has exactly three singular points, and these are A 1 , A 2 , A 6 singularities. So, χ top ( Y ) = 24 − 1 − 2 − 6 = 15 = µ E 12 + 3.
3.6. The gluing curve Y ∩ Z. To describe the curve along which the stable surfaces Y and Z are glued, we view G := Y ∩ Z ⊆ Y as the double cover of G ⊆ Y . The advantage is that for G we have an explicit equation in P(p, q, d/2) or P(2p, 2q, d), depending whether d is even or odd respectively. We will prove that G is isomorphic to P 1 . If the singularity type is E 12 , E 13 , Z 11 , or Z 12 (which correspond to d odd), then the isomorphism P(2p, 2q, d) ∼ = P(p, q, d) induces an isomorphism of the curve (3.4) with w = θ(π 0 (u)). This proves that G is in the branch locus of the cover Y → Y , as each point on it has only one preimage. In particular, G ∼ = P 1 .
We now analyze the case of E 14 , Z 13 , W 12 , W 13 (which correspond to d even). In these cases, the restriction to G of the projection P(p, q, d/2) P(p, q) such that [α : β : w] → [α : β] gives a 2 : 1 morphism branched at two distinct points (this can be checked inspecting the four cases). In conclusion, G is isomorphic to P 1 also if d is even. We illustrate this strategy with one of the cases, since the other ones are analogous. For E 14 and under the isomorphisms P(3, 8, 12) ∼ = P(3, 2, 3) ∼ = P (1, 2, 1), the curve G becomes identified with The restriction to C of the projection P(1, 2, 1) P(1, 2) such that [α : β : w] → [α : β] is 2 : 1 and it is branched at the points [0 : 1] and [1 : 3.7. The double cover Z → Z. We now study the geometry of the double cover Z → Z = Bl (p,q) ξ P (1, 1, 2). Proof. Recall from § 3.6 that Y ∩ Z is part of the ramification divisor if and only if d is odd. Therefore, we distinguish two cases. If d is even, the branch divisor of Z → Z equals B ′ | Z . Using the expressions for B ′ | Z , D y , D z computed in the proof of Proposition 3.25, we obtain that from which we obtain the claimed values of K 2 Z . To compute the cohomology of O Z we use . We have that h 1 (O Z ) = h 2 (O Z ) = 0 because Z is a rational surface with rational singularities. Hence, we obtain that where we used [CLS11, Proposition 9.1.6]. Alternatively, one can use the following Macaulay2 code: If d is odd, the branch divisor of Z → Z equals B ′ | Z + E instead. Hence, from which we obtain the remaining values of K 2 Z in the table. For the cohomology of O Z we use that by [CLS11, Proposition 9.1.6], or by the same Macaulay2 code as above with d replaced by d − 1.
Next, we compute the topological Euler characteristic of Z across the eight singularity types. Preliminarily, we find the Euler characteristic of the singular curve B 0 ⊆ P(1, 1, 2) in (3.2). To do this, we start by recalling the following geometric genus formula for plane curves (see [CAMMOG14]): Let D ⊆ P 2 be a smooth curve of degree d and C ⊆ P 2 be an integral curve of degree d with normalization π : C → C. Then, is the genus of the curve E; • 2δ p = µ p + |π −1 (p)| − 1; • µ p is the Milnor number of C at p. Let P ω = P(w 0 , w 1 , w 2 ) where gcd(w i , w j ) = 1 for i = j. Assume that P ω contains a smooth curve of D degree d. Then, by [CAMMOG14, Theorem 5.6], the geometric genus formula (3.5) holds for any integral curve C ⊆ P ω of degree d which does not pass through a singular point of P ω . Moreover, by [CAMMOG14, Corollary 5.4], in this case which simplifies to the genus-degree formula for curves in P 2 , but need not be an integer if there are no smooth curves in P ω .
This implies that By (3.5), we have that g( B 0 ) = 16 − δ ξ . Here we chose as D the Fermat curve of degree 10 in P (1, 1, 2), which is smooth of genus g(D) = 16, and we used that the generic curve B 0 is singular only at the point ξ = [1 : 0 : 0], which is not an orbifold point of P (1, 1, 2). By substituting this in (3.7) together with the equality 2δ p = µ p + |π −1 (p)| − 1, we obtain Remark 3.29. The sequence (3.6) is an exact sequence of mixed Hodge structure in which every term except H 1 (B 0 ) is pure of weight equal the cohomological degree. Consequently, Moreover, the normalization π : B 0 → B 0 in this case is given by the strict transform of B 0 relative to the weighted blow up Z → Bl (p,q) ξ P(1, 1, 2), and hence |π −1 (ξ)| is just the number of times the red curve intersects the exceptional divisor in Table 4.  1, 1, 2 where we used Lemma 3.28 for χ top (B 0 ). As Z is a weighted blow up with exceptional divisor E ∼ = P 1 of S 0 at a single point, using again the additivity of the topological Euler characteristic we obtain that χ top ( Z) = χ top (S 0 ) + 1 = 36 − µ Σ .
3.8. The singularities of Y and Z. We conclude the proof of Theorem 3.15 with the following proposition. Proof. Consider the curves G ⊆ Y and E ⊆ Z, which are double covers of G ⊆ Y and E ⊆ Z respectively. From the work we carried out so far, we know that the pairs ( Y , G) and ( Z, E) are log canonical. The singularities of Y and Z only occur along the gluing curves G and Z. As the pairs ( Y , G) and ( Z, E) are log canonical, we can apply [Ish00, Lemma 5.5] to conclude that the isolated singularities of Y and Z along the respective gluing loci are log terminal singularities. Furthermore, by [KSB88, § 4], we know these singularities are cyclic quotient ones.
3.9. Summary of the construction of the stable surface Y ∪ Z. The goal of this subsection is to summarize the construction of the limit surfaces Y ∪ Z described so far. Along the way, we use the case of W 12 as a guiding example. Let [x : y : z] be the coordinate of P (1, 1, 2). Let V 10 denote the complex vector space of degree 10 polynomials in x, y, z. Let M the basis of V 10 consisting of the possible degree 10 monomials. For each singularity type Σ considered above, let (p, q), d as in Table 3. For W 12 , we have (p, q) = (4, 5) and d = 20. Consider the weight function wt Σ (x a y b z c ) = pb + qc − d. Let m 1 , m 2 ∈ M be the only two monomials of weight 0. For W 12 these are x 5 y 5 , x 2 z 4 . Let U Σ denote the subspace of V 10 consisting of elements such that m 1 and m 2 have the same coefficient. Given a Σ-generic u ∈ U Σ in the sense of Definition 3.9, decompose it as u = π − (u) + π 0 (u) + π + (u), see Definition 3.3 -This notation is nothing more than the monomials of negative, zero, and positive degree with respect to the weight functions.
We consider the one-parameter family S(t ⋆ u) → ∆ (see Definitions 3.5 and 3.7), of which we want to compute the stable replacement of the central fiber, which is given by Y (u)∪ Z(u).
• If d is even consider P(1, p, q, d/2) with coordinate [t : α : β : w]. Let Y (u) is the hypersurface of degree d in P(1, p, q, d/2) given by the polynomial equation where θ was introduced in Definition 3.11. For an example in the case of W 12 , see Example 3.12. If d is odd consider instead P(1, 2p, 2q, d) again with coordinate [t : α : β : w]. Let Y (u) be the hypersurface of degree 2d in P(1, 2p, 2q, d) given by In either case, let G ⊆ Y (u) be the curve V (t) ∩ Y (u), which is isomorphic to P 1 as shown in § 3.6. • The surface Z(u) is the double cover of Bl (p,q) ξ P (1, 1, 2), where ξ = [1 : 0 : 0], with branch curve V ((π 0 + π + )(u)) if d is even, or V ((π 0 + π + )(u)) union the exceptional divisor of the blow up E ⊆ Bl The surfaces Y (u) and Z(u) are glued along the curves G ∼ = P 1 ∼ = E.
3.10. One-parameter degenerations over a DVR. Thus far, we have computed the stable replacement of the central fiber of the families S(t⋆u) → ∆ described in Definition 3.7. Such stable replacement can be understood as in Remark 3.16. Although this is not going to be used later in the paper, we point out that this actually extends to other more general families.
Suppose that R is a DVR with residue field C and D = Spec(R) with uniformizing parameter s. In this section, we explain how to modify our previous work to determine the KSBA stable replacement for one-parameter degenerations of Horikawa surfaces of type Σ over D. For simplicity of exposition, we focus on the case where ∆ ⊆ C is a disk and R = O p is the ring of germs of holomorphic functions at p ∈ ∆.
Given g ∈ O p , let k = ord(g) denote the order of vanishing of g at p and let τ (g) = g (k) (0)s k /k! be the truncation of g to its lowest order term. Analogously, the truncation of f ∈ V 10 ⊗O p is defined component by component relative to the basis M of degree 10 monomials in P (1, 1, 2 Definition 3.32. Given a singularity type Σ, we say that an element f ∈ U Σ ⊗ O p is Σ-generic if there exists u ∈ (U Σ ) reg which is Σ-generic and satisfying τ (f ) = s ⋆ u.
is a one-parameter degeneration of smooth Horikawa surfaces. It turns out that the stable replacement of the central fiber of S(f ) is computed in the same way as for the degenerations in Definition 3.7. Before proving this we first introduce the following notation. We denote by S(f ) the base change of S(f ) with respect to s → s 2 .
Proposition 3.33. Let f be Σ-generic. Consider the one-parameter families S(f ) and S(τ (f )). As constructed in Remark 3.16, consider the following modified families: • Assume d is even. Let S(f ) ′ and S(τ (f )) ′ be respectively the weighted blow up of S(f ) and S(τ (f )) with respect to the ideal (s d , y q , z p , w 2 ). • Assume d is odd. Let S(f ) ′ and S(τ (f )) ′ be respectively the weighted blow up of S(f ) and S(τ (f )) with respect to the ideal (s 2d , y 2q , z 2p , w 2 ). Proof. Let Z τ ∪ Y τ and Z ∪ Y be the central fibers of S(τ (f )) and S(f ) respectively, where Y τ , Y denote the exceptional divisors. We already know that Z τ ∪ Y τ is a stable surface by the discussion in § 3.2 (see in particular Remark 3.16). As V (lim s→0 τ (f )) = V (lim s→0 f ) =: C, then we have that Z τ and Z are isomorphic because they are both the double cover of Bl (p,q) ξ P(1, 1, 2) with branch divisor the strict transform of the curve C (union the exceptional divisor if d is odd). Let ][x, y, z] denote the map obtained by setting φ(s) = t, φ(x) = 1, φ(y) = u, and φ(z) = v (φ is the analogue of θ in Definition 3.11). We distinguish two cases. If d is even, then Y τ is given by V ((π 0 + π − )(φ(τ (f )))) ⊆ P(1, p, q, d/2). On the other hand, Y is given by the vanishing of the lower degree part of (π 0 + π − )(φ(f )), which is precisely (π 0 + π − )(φ(τ (f ))), so Y τ and Y coincide. If d is odd, then the argument is analogous to the previous one, with the difference that Y τ and Y are hypersurfaces in the weighted projective space P (1, 2p, 2q, d).

Dimension count of boundary strata
We now use the families in Definition 3.7 to define eight closed and irreducible subsets of the boundary of M, one for each singularity type Σ. The starting point is the construction of a family of degenerate stable surfaces over P(U Σ ) reg .
Definition 4.1. In a slight abuse of notation with our conventions so far, we let ∆ =
If C is the scheme associated to the ideal (t d , y q , z p ), then define F ′ := Bl C F, and let D ′ ⊆ F ′ be the strict transform of D. Then the fiber of F ′ , 1 2 D ′ → P(U Σ ) reg × ∆ over (u, 0) ∈ P(U Σ ) reg × ∆ is the gluing of Y, G + 1 2 B ′ | Y and Z, E + 1 2 B ′ | Z as in § 3.2. We have that K F ′ and D ′ are both Q-Cartier.
In particular, for N large enough and divisible by p and q across the eight singularity types, we have that N K F ′ + 1 2 D ′ is Cartier and it restricts to the fibers F ⊆ F ′ giving the Cartier divisor N K F + 1 2 D . Then F ′ , 1 2 D ′ → P(U) reg × ∆ is a family of KSBA stable pairs as in § 3.2. Both F ′ → P(U) reg × ∆ and D ′ → P(U) reg × ∆ are flat as they are dominant morphisms from integral schemes to normal schemes with reduced fibres of constant dimension [HKT09, Lemma 10.12]. In particular, F ′ , 1 2 D ′ → P(U) reg × ∆ is a well defined family of KSBA stable pairs for the Viehweg's moduli stack with N as above (see Definition 2.5). In particular, it induces a morphism f Σ : P(U) reg × ∆ → M to the KSBA compactification of the moduli space of Horikawa surfaces. Then, we define the boundary stratum D Σ ⊆ M as the Zariski closure with the reduced scheme structure of the image For the rest of this section, the goal is to prove the following result. Let Σ be one of the eight singularity types and denote by µ Σ its Milnor number. The proof of Theorem 4.2, which we are about to discuss, boils down to checking the following for each singularity Σ: • The dimension of the space of Y will be shown to be µ Σ − 2; • The dimension of the space of Z is 29 − µ Σ , where 29 is the rank of h 1,1 of a smooth Horikawa surface; • The deformations of Y and Z are independent. We first need some preliminaries.  (1, a, b))) = 4 + ⌊b/a⌋, dim(Aut (P(1, 1, a))) = 5 + a.
E 12 E 13 E 14 Z 11 Z 12 Z 13 W 12 W 13 dim(Γ Σ ) 2  2  2  3  3  3  3  3 Proof. In the current proof, we denote d in Table 3 by deg instead. Let us start by describing the automorphisms in Γ Σ . A generic automorphisms of P(1, 1, 2) has the following form: with a = 0. To preserve P(U Σ,+ ⊕ U Σ,0 ) reg , first we must have that for each monomial x i y j z k such that i + j + 2k = 10 and wt Σ (x i y j z k ) ≥ 0, the monomials appearing iñ c ℓmn x ℓ y m z n also have non-negative weight. With this we can explicitly describe ϕ: for each x i y j z k , let C ijk be the set of coefficients c ℓmn such that wt Σ (x ℓ y m z n ) < 0. Define I Σ to be the ideal generated by the sets C ijk for all i, j, k. The construction of I Σ and a primary decomposition for it can be automatized with a computer using the following SageMath code [Sag22]: (Here we use p = 3, q = 4, deg = 15 as an example, which correspond to Σ = Z 11 . These values can be changed according to Σ.) If J Σ denotes the radical of I Σ , we obtain that where recall a = 0. On the other hand, to be an element of Γ Σ we must also have that the weight zero monomials of the transformed polynomial have equal coefficients. This imposes one condition on the coefficients of ϕ. To understand this, it is enough to prove this for the associated Lie algebra. More precisely, the action of Γ Σ on P(U Σ,+ ⊕ U Σ,0 ) reg gives a representation of the corresponding Lie algebra γ Σ on the tangent space T u (P(U Σ,+ ⊕U Σ,0 ) reg ) for any u ∈ P(U Σ,+ ⊕ U Σ,0 ) reg . So, the Lie algebra γ Σ acts linearly on T u (P(U Σ, showing that we only need to impose one linear condition of the fact that these two coefficients are equal. These considerations together give the dimension count for Γ Σ in the statement. Proof of Theorem 4.2. As discussed in § 3.2, it will be equivalent to count the dimension of the space of isomorphism classes of stable pairs X ′ 0 , 1 2 B ′ 0 , which recall is the gluing of two pairs: Y, G + 1 2 B ′ | Y and Z, E + 1 2 B ′ | Z . The dimension of the space of pairs Y, G + 1 2 B ′ | Y is equal to the dimension of the projectivized vector space V p,q,d of degree d curves in P(1, p, q) with equal nonzero coefficient for θ(m 1 ) and θ(m 2 ) (see Lemma 3.18) minus the dimension of the subgroup G Σ ≤ Aut (P(1, p, q)) which preserves the equality of these two coefficients, and hence has codimension 1 in Aut (P(1, p, q)) (for the dimension of the latter see Lemma 4.3). Therefore, we obtain Sing.
E 12 E 13 E 14 Z 11 Z 12 Z 13 W 12 W 13 dim(P(V p,q,d )) − dim G Σ 10 11 12 9 10 11 10 11 The dimension of the space of pairs Z, E + 1 2 B ′ | Z is equal to the dimension of the projectivized vector space of coefficients U Σ,+ ⊕ U Σ,0 (for this dimension we refer to the tables in Proposition 3.2) minus the dimension of the group Γ Σ which was computed in Lemma 4.4. Therefore, we obtain Sing. 17 16 15 18 17 16 17  16 Finally, let us discuss the gluing of the two pairs along G and E. Let g p , g q ∈ G (resp. e p , e q ∈ E) be the torus fixed points with singularities of type 1 p (1, q), 1 q (1, p) (resp. 1 p (1, −q), 1 q (1, −p)). Denote by g b ∈ G (resp. e b ∈ E) the point in B ′ | Y ∩ G (resp. B ′ | Z ∩ E) different from g p , g q (resp. e p , e q ) (see Lemma 3.19). Then, the pointed curves (G; g p , g q , g b ) and (E; e p , e q , e b ) are identified via the unique isomorphism such that g p → e p , g q → e q , g b → e b . Having that the coefficients of the monomials m 1 , m 2 are equal, implies that we fix the points g b and e b . In particular, there is no moduli associated with the gluing.
Adding up the two contributions for each Σ, we obtain 27 as the dimension of the KSBA boundary stratum D Σ .
We have a natural action GL 2 W which is induced by linear change of coordinates in x and y. It turns out that the isomorphism classes of Horikawa surfaces coincide with the orbits of this GL 2 -action [Wen21, Lemma 7]. Therefore, we obtain the quotient described in Definition 2.3: Theorem 5.1. Let X be a double cover of P(1, 1, 2) with branch curve of degree 10. If X has isolated log canonical singularities or isolated singularities of type E 12 , E 13 , E 14 , Z 11 , Z 12 , Z 13 , W 12 , W 13 , then X is GIT stable.
Up to a change of coordinates, we can suppose that λ(t) = diag(t a , t −a ) with a > 0. The existence of the limits implies that q 4 (x, y), . . . , q 10 (x, y) can be written as follows: Let X h be the hypersurface in P(1, 1, 2, 5) given by The above surface has a singularity at [0 : 1 : 0 : 0]. So, if we consider the affine patch associated to y = 0, then the affine equation of our singularity w 2 = z 5 + x 2 h 2 (x, 1)z 3 + x 3 h 3 (x, 1)z 2 + x 4 h 4 (x, 1)z + x 5 h 5 (x, 1), can be written as where p 5 (x, z) is a homogeneous polynomial of degree 5. Therefore, by definition, we have that X h has either a N 16 singularity [Arn76, Page 13] or a degeneration of it. By using Arnold's work in [Arn76], we know that the Milnor number and the modality of the N 16 singularity are 16 and 3 respectively. As the Milnor number µ and the modality m are upper semicontinuous invariants, see [GLS07, §I.2.1], any non-stable surface with isolated singularities must have a singularity that satisfies µ ≥ 16 and m ≥ 3. On the other hand, the classification in [Arn76] and [LR12] imply that the isolated log canonical singularities and the eight singularity types in the statement satisfy the inequalities µ ≤ 14 and m ≤ 1. Therefore, they are stable.

5.2.
Extending the morphism from KSBA to GIT. To prove this we need a preliminary lemma, which is a slight generalization of [AET23, Lemma 3.18] (see also [GG14,Theorem 7.3]).
Lemma 5.3. Let X and Y be proper varieties with X normal. Let ϕ : X Y be a rational map which is regular on an open dense subset U ⊆ X. Let (C, 0) be a regular curve and f : C → X a morphism whose image meets U. Let g f : C → Y be the unique extension of ϕ • f , which exists by the properness of Y .
Let V ⊆ X be another dense open subset containing U. Assume that for all f with the same f (0) ∈ V , there are only finitely many possibilities for g f (0). Then ϕ can be extended uniquely to a regular morphism V → Y .
Proof. Following the proof of [AET23, Lemma 3.18], let Z ⊆ X × Y be the closure of the graph of U → Y . By hypothesis the proper birational morphism Z → X is finite on V , so the base change Z V := Z × X V → V is also proper, finite, and birational. As V is normal, Z V → V is an isomorphism by the Zariski Main Theorem, hence we obtain the claimed extension V → Y by composing V → Z V with the restriction to Z V of Z → Y . Then there is only one possibility for g f (0), which parametrizes the following GIT stable orbit. The point x parametrizes a pair X ′ 0 , 1 2 B ′ 0 given by the gluing of Y, G + 1 2 B ′ | Y and Z, E + 1 2 B ′ | Z . Recall that Z is the weighted blow up of P(1, 1, 2) at the point ξ = [1 : 0 : 0], 28 and under this blow up the curve B ′ | Z is mapped to a curve in P(1, 1, 2) with equation given by (π + (u) + π 0 (u))(x, y, z) = 0, where u ∈ P(U Σ ) reg (see § 3.1). This has a unique singularity of type Σ at [1 : 0 : 0], which we know is GIT stable by Theorem 5.1. In other words, g f (0) can be uniquely reconstructed from Z, E + 1 2 B ′ | Z , which only depends of the point x an not from the choice of f : (C, 0) → (M, x). Since M is normal (see Definition 2.6), we are done by Lemma 5.3. 6. Limit mixed Hodge structure of the degenerations Next, we study the behavior of the Hodge structure associated with our stable surfaces. Let f : X → ∆ be a semistable degeneration with central fiber X 0 = f −1 (0). Let X η = f −1 (η) be a generic fiber of f and H k lim (X η , Q) denote the Q-limit mixed Hodge structure of R k f * (Q), i.e. the underlying Q-vector space is H k (X η , Q), but the Hodge and weight filtrations arise from the asymptotic behavior of the period map. See [Mor84,PS08] for an introduction.
(b) The central fiber S 0 = π −1 (0) is the union of two irreducible components Y and Z, each of which has h 2 (O) = 1 and at worst rational singularities. Then, the local system V Q = R 2 π * (Q) over ∆ * has finite monodromy.
Proof. Let us consider a semistable degeneration where ∆ → ∆ is a morphism of the form t → t n for some n ≥ 1 and the central fiber S 0 is reduced and simple normal crossing. In particular, in order to prove that V Q has finite local monodromy it is sufficient to prove that the corresponding local system attached to S → ∆ has trivial local monodromy operator T . Let be the decomposition into irreducible components of the central fiber of S → ∆. Given η ∈ ∆ \ {0}, by [Mor84, Page 118] we have that and equality holds if and only if N = log(T ) = 0. So let us prove that equality holds. By the semistable reduction process, we have that the surfaces Y and Z are birational to S 0j and S 0k for some distinct j, k ∈ {1, . . . , n}. Since Y and Z have rational singularities, 29 we can conclude that p g ( S 0j ) = h 2 (O Y ) and p g ( S 0k ) = h 2 (O Z ), which are both equal to 1 by hypothesis. Thus, and hence equality holds in (6.2).
In particular, this theorem applies to the one-parameter stable degeneration of Horikawa surfaces whose central fiber S 0 is in the form Y ∪ Z as described in § 3.2. In this case, • The generic surface is a smooth Horikawa surface which has p g = 2; • The surface Y is an ADE K3 surface by Proposition 3.27; • The surface Z has only finite cyclic quotient singularities by Proposition 3.31 (hence rational singularities by [KM98,Proposition 5.15]) and h 2 (O Z ) = 1 by Proposition 3.27.
Looking ahead to Theorem 6.10, we note that the mixed Hodge structures on H 2 ( Y , Q) and H 2 ( Z, Q) are pure of weight 2. This is a well-known result in the case of ADE K3 surfaces. On the other hand, since Z has only finite cyclic quotient singularities, it is a Kähler V-manifold, and hence H 2 ( Z, Q) admits a pure Hodge structure of weight 2.
To continue, we recall the following result of Griffiths. Proof. The proof of Theorem 6.1 shows that both S and S have finite local monodromy. Hence, apply Theorem 6.2. In the case of S, the local monodromy T = e N = id and hence the limit mixed Hodge structure is pure by Theorem 6.16 of [Sch73].
Returning to the first paragraph of this section, let f : X → ∆ be a semistable degeneration with central fiber X 0 and X η be a generic fiber of defined via the inclusion X η ֒→ X and the retraction X → X 0 .
Theorem 6.4 (Clemens-Schmid Sequence, [Mor84,PS08,dCM14]). Let f : X → ∆ be a semistable degeneration and d = dim C (X ). Then, is an exact sequence of mixed Hodge structures (after appropriate Tate twists), where T = e N denotes the local monodromy of R k f * (Q).
We now specialize the previous theorem to the case where d = 3, k = 2 and N = 0 on H 2 lim (X η , Q). Let Q(ℓ) denote the pure Q-Hodge structure of type (−ℓ, −ℓ) of rank 1 with Q-structure (2πi) ℓ Q ⊆ C. Then, → 0 is an exact sequence. The local system R 0 f * (Q) over ∆ * is the constant variation of Hodge structure Q(0), and hence the previous sequence becomes Adding the correct Tate twists [Mor84, Page 108], the sequence becomes: To simplify the previous sequence, we note that since we are considering a degeneration of surfaces, it follows that (see [Mor84,Page 117]) for some integer r ≥ 0. Therefore, combining (6.3) with (6.4), we obtain an exact sequence of pure Hodge structures of weight 2 To show that this sequence splits, we recall the following.
Theorem 6.5 ( [Del71,Del74]). The mixed Hodge structure on the rational cohomology of a complex algebraic variety is graded-polarizable.
Accordingly, after selecting a choice of polarization of Gr W 2 H 2 (X 0 , Q) we obtain a direct sum decomposition (6.6) Definition 6.6. Let A be a Q-Hodge structure of weight 2 with F 3 A = 0. Then, the transcendental part of A, denoted by T [A], is the smallest Q-sub-Hodge structure of A such that Lemma 6.7. Suppose that A and B are pure Q-Hodge structures of weight 2 such that Proof. The Hodge filtration is an exact functor from the category of mixed Hodge structures to the category of C-vector spaces. In particular, By applying Lemma 6.7 to (6.6) we obtain the following result.
Corollary 6.8. In the setting of Equation (6.6), As a prelude to the next result, we recall that if S = S 1 ∪ S 2 is the union of non-singular projective surfaces intersecting transversely then there exists a Mayer-Vietoris sequence With the exception of H 2 (S ), all of the terms in this sequence carry pure Hodge structures of weight equal to the cohomological degree. Moreover, all of these maps are morphisms of mixed Hodge structure. Therefore, . Equivalently, after extending the definition of the Néron-Severi group additively across disjoint unions, the previous equation becomes . More generally, we have the following.
Lemma 6.9. Let S be a projective surface which has only simple normal crossing singularities. Let S = ∪ i S i denote the decomposition of S into irreducible components. Then, Then, via the theory of semisimplical varieties (see [Car85,§ 11]), where the map δ * is constructed from an alternating sum of pullbacks along the inclusion maps S i ∩ S j ֒→ S i . In analogy with our previous discussion of the case where S had only two irreducible components, Theorem 6.10. Let f : X → ∆ be a semistable degeneration of projective surfaces with trivial local monodromy (as in the paragraph above (6.5)). Let X 0 = ∪ j D j be the decomposition of the central fiber X 0 into irreducible components. Then, Proof. The first isomorphism is Corollary 6.8. The second isomorphism is Lemma 6.9.
Corollary 6.11. Let π : S → ∆ be as in Theorem 6.1 and S → ∆ be the corresponding semistable degeneration (6.1). Then, Proof. By Theorem 6.10, the left hand side is equal to the sum of the transcendental parts of the irreducible components of the central fiber S 0 of S → ∆. If D is an irreducible component of S 0 with geometric genus zero then T [H 2 (D, Q)] = 0. By the proof of Theorem 6.1, this is true for every irreducible component of S 0 except for the two corresponding to Z and Y . Since the transcendental part of H 2 of a surface is a birational invariant, the result follows.
Remark 6.12. In [KLS21], the authors consider various generalizations of the Clemens-Schmid sequence using the decomposition theorem. Of particular relevance to the class of degenerations considered in this paper is Corollary 9.9 (i), which asserts the following: Let f : X → ∆ be a flat projective family with X − X 0 smooth. Assume that X 0 is reduced with semi-log canonical singularities and X is normal and Q-Gorenstein. Then, One consequence of this result is the equality of the Hodge-Deligne numbers h k (X 0 ) p,q = h k lim (X t ) p,q for pq = 0. 7. Birational type of limit surfaces In this section, we show that the minimal model of a generic surface S 0 in the sense of Definition 3.9 of type Z 11 , Z 12 , Z 13 , W 12 , W 13 is a K3 surface. In fact, our method constructs the minimal model as the minimal resolution of a double sextic. Our techniques do not apply to the E 12 , E 13 , E 14 cases because the method does not produce such plane curve.
Proposition 7.1. Let Σ ∈ {Z 11 , Z 12 , Z 13 , W 12 , W 13 } and let S 0 be the central fiber of the one-parameter degeneration S = S(t ⋆ u) → ∆ as in Definition 3.7 with u ∈ (U Σ ) reg Σgeneric. In particular, S 0 is the double cover of P(1, 1, 2) with branch curve B 0 in (3.2). Then, S 0 is birational to a K3 surface with ADE singularities which is the double cover of P 2 branched along a plane sextic V (H Σ ). Furthermore, a plane sextic C is projectively equivalent to V (H Σ ) if and only if (i) Σ = Z 11 , C is smooth, and there exists a line L such that L ∩ C is the union of three points of multiplicities three, two, and one. (ii) Σ = W 12 , C is smooth, and there exists a line L such that L ∩ C is the union of two points of multiplicities two and four. (iii) Σ = W 13 , C has an A 1 singularity at a point p, and there exists a line L such that L ∩ C is the union of p with multiplicity four and another double point. (iv) Σ = Z 12 , C has an A 1 singularity at a point p, and there exists a line L such that L ∩ C is the union of p with multiplicity three, and other two points with multiplicity two and one. (v) Σ = Z 13 , C has an A 2 singularity at a point p, and there exists a line L such that L ∩ C is the union of p with multiplicity three, and other two points with multiplicity two and one. Moreover, let L(Σ) ⊆ P(H 0 (P 2 , O(6))) be the locus parametrizing plane sextics that are projecively equivalent to V (H Σ ). Then, it holds that 33 Sing.
The transformed polynomial defines a plane sextic curve (this is also verified by inspection using Proposition 3.2), and the surface S 0 is birational to the double cover of P 2 branched along V (H Σ ). (As remarked at the beginning of § 7, this construction does not give a plane sextic for the E series because the monomial x 4 z 3 has weight zero and is transformed to µ(x 4 z 3 ) · x 1 = x 4 0 x −1 1 x 3 2 .) To describe S 0 , we study the singularities of this plane sextic as follows. We choose a specialization F Σ of the polynomial H Σ . If the curve V (F Σ ) is smooth, then the general V (H Σ ) is also smooth. Or, if the particular curve V (F Σ ) has exactly one isolated A n singularity at p and the general V (H Σ ) also has one A n singularity at p, then the singularity of V (H Σ ) is unique as well. The reason is that any other singularity of V (H Σ ) would degenerate to p along with the A n one. Therefore, the Milnor number of the singularity at p of the degeneration has to be strictly larger than n. This is in contradiction with the characterization of V (F Σ ), by considering the sum of the Milnor numbers of the general curve V (H Σ ) and the fact that the Milnor number is upper semicontinuous, see [GLS07, Theorem 2.6].
The special polynomials F Σ are provided below, together with the Macaulay2 code [GS] that computes their singularities (we use the packages [LK, Sta] The chosen special polynomials F Σ we listed satisfy the condition that the two monomials m 1 , m 2 of weight zero (see Proposition 3.2) have equal non-zero coefficient. We have that V (H Σ ) is smooth for Σ = Z 11 , W 12 and it has exactly one singular point if Σ = Z 12 , Z 13 , W 13 . More specifically, we prove that V (H Z 12 ), V (H Z 13 ), V (H W 13 ) have an A 1 , A 2 , A 1 singularity at p respectively. This will be done on the way as we prove the claimed geometric characterization of the plane sextics. We now prove the geometric characterization of the plane sextic V (H Σ ) given in the statement. In what follows, for a positive integer d, p d and q d denote homogeneous polynomials of degree d.
First, observe that H Z 11 in (7.1) satisfies the claimed condition with respect to the line L = V (x 1 ). For the converse, suppose that C and L are as in part (i). Let us write the homogeneous polynomial of degree 6 describing C as Up to projectivity, we can suppose that L = V (x 1 ), and that the intersection points C ∩ L with multiplicities two and three are at [0 : 0 : 1] and [1 : 0 : 0] respectively. In this case, q 6 (x 0 , x 2 ) = x 2 0 x 3 2 q 1 (x 0 , x 2 ), which shows C is described by the vanishing of an equation in the form (7.1).
In the other direction, by using the action of SL 3 we can suppose that the line L is V (x 1 ), that the intersection points C ∩ L are [1 : 0 : 0] and [0 : 0 : 1] with multiplicities four and two respectively, so that the singular point p ∈ C is supported at [1 : 0 : 0]. Any degree 6 polynomial can be written as x 1 p 5 (x 0 , x 1 , x 2 ) + p 6 (x 0 , x 2 ).
Alternatively, fixed the plane sextic, the line L is determined up to a finite choice, we choose two points on this line, and then subtract the dimension of the subgroup of automorphisms of P 2 that preserve the two points. An analogous argument gives the dimension of L(Σ) for Σ = Z 12 , Z 13 , W 12 , W 13 .
Remark 7.2. Let M be a complex projective surface. Then, by [Shi08, Lemma 3.1], the transcendental lattice of M is a birational invariant of M. This implies that the transcendental part T [H 2 ( S 0 , Q)] considered in § 6 depends only on the birational type the component surfaces Z and Y . Let S 0 be the central fiber of S → ∆ as in the statement of Proposition 7.1. We note that S 0 is a simply connected variety [Dim92, (B21) Corollary]. As an application of the fact that S 0 is birational to a K3 surface we show that Z is also simply connected.
Proof. For a fixed Σ as in our hypothesis, Proposition 7.1 the surface Z is birational to a K3 surface P with canonical singularities. Therefore, we have a smooth surface W such that Z ← W → P . The surfaces P and Z have log terminal singularities (P has ADE singularities and for Z see the discussion in the proof of Proposition 3.31), and W is smooth. Then, their fundamental groups are isomorphic by [Tak03, Theorem 1.1]. Simply connectedness implies H 1 ( Z) = 0, and since our surfaces have cyclic quotient singularities, Serre duality applies and H 3 ( Z) = 0 by [PS08, Corollary 2.48]. By Corollary 3.30, we have χ top ( Z) = 36 − µ Σ and p g = 1 by Proposition 3.27. Therefore, we obtain h 1,1 ( Z) = 32 − µ Σ .