2. Periods for Looijenga pairs

For a Looijenga pair $(Y,D)$ , a blowup at a point in $D$ is a toric blowup if the blowup point is a node. Otherwise, it is called a non-toric blowup. Many constructions we consider are not affected by toric blowups. For convenience, we will use the following proposition.

Fix a Looijenga pair $(Y,D)$ . Let $\textrm {Pic}^{0}(D)$ be the connected component of the identity of $\textrm {Pic}(D)$ . For each $i$ , let $D_i^{0 }$ be the intersection of $D_i$ with the smooth locus of $D$ . We now cover the notion of periods for $(Y,D)$ . Let us first recall the following widely known identification $\textrm {Pic}^0(D)\simeq \mathbb {G}_m$ determined by an orientation of $D$ .

Proof. Let $\nu :\tilde {D}\rightarrow D$ be the normalization map and $\tilde {D}_i$ the irreducible component of $D$ that is mapped to $D_i$ . The choice of an orientation of $D$ determines a generator $\gamma$ of $H_1(D, \mathbb {Z})$ . For each $i$ , let $0_i$ and $\infty _i$ be points on $\tilde {D}_i$ such that $\gamma |_{D_i}$ is oriented from $\nu (0_i)$ to $\nu (\infty _i)$ . In particular, since the ordering of $D_1, \ldots , D_n$ is compatible with $\gamma$ , $\nu (0_i) \in D_i \cap D_{i-1}$ and $\nu (\infty _i) \in D_i \cap D_{i+1}$ . Let $x_i = \nu (0_i)$ for each $i$ .

We have the short exact sequence

\begin{equation*} 1 \rightarrow \mathcal {O}^{*}_{D} \rightarrow \nu _{*}{\mathcal {O}^{*}_{\tilde {D}}} \rightarrow \prod _i \mathcal {O}^{*}_{x_i} \simeq \mathbb {G}^{n}_{m} \rightarrow 1, \end{equation*}

which induces the following long exact sequence in cohomology

\begin{equation*} 1 \rightarrow \mathbb {G}_m\rightarrow \mathbb {G}_{m}^{n} \rightarrow \mathbb {G}_{m}^{n} \rightarrow \textrm {Pic} (D) \rightarrow \textrm {Pic} (\tilde {D}) \rightarrow 0. \end{equation*}

Here, the second last arrow in the long exact sequence is the multi-degree map given by $L \mapsto (\deg (L|_{D_1}),\ldots , \deg (L|_{D_n}))$ . In particular, we have the exact sequence

\begin{equation*} 1 \rightarrow \mathbb {G}_m \stackrel {\delta _1}{\rightarrow } \mathbb {G}_{m}^{n} \stackrel {\delta _2}{\rightarrow } \mathbb {G}_{m}^{n} \stackrel {\delta _3} {\rightarrow } \textrm {Pic}^{0}(D) \rightarrow 0. \end{equation*}

The map $\delta _1$ is given by $t \mapsto (t,\ldots ,t)$ and the map $\delta _2$ is given by $(t_1, \ldots , t_n)\mapsto (t_n^{-1}t_1, \ldots ,$ $t_{n-1}^{-1}t_n)$ .

Given $L\in \textrm {Pic}^0(D)$ , let $L_i=L|_{\tilde {D}_i}$ . Then, we can recover $L$ from the descent data $\phi _i:L_{i-1} |_{\infty _{i-1}} \stackrel {\sim }{\rightarrow } L_i|_{0_i}$ . By choosing a non-vanishing section $\sigma _i \in \Gamma (\tilde {D_i},L_i)$ and therefore an explicit trivialization $L_i \stackrel {\sim }{\rightarrow } {\mathcal {O}}_{\tilde {D}_i}$ , we get an explicit identification of the set of descent data with $\mathbb {G}_m^{n}$ by identifying $\phi _i$ with $\sigma _i(0_i)/ \sigma _{i-1}(\infty _{i-1}) \in \mathbb {G}_m$ . Under this identification, the isomorphism class of $L$ is determined by $\prod _i \sigma _i(0_i)/ \sigma _{i-1}(\infty _{i-1})$ , which is independent of the choice of $\sigma _i$ ’s; moreover, the image of $\delta _2$ , viewed as inside the set of descent data, is identified with $\ker \delta _3$ . Thus, we get a canonical isomorphism $\psi : \textrm {Pic}^{0}(D) \stackrel {\sim }{\rightarrow } \mathbb {G}_m$ given the choice of an orientation of $D$ such that via $\psi$ , $\delta _3$ is explicitly given by $(\mu _1, \ldots , \mu _n) \mapsto \mu _1 \cdots \mu _n$ .

Consider $L=\mathcal {O}(p-q)$ where $p$ and $q$ are on $D_i^{0 }$ . Let $z:D_i^{0 } \stackrel {\sim }{\rightarrow } \mathbb {G}_m$ be a coordinate so that $z(0_i)=0$ . Following the notation above, trivialize $L_i$ by $\sigma _i = {(z-z(q))}/{(z-z(p))}$ and, for $j \neq i$ , use $\sigma _j =1$ to trivialize $L_j$ . Then, we see that $\mu _i={z(q)}/{z(p)}$ and $\mu _j=1$ for $j\ne i$ . Thus, $\psi (\mathcal {O}(p-q)) = {z(q)}/{z(p)}$ .

Definition 2.6. For a Looijenga pair $(Y,D)$ , define

\begin{equation*} D^{\perp }:=\{\alpha \in \textrm {Pic}(Y):\alpha \cdot [D_i]=0 \quad \text{for all } i\}.\end{equation*}

Define $D^{\perp }_{\textrm { eff}} \subset D^{\perp }$ to be the sublattice generated by classes that can be realized by effective curves on a pair deformation equivalent to $(Y,D)$ .

We have an exact sequence

\begin{equation*} 0 \rightarrow \mathbb {Z} \rightarrow H_2(U, \mathbb {Z}) \rightarrow D^{\perp } \rightarrow 0. \end{equation*}

The kernel of this exact sequence is generated by the homology class $\Gamma$ of a real 2-torus in $U$ . Let $\omega$ be the canonical holomorphic volume (unique up to scaling) form of $U$ as in Definition 2.1 In this paper, following the notation in the proof of Lemma 2.5, we always assume that we normalize $\omega$ so that it has residue $1$ at $0_i \in D_i$ for each $i$ . In particular,

\begin{equation*} \int _{\Gamma } \omega = (2\pi i )^{2}. \end{equation*}

Definition 2.7. Let $T_{(D^{\perp })^*}={\textrm {Hom}}(D^{\perp },\mathbb {G}_m)$ . The period point $\varphi _Y\in T_{(D^{\perp })^*}$ of $(Y,D)$ is the homomorphism given by

\begin{equation*} \varphi _Y:D^{\perp }\rightarrow \textrm {Pic}^0(D)=\mathbb {G}_m, \hspace {.5cm}L\mapsto L|_D.\end{equation*}

In justification of calling these homomorphisms periods, the mixed Hodge structure of $H^2(Y\setminus D)$ is classified by the period (see [Reference FriedmanFri16]). Briefly, given $\gamma \in D^{\perp }$ , $\gamma$ can be represented by a class $\Sigma =\sum _k\pm [C_k]$ where $[C_k]$ are smooth curves in $Y$ meeting $D$ transversally at distinct points of the smooth locus of $D$ . A cycle, $\Sigma '$ , that is contained in $U$ and homologous to $\Sigma$ , can be formed and used to define the period map $\varphi _Y':D^{\perp }\rightarrow \mathbb {G}_m$ by

\begin{equation*} \gamma \mapsto \exp \left (-\frac {1}{2\pi i}\int _{\Sigma '}\omega \right ).\end{equation*}

The mixed Hodge structures on $H^2(U)$ are indeed classified by such integrals. Then, [Reference FriedmanFri16, Proposition 3.12] shows that these integrals are equal to the periods as defined in Definition 2.7.

We will use an extended notion of the period point, which is that of a marked period point.

By [Reference Gross, Hacking and KeelGHK15b, Proposition 3.4], we have $\Phi _Y =W_Y \cdot \Delta _Y$ .

Proof. Recall that $\mathrm {Adm}$ is invariant under toric blowups. So we can assume that $(Y,D)$ has a toric model $p: (Y,D) \rightarrow (\bar {Y},\bar {D})$ . We use classifications of positive Looijenga pairs given by [Reference MandelMan19, Theorems 4.4 and 4.5]. It follows from these theorems that $D^{\perp }_{\textrm { eff}}$ has the following three possibilities:

1. (i) $D^{\perp }_{\textrm { eff}} = D^{\perp } = D_n (n \geqslant 4)$ ;

2. (ii) $D^{\perp }_{\textrm { eff}} = D^{\perp } = E_n (n=6,7,8)$ ; and

3. (iii) $D^{\perp }_{\textrm { eff}} = A_{d_1-1}\oplus \cdots \oplus A_{d_m-1}$ .

In the first two cases, by [Reference MandelMan19, Theorem 4.4], we can assume $(Y,D)$ has a toric model of $\mathbb {P}^2$ together with its toric boundary. Then, that $\mathrm {Adm}_{Y}=W$ follows from [Reference Gross, Hacking and KeelGHK15b, Lemma 5.4].

In the third case, let $\bar {D}_{i_1}, \ldots , \bar {D}_{i_k}$ be the toric boundary divisors where we perform non-toric blowups. Let $N_{\bar {Y}}$ be the cocharacter lattice of the Zariski-open dense torus orbit of $\bar {Y}$ , and $\nu _i$ be the primitive generator of the ray in the toric fan of $\bar {Y}$ corresponding to $\bar {D}_i$ . By [Reference MandelMan19, Theorem 4.5], we can choose $p$ so that the rays generated by $\nu _{i_1}, \ldots , \nu _{i_k}$ are contained in one half of $N_{\bar {Y},\mathbb {R}}$ , and $d_{k} (1\leqslant k \leqslant m)$ is just the number of non-toric blowups on $\bar {D}_{i_k}$ in our toric model $p$ . Let $\{E_{i_k \,j}\}$ be an exceptional configuration of $p$ . By [Reference Gross, Hacking and KeelGHK15b, Theorem 4.7], $\mathrm {Adm}_Y$ acts simply transitively on the set of exceptional configurations of the same combinatorial type as $\{E_{i_k\,j}\}$ . First, we show that the only interior irreducible curves contained in $Y$ that intersect only $D_{i_1}, \ldots , D_{i_k}$ are the irreducible exceptional curves of the toric model $p$ . Then, given an element $g$ in $\mathrm {Adm}_Y$ , for each $k$ , $g$ must permute the set of exceptional curves in the exceptional configuration that intersects $D_{i_k}$ , and therefore $\mathrm {Adm}_Y = W$ . We use the same argument as used in the proof of [Reference MandelMan19, Theorem 4.5]. Indeed, let $C$ be such an interior curve in $Y$ . Let $\bar {C}$ be its image under $p$ . Then, by standard toric geometry, $\sum (\bar {C} \cdot \bar {D}_i) \nu _i =0$ . For $i$ not equal to one of $i_1, \ldots , i_k$ ,

\begin{equation*} \bar {C} \cdot \bar {D}_i = p^{*}(\bar {C}) \cdot p^{*}(\bar {D}_i) = C \cdot D_i =0. \end{equation*}

Since $\sum (\bar {C} \cdot \bar {D}_i) \nu _i =0$ and $\nu _{i_1}, \ldots , \nu _{i_k}$ are contained in one half of $N_{\bar {Y},\mathbb {R}} \simeq \mathbb {R}^2$ , this is possible only if $\bar {C} \cdot \bar {D_i}=0$ for all $i$ . Thus, $C$ must be supported on one of the exceptional curves of our toric model $p$ .

The marked period can now be defined as follows.

Definition 2.15. Given a tuple $((Y,D),p_i,\mu )$ , the marked period point is the homomorphism $\varphi _{((Y,D),p_i,\mu )}\in {\textrm {Hom}}(\textrm {Pic}(Y_0),\textrm {Pic}^0(D))$ given by

\begin{equation*} \varphi (L):=(\mu ^{-1}(L)|_D)^{-1}\otimes {\mathcal {O}}_D\left (\sum (L\cdot D_i)p_i\right ).\end{equation*}

In [Reference Gross, Hacking and KeelGHK15b, Theorem 1.8], a global Torelli theorem for Looijenga pairs is given. We will use the following consequence of the theorem.

Fix a Looijenga pair $(Y,D)$ . The notion of marked Looijenga pairs extends to families and gives rise to a marked period map. Meaning, for a family of Looijenga pairs

\begin{equation*} ((\mathcal {Z},\mathcal {D}),p_i,\mu )\rightarrow S, \end{equation*}

where for the fiber $\mathcal {Z}_s$ over $s$ , $\mu _s$ is the marking of $\textrm {Pic}(\mathcal {Z}_s)$ by $\textrm {Pic}(Y)$ and $p_{i}(s)$ the marking of $\mathcal {D}_{i,s}$ , there is a marked period map

\begin{equation*} \Phi :S\rightarrow {\textrm {Hom}}(\textrm {Pic}(Y),\textrm {Pic}^0(D)), \end{equation*}

where $s\in S$ gets mapped to the homomorphism

\begin{equation*} L\mapsto (\mu ^{-1}_s(L)|_D)^{-1}\otimes {\mathcal {O}}\left (\sum (L\cdot D_i)p_i(s)\right ).\end{equation*}

The corresponding integral $\int _{\tilde {\beta }} \omega$ can be computed. Using the standard Stokes’ theorem and the fact that the restriction of $\omega$ to $E_{ij}\setminus D$ is zero (since it is holomorphic on $U$ ), it follows that

(2.1) \begin{equation} \exp \bigg (-\frac {1}{2\pi i}\int _{\tilde {\beta }} \omega \bigg )= \exp \bigg(-\int _{v_{ij}}^{p_i} \frac {dz}{z} \bigg) = \frac {z(v_{ij})}{z(p_i)}, \end{equation}

where $z: D_i^{0 } \xrightarrow {\sim } \mathbb {C}^{*}$ is a coordinate we choose as in Lemma 2.5.

Lemma 2.19. We have

(2.2) \begin{equation} \exp \left (-\frac {1}{2\pi i}\int _{\tilde {\beta }}\omega \right )=\varphi _{((Y,D),p_i,\mu )}({\mathcal {O}}_{Y}(E_{ij})), \end{equation}

where we assume that $\mu :\textrm {Pic}(Y)\rightarrow \textrm {Pic}(Y)$ is the identity.

Proof. Let $z: D_i^{0 } \xrightarrow {\sim } \mathbb {C}^{*}$ be a coordinate as in Lemma 2.5. By Equation (2.1), the left-hand side of equality (2.2) is equal to ${z(v_{ij})}/{z(p_i)}$ . The marked period for $E_{ij}$ is given by

\begin{align*} \varphi _{(Y,D,p_i,\mu )}({\mathcal {O}}_{Y}(E_{ij}))&=({\mathcal {O}}_{Y}(E_{ij}))|_D^{-1}\otimes \mathcal {O}_Y\bigg(\sum _k (E_{ij}\cdot D_k) p_k\bigg)\\ &=\mathcal {O}(-v_{ij})\otimes \mathcal {O}(p_i)\\ &=\mathcal {O}(p_i-v_{ij})\\ &=\frac {z(v_{ij})}{z(p_i)}, \end{align*}

where the last equality comes from the canonical identification $\textrm {Pic}^{0 }(D)=\mathbb {G}_m$ in Lemma 2.5.

We now construct universal families as in [Reference Gross, Hacking and KeelGHK15b]. Let $(Y,D)$ be a Looijenga pair, with $\pi :(Y,D)\rightarrow (\bar {Y},\bar {D})$ a toric model with an exceptional configuration $E_{ij}$ . Denote $T_{Y}^{*}:={\textrm {Hom}}(\textrm {Pic}(Y),\mathbb {G}_m)$ . Then there are sections (as in [Reference Gross, Hacking and KeelGHK15b, Proposition 2.9])

\begin{equation*} p_i:T_{Y}^{*}\rightarrow T_{Y}^{*} \times D_i^{0 }, \end{equation*}

\begin{equation*} q_{ij}:T_{Y}^{*}\rightarrow T_{Y}^{*}\times D_i^{0 }, \end{equation*}

such that for $\varphi \in T_{Y}^{*}$ , we have

\begin{equation*} \varphi (E_{ij})={\mathcal {O}}_D(q_{ij}(\varphi ))^{-1}\otimes {\mathcal {O}}_D(p_i(\varphi ))\in \textrm {Pic}^0(D).\end{equation*}

Let $\Pi :(\mathcal {Y}_{\left \{ {E_{ij}}\right \}},\mathcal {D})\rightarrow T_{Y}^{*}\times \bar {Y}$ be the blowup along the sections $\left \{q_{ij}\right \}$ . This family then comes with a marking of the Picard group by $\textrm {Pic}(Y)$ on each fiber. Then

\begin{equation*} \lambda :((\mathcal {Y}_{\left \{E_{ij} \right \}},\mathcal {D}),p_i,\mu )\rightarrow T_{Y}^{*}, \end{equation*}

is what [Reference Gross, Hacking and KeelGHK15b] calls a universal family. Note that these universal families are not unique since there is a choice of order of blowup and changing the order will give rise to a non-isomorphic family over the torus, though they will be birational to any other universal family. To avoid confusion, we call these families in this paper by the alternative term, almost-universal families.

2.1 Generalized marked pairs

The fibers of the mirror family are not necessarily smooth, and this section presents adjustments to the definition of a Looijenga pair in a way that allows for the singularities that appear. The construction here is similar to the situation for the moduli space of polarized K3 surfaces with simple surface singularities (see [Reference MorrisonMor83]).

Assume that the Looijenga pair $(Y,D)$ is positive. As in [Reference Gross, Hacking and KeelGHK15a, Lemma 6.9], being positive is equivalent to the existence of $c_i$ such that the divisor $-(K_Y+\sum c_iD_i)$ is big and nef. It follows that for $m\in \mathbb {N}$ sufficiently large, the linear system $|\!-\!m(K_Y+\sum c_iD_i)|$ defines a birational morphism with the exceptional locus given by all curves disjoint from $D$ (which is a configuration of $(-2)$ -curves). After contracting the internal curves, we get a pair $(Y',D)$ where $Y'$ has du Val singularities and the push forward of $-(K_Y+\sum c_iD_i)$ is supported on $D$ and is ample. Motivated by this, we now consider generalized pairs $(Y,D)$ as in Definition 1.2.

Recall now Definitions 1.5 and 1.7. Suppose $(Y,D)$ is a generalized pair such that $Y$ has a minimal resolution given by $X\rightarrow Y$ and a marking of $\textrm {Pic}$ given by $\mu$ . Note that when $X$ has no internal $(-2)$ curves, $\mu$ gives a marking of all of $\textrm {Pic}(X)$ so the regular definition of a marked pair is recovered.

Definition 2.20. An isomorphism of generalized marked pairs $((Y,D),p_i,\mu )$ and $((Y',D'),$ $p'_i,\mu ')$ is given by an isomorphism $f:Y\rightarrow Y'$ that lifts to an isomorphism $\tilde {f}:X\rightarrow X'$ between corresponding minimal resolutions such that $f(D_i)=D'_i$ , $f(p_i)=f(p'_i)$ , and $\tilde {f}$ induces an isomorphism of lattices $\tilde {f}^*:I(Y')\rightarrow I(Y)$ such that $\tilde {f}^*=\mu ^{-1}\circ \mu '$ .

Definition 2.21. The marked period point of a generalized marked pair $((Y,D),p_i,\mu )$ with minimal resolution $X\rightarrow Y$ , is given by

\begin{equation*} \varphi _{((Y,D),p_i,\mu )}:=\varphi _{((X,D),p_i,\tilde {\mu })},\end{equation*}

where the right-hand side is as defined in Definition 2.15.

Lemma 2.22. The marked period point in Definition 2.21 of a generalized marked pair $((Y,D),p_i,\mu )$ is well defined.

Proof. Let $(X,D) \rightarrow (Y,D)$ be the minimal resolution of $(Y,D)$ . In particular, $(X,D)$ is a positive Looijenga pair. Let $W_X$ , $W$ , $\Delta _X$ , $\Phi _X$ , and $\Phi$ be as in Definition 2.11. Let $W(Y)$ be the Weyl group of $(Y,D)$ as in Definition 1.5. By [Reference Gross, Hacking and KeelGHK15b, Proposition 3.4], $\Phi _X = R(Y)$ and $W(Y)=W_X$ .

Let $\tilde {\mu }: \textrm {Pic}(X) \rightarrow \textrm {Pic}(Y_0)$ and $\tilde {\mu }': \textrm {Pic}(X) \rightarrow \textrm {Pic}(Y_0)$ be two markings of $\textrm {Pic}(X)$ extending $\mu : I(Y) \hookrightarrow \textrm {Pic}(Y_0)$ . By Lemma 2.14, there exists a unique $w \in W$ such that $w\circ \tilde {\mu } = \tilde {\mu }'$ . We want to show that $w \in W_X$ . Then, the statement of the lemma will follow from the fact that roots in $\Phi _X$ all have marked periods equal to $1$ . Let $D_{\textrm {eff}, \mathbb {R}}^{\perp } = D_{\textrm { eff}}^{\perp }\otimes \mathbb {R}$ and $I(Y)_{{\mathbb {R}}} = I(Y)\otimes {\mathbb {R}}$ . The three possibilities for $D_{\textrm { eff}}^{\perp }$ are as described in the proof of Lemma 2.14. Observe that

\begin{equation*} D_{\textrm {eff},\mathbb {R}}^{\perp } \cap I(Y)_{{\mathbb {R}}} = \bigcap _{\alpha \in \Delta _X} ( \alpha ^{\perp } \cap D_{\textrm { eff},\mathbb {R}}^{\perp }). \end{equation*}

Suppose $w=s_{\beta }$ for a root $\beta \in \Phi \setminus \Phi _X$ . Then $\beta ^{\perp } \cap D_{\textrm {eff}, \mathbb {R}}^{\perp }$ is not contained in $D_{\textrm {eff},\mathbb {R}}^{\perp } \cap I(Y)_{{\mathbb {R}}}$ . In particular, there exists an element $x$ in $D_{\textrm {eff}}^{\perp } \cap I(Y)$ such that $x$ is not fixed under the reflection $s_{\beta }$ , which is a contradiction that $w$ fixes elements in $I(Y)$ pointwise. Hence, the assumption is false and $w \in W_X$ .

Theorem 2.23 (Global Torelli theorem for generalized marked pairs). Two generalized marked pairs, $((Y_1,D_1),p_1,\mu _1)$ and $((Y_2,D_2),p_2,\mu _2)$ , are isomorphic if and only if $\varphi _{((Y_1,D_1),p_1,\mu _1)}=\varphi _{((Y_2,D_2),p_2,\mu _2)}$ .

Proof. Let $f_i: X_i\rightarrow Y_i$ be the minimal resolution of $Y_i$ . If the two generalized marked pairs are isomorphic, then it is clear that the marked periods must coincide.

Suppose the two generalized marked pairs have the same period. Consider the induced isometry on $I(Y_i)^{\perp }$ which gives an isomorphism of root systems. Let $\Phi$ be the set of roots of $\textrm {Pic}(Y_0)$ . Let $\Phi ' = \{ \delta \mid \delta \in \Phi ,\, \varphi (\delta ) =1\}$ . By [Reference Gross, Hacking and KeelGHK15b, Proposition 3.4], $\Phi '$ is contained in $\tilde {\mu }_i(R({Y_i}))$ . By Lemma 2.15 and [Reference Gross, Hacking and KeelGHK15b, Theorem 3.2], each $ \tilde {\mu }_i(\textrm {Nef}(X_i))$ is the closure of a connected component of

\begin{equation*} C_{D}^{++} \setminus \bigcup _{\alpha \in \Phi '}{\alpha }^{\perp }, \end{equation*}

and the action of the Weyl group $W(\Phi ')$ on the set of connected components of $C_{D}^{++} \setminus \bigcup _{\alpha \in \Phi '}{\alpha }^{\perp }$ is transitive. Thus, there must exist $w\in W (\Phi ')$ such that $w \circ \tilde {\mu }_1(\textrm {Nef}(X_1))=\tilde {\mu }_2(\textrm {Nef}(X_2))$ . Composing an isometry with $w$ is still an isometry, and $w$ does not affect the isometry on the level of $I(Y_i)=R(Y_i)^{\perp }$ . Thus, replacing $\tilde {\mu }_1$ with $w\circ \tilde {\mu }_1$ keeps the marked period points for $((X_i,D_i),p_i,\tilde {\mu }_i)\ (i=1,2)$ the same but their Nef cones are identified. By [Reference Gross, Hacking and KeelGHK15b, Theorem 1.8], they are isomorphic marked tuples. Thus, the same is true for $((Y_i,D_i),p_i,\mu _i)$ $(i=1,2)$ .

Lemma 2.24. There are no non-trivial automorphisms of generalized marked pairs $((Y,D),p_i,\mu )$ .

Proof. Let $((Y,D),p_i,\mu )$ be a generalized marked pair with minimal resolution $((X,D),p_i,\tilde {\mu })$ . Suppose $f\in {\textrm {Aut}}(Y,D)$ fixes $D$ pointwise, and $f^*$ induces the identity on $I(Y)$ which extends to an isometry on $\textrm {Pic}(X)$ . Consider a lift to an automorphism of $(X,D)$ given by $\tilde {f}$ . Let $N'$ be the cokernel of the map

\begin{equation*}\textrm {Pic}(Y) \rightarrow \mathbb {Z}^n,\quad L \mapsto (L\cdot D_i)_{1 \leqslant i \leqslant n}.\end{equation*}

Consider the following exact sequence

\begin{equation*} 1\rightarrow {\textrm {Hom}}(N', \mathbb {G}_m)\rightarrow {\textrm {Aut}}(X,D)\rightarrow H_{X}/W_{X}\rightarrow 1,\end{equation*}

where $H_X\subset {\textrm {Aut}}(\textrm {Pic}(X))$ is the subgroup that fixes the period point $\varphi _X$ . If $\tilde {f}$ maps to an element in $H_X$ that is not contained in $W_X$ , then it will not fix $R(Y)$ and therefore $I(Y)$ , in contradiction with $I(Y)$ being fixed. Therefore, the automorphism $\tilde {f}$ must lie in ${\textrm {Hom}}(N', \mathbb {G}_m)$ . By [Reference Gross, Hacking and KeelGHK15b, Proposition 2.6], ${\textrm {Hom}}(N', \mathbb {G}_m)$ is identified with

\begin{equation*}K:= \textrm {ker}[{\textrm {Aut}}(X,D)\rightarrow {\textrm {Aut}}(\textrm {Pic}(X))], \end{equation*}

and the natural map $K\rightarrow {\textrm {Aut}}^0(D)$ given by restriction is an injection. Then for $D$ to be fixed pointwise, $f$ must be the identity.

We can also make sense of families of generalized marked pairs. To make this precise, we first quote a result from [Reference MorrisonMor83]. First recall Definition 1.5 for $I(Y)$ .

Lemma 2.25 [Reference MorrisonMor83, Corollary 3]. If $p:\mathcal {X}\rightarrow S$ is a proper family of surfaces with only rational double points as singularities, then there is a sheaf $I(\mathcal {X}/S)$ such that for each $s\in S$ , with $U$ a sufficiently small neighborhood of $s\in S$ , and $X_s$ the fiber of $s$ , it follows that $\Gamma (U,I(\mathcal {X}/S))=I(X_s)$ .

Now fix a generic Looijenga pair $(Y_0,D)$ . We want to show that $T_{Y_0}^{*}={\textrm {Hom}}(\textrm {Pic}(Y_0),\mathbb {G}_m)$ represents a fine moduli space. We will work in the analytic category and the stack we define will be over the category of analytic spaces. The stack we consider is similar to the moduli stack of marked Looijenga pairs considered in [Reference Gross, Hacking and KeelGHK15b, § 6].

Let $\mathcal {M}_{Y_0}$ be the moduli stack of families of generalized pairs $(Y,D)$ together with a marking of $D$ and a marking of $\textrm {Pic}(Y)$ by $\textrm {Pic}(Y_0)$ that allow simultaneous resolutions. More precisely, for an analytic space $S$ , the objects of the category $\mathcal {M}_{Y_0}(S)$ are a morphism $f:(\mathcal {Y},\mathcal {D})\rightarrow S$ together with sections $p_i:S\rightarrow \mathcal {D}_i$ and injections $\mu :I(\mathcal {Y}/S)\rightarrow {\underline {\textrm {Pic}(Y_0)}}_S$ where $I(\mathcal {Y}/S)$ is as in Lemma 2.25 such that:

1. (I) $\mathcal {Y}/S$ is a flat family of surfaces and it admits a simultaneous resolution;

2. (II) $\mathcal {D}_i$ is a Cartier divisor on $\mathcal {Y}/S$ for each $i$ ;

3. (III) each closed fiber $(\mathcal {Y}_s,\mathcal {D}_s,p_i(s),\mu |_s)$ is a generalized marked pair with the marking of the boundary given by $p_i(s)$ and the marking of $\textrm {Pic}$ by $\mu |_s$ ; and

4. (IV) in the case $n=1,2$ , we assume that we are given an orientation of $\mathcal {D}$ , that is, an identification $\mathbb {Z} \times S \xrightarrow {\sim } R^{1}f_{*}\mathbb {Z}_{\mathcal {D}}$ .

Notice that the sections $p_i$ and the orientation $\mathcal {D}$ determine a canonical isomorphism $D\times S \xrightarrow {\sim } \mathcal {D}$ . This is used to define the marked period mapping for families over an arbitrary base $S$ . Then given a family of generalized marked pairs, $((\mathcal {Y},\mathcal {D}),p_i,\mu )\rightarrow S$ , we have an induced marked period map to ${\textrm {Hom}}(\textrm {Pic}(Y_0),\mathbb {G}_m)$ .

Proposition 2.26. The stack $M_{Y_0}$ is represented by $T_{Y_0}^{*} = \textrm{Hom}(\textrm{Pic}(Y_0),\mathbb {G}_m)$ .

Proof. Since objects of $\mathcal {M}_{Y_0}$ have no non-trivial automorphisms by Proposition 2.24, the stack $\mathcal {M}_{Y_0}$ is represented by an analytic space. We have the morphism

\begin{equation*} \Psi : \mathcal {M}_{Y_0} \rightarrow T_{Y_0}, \end{equation*}

which is defined as follows. Given $((\mathcal {Y,D})/S,\mu , {p_i})$ , we have the associated marked period mapping $\phi : S \rightarrow T_{Y_0}$ as in Definition 2.21.

We want to show that $\Psi$ is an isomorphism. That the period mapping $\Psi$ is étale follows from [Reference LooijengaLoo81, II 2.5] and [Reference MorrisonMor83, § 3, Corollary 1]. That $\Psi$ is injective on points follows from Theorem 2.23. To show that $\Psi$ is surjective, consider an almost-universal family $\mathcal {Y}\rightarrow T_{Y_0}^{*}$ of Looijenga pairs $(Y,D)$ with a marking of the Picard group by $\textrm {Pic}(Y_0)$ as well as a marking of the boundary $D$ . There is a morphism $(\mathcal {Y},\mathcal {D}) \rightarrow (\check {\mathcal {Y}},\mathcal {D})$ contracting all the internal $(-2)$ -curves on each fiber in $(\mathcal {Y},\mathcal {D})$ . This gives rise to a family $((\check {\mathcal {Y}},\mathcal {D}),p_i)\rightarrow T_{Y_0}^{*}$ , where the fibers $\check {\mathcal {Y}}_s$ have, at worst, du Val singularities. Furthermore, the original marking by $\textrm {Pic}(Y_0)$ contains the data for a marking on the generalized family and the boundary marking descends as well. Thus, we get a family of generalized marked pairs $((\check {\mathcal {Y}},\mathcal {D}),p_i,\mu )\rightarrow T_{Y_0}^{*}$ . Furthermore, by construction, the period of the fiber over $\varphi \in T_{Y_0}^{*}$ is $\varphi$ . Note that the choice of the almost-universal family does not affect the construction. Thus $\Psi$ is an isomorphism as we have claimed.

3. Review of mirror symmetry for log Calabi–Yau surfaces

We review notions and results related to the construction of the mirror family in [Reference Gross, Hacking and KeelGHK15a]. Let $(Y,D)$ be a positive Looijenga pair and denote $U=\,Y\setminus D$ . To each pair, we will associate to it an integral affine manifold with singularities.

The spaces we consider do not have a global integral affine structure, so we consider integral affine manifolds with singularities. These are real manifolds $B$ such that there is an open subset $B_0\subset B$ that is an integral affine manifold such that the singular locus of $B$ (where the affine structure is not defined) given by $B\setminus B_0$ is a locally finite union of closed submanifolds of codimension at least two. There is a local system $\Lambda$ on $B_0$ given by flat integral vector fields. Given any simply connected subset $\tau \subset B_0$ , we write $\Lambda _{\tau }$ for the stalk of $\Lambda$ at any point of $\tau$ (any two such stalks are canonically identified via parallel transport) and let $\Lambda _{\tau , \mathbb {R}} = \Lambda _{\tau }\otimes _{\mathbb {Z}} \mathbb {R}$ . We also use $\Lambda _{\tau }$ to denote the local system of flat integral vector fields on $\tau$ since what we mean will be clear from the context. We denote by $\check {\Lambda }$ the local system dual to $\Lambda$ .

To a fixed Looijenga pair $(Y,D)$ , we will associate an integral affine manifold, $B_{(Y,D)}$ . Fix a lattice $M\simeq \mathbb {Z}^2$ . Recall $M_{{\mathbb {R}}}=M\otimes _{\mathbb {Z}}{\mathbb {R}}$ . Let $N = \mathrm {Hom}_{\mathbb {Z}} (M, \mathbb {Z})$ be the dual lattice of $M$ . Let $D= D_1+\cdots +D_n$ . For each node $p_{i,i+1}=D_i\cap D_{i+1}$ , we take a rank $2$ lattice $M_{i,i+1}$ with a basis $v_i,v_{i+1}$ and the cone $\sigma _{i,i+1}\subset {M_{i,i+1}}\otimes _{\mathbb {Z}}{{\mathbb {R}}}$ generated by the two basis vectors. We then form $B_{(Y,D)}$ by gluing $\sigma _{i,i+1}$ to $\sigma _{i-1,i}$ along the ray $\rho _i := \mathbb {R}_{\geqslant 0} v_i$ . Note that $B_{(Y,D)}$ is homeomorphic to $\mathbb {R}^2$ . Now we define the integral affine structure on $B_{(Y,D)}\setminus \{0\}$ . Let $U_i$ be the interior of $\sigma _{i,i+1}\cup \sigma _{i-1,i}$ . Then, we define a chart $\psi _i:U_i\rightarrow M_{{\mathbb {R}}} \simeq {\mathbb {R}}^2$ given by

\begin{equation*} \psi _i(v_{i-1})=(1,0), \quad \psi _i(v_i)=(0,1), \quad \psi _i(v_{i+1})=(-1,-D_i^2).\end{equation*}

Therefore, $B_{(Y,D)}$ is actually an integral linear manifold with a singularity at $0$ . Note that $B_{(Y,D)}$ also has a fan structure $\Sigma$ given by

\begin{equation*}\Sigma = \left \{ \sigma _{i,i+1} \mid 1 \leqslant i \leqslant n\right \} \cup \left \{ \rho _i \mid 1 \leqslant i \leqslant n \right \} \cup \left \{0 \right \}.\end{equation*}

When $(Y,D)$ is clear from context, we will often drop the subscript and just denote the associated manifold $B$ instead of $B_{(Y,D)}$ . Note that if the pair $(Y,D)$ is toric, the affine structure extends over the origin (see [Reference Gross, Hacking and KeelGHK15a, Lemma 1.8]).

Denote by $B(\mathbb {Z})$ the set of integer points on $B$ . Then, we also have the identification $B(\mathbb {Z}) \simeq U^{\mathrm {trop}}(\mathbb {Z})$ . However, we will not use this perspective until § 7.

Let $(Y,D)$ be positive, $P={\textrm {NE}}(Y)$ and $R=\mathbb {C}[P]$ . Let $P^{\textrm {gp}}$ be the group completion of $P$ and $P^{\textrm {gp}}_{{\mathbb {R}}} = P^{\textrm {gp}}\otimes _{\mathbb {Z}}{\mathbb {R}}$ .

Notions from [Reference Gross, Hacking and KeelGHK15a, § 2.1] will be used and we recall them here.

Definition 3.3. For each cone $\tau \in \Sigma$ and each cone $\sigma \in \Sigma$ containing $\tau$ , there is a canonical inclusion $\sigma \subset \Lambda _{\tau , \mathbb {R}}$ . We define the localized fan $\tau ^{-1}\Sigma$ in $\Lambda _{\tau , \mathbb {R}}$ as the collection of cones

\begin{equation*}\tau ^{-1}\Sigma = \left \{\sigma + \mathbb {R}\tau \subset \Lambda _{\tau , \mathbb {R}} \mid \sigma \in \Sigma , \tau \subset \sigma \right \}.\end{equation*}

Definition 3.4. Let $\sigma _{\pm }$ be the two maximal cones of $\Sigma$ contained in $U_i$ . A $\Sigma$ -piecewise linear function on $U_i$ with values in $P_{{\mathbb {R}}}^{\textrm {gp}}$ is a continuous function $\varphi : U_i \rightarrow P_{{\mathbb {R}}}^{\textrm {gp}}$ such that $\varphi \vert _{\sigma _{\pm }}$ is given by an element $\varphi _{\pm }$ in $\mathrm {Hom}_{\mathbb {Z}} (M, P^{\textrm {gp}}) = N \otimes _{Z} P^{\textrm {gp}}$ . Write

\begin{equation*} \varphi _{+} - \varphi _{-} = \eta _{\rho _i} \otimes \kappa _{\rho _i, \varphi }, \end{equation*}

where $\eta _{\rho _i} \in N$ is the unique primitive element annihilating $\rho _i$ and positive on $\sigma _{+}$ and $\kappa _{\rho _i, \varphi } \in P^{\textrm {gp}}$ . We call $\kappa _{\rho _i, \varphi }$ the bending parameter of $\varphi$ . Notice that $\kappa _{\rho _i, \varphi }$ does not depend on the ordering of $\sigma _+, \sigma _-$ .

From a $\Sigma$ -piecewise linear multivalued function, $\varphi =\{\varphi _i\}$ , we can form a linear locally trivial principal $P^{\textrm {gp}}_{{\mathbb {R}}}$ -bundle $\pi :\mathbb {P}_0\rightarrow B_0$ together with a section $\varphi :B_0\rightarrow \mathbb {P}_0$ . This is constructed by gluing $U_i\times P_{{\mathbb {R}}}^{\textrm {gp}}$ to $U_{i+1}\times P_{{\mathbb {R}}}^{\textrm {gp}}$ along $U_i\cap U_{i+1}\times P_{{\mathbb {R}}}^{\textrm {gp}}$ by

\begin{equation*} (x,p)\mapsto (x,p+\varphi _{i+1}(x)-\varphi _i(x)).\end{equation*}

In particular, $\mathbb {P}_0$ is an integral linear manifold and $\pi$ is linear. The section $\varphi$ is formed by local sections $x\mapsto (x,\varphi _i(x))$ . Let $\Lambda _{\mathbb {P}_0}$ be the sheaf of integral tangent vectors on $\mathbb {P}_0$ . We define a local system $\mathcal {P}$ on $B_0$ by

\begin{equation*} \mathcal {P}:=\pi _{*}\Lambda _{\mathbb {P}_0}, \end{equation*}

which fits into the short exact sequence of local systems given by

(3.1) \begin{align} 0\rightarrow \underline {P}^{\textrm {gp}}\rightarrow \mathcal {P}\xrightarrow {r} \Lambda \rightarrow 0, \end{align}

where $r$ is the derivative of $\pi$ .

There is a unique (up to a linear function) $\Sigma$ -piecewise linear multivalued function

\begin{equation*} \varphi : |\Sigma |\rightarrow P_{{\mathbb {R}}}^{\text {gp}}, \end{equation*}

where the bending parameter (as defined in [Reference Gross, Hacking and KeelGHK15a, Definition 1.11]) at $\rho _i$ , denoted by $\kappa _{\rho _i}$ , is given by $[D_{\rho _i}]$ . This $\Sigma$ -piecewise linear multivalued function is what we use throughout this paper to define $\mathcal P$ .

Given $\tau \in \Sigma$ with $\dim \tau \gt 0$ , write $\mathcal {P}_{\tau }$ for the stalk of $\mathcal {P}$ at any point of $\tau$ . Then, $\varphi$ determines a $\tau ^{-1}\Sigma$ -piecewise linear section $\varphi _{\tau }: \Lambda _{\tau , {\mathbb {R}}} \rightarrow \mathcal {P}_{\tau , {\mathbb {R}}}:= \mathcal {P}_{\tau } \otimes _{\mathbb {Z}}{\mathbb {R}}$ of the projection $\mathcal {P}_{\tau , {\mathbb {R}}} \rightarrow \Lambda _{\tau , {\mathbb {R}}}$ .

A scattering diagram on $B$ is the data of a collection of rays with rational slopes called walls on $B$ together with formal power series satisfying certain conditions attached to these walls. See [Reference Gross, Hacking and KeelGHK15a, § 2.2] for details. Below, we recall the definition of the canonical scattering diagram in [Reference Gross, Hacking and KeelGHK15a].

Fix a ray $\mathfrak {d}\subset B$ with a rational slope with the origin at its endpoint. If $\mathfrak {d}$ is not a ray of $\Sigma$ , then refine $\Sigma$ by adding $\mathfrak {d}$ and also other rays such that each maximal cell of $\Sigma '$ is affine isomorphic to the first quadrant in $\mathbb {R}^2$ . This corresponds to a toric blowup $\pi : (\tilde {Y}, \tilde {D})\rightarrow (Y,D)$ . Let $C$ be the irreducible component of $\tilde {D}$ corresponding to $\mathfrak {d}$ . Let $\tau _{\mathfrak {d}} \in \Sigma$ be the smallest cone containing $\sigma$ and $m_{\mathfrak {d}}$ the primitive generator of the tangent space to $\mathfrak {d}$ , pointing away from the origin. Then define

\begin{equation*} f_{\mathfrak {d}}:=\exp \bigg( \sum _{\beta } k_{\beta }N_{\beta }z^{\pi _{*}(\beta )-\varphi _{\tau _{\mathfrak {d}}}(k_{\beta }m_{\mathfrak {d}})}\bigg), \end{equation*}

where the sum is made over $\beta \in A_1(\tilde {Y})$ such that

\begin{equation*} \beta \cdot \tilde {D}_i= \begin{cases} k_{\beta } \quad \text{for }\tilde {D}_i=C,\\ 0 \quad \text{for }\tilde {D}_i\ne C, \end{cases} \end{equation*}

for some $k_{\beta }\gt 0$ . Here, the $N_{\beta }$ are the relative Gromov–Witten invariants counting stable relative maps of genus zero curves representing the class $\beta$ with tangency of order $k_{\beta }$ at an unspecified point of $C^{0 }$ , the complement of nodes in $C$ . See [Reference Li and RuanLR01, Reference Ionel and ParkerIP03] for the original symplectic definitions of these relative Gromov–Witten invariants and [Reference LiLi01, Reference LiLi02] for the algebraic definitions. Recently, the general main result of [Reference Gross and SiebertGS22] implied that $N_{\beta }$ can also be defined using punctured log Gromov–Witten invariants as developed in [Reference Abramovich, Chen, Gross and SiebertACGS20].

A class $\beta \in A_{1}(\tilde {Y})$ is called an $\mathbb {A}^1$ -class if $N_{\beta }\ne 0$ .

The canonical scattering diagram is given by

\begin{equation*} \mathfrak {D}^{\text {can}}:=\{ (\mathfrak {d},f_{\mathfrak {d}}):\mathfrak {d}\subset B, \text { a ray of rational slope}\}.\end{equation*}

We call a pair $(\mathfrak {d}, f_{\mathfrak {d}})$ in $\mathfrak {D}^{\text {can}}$ a scattering ray.

Definition 3.7. For a broken line $\gamma$ with endpoint $Q \in \tau \in \Sigma$ , define $s(\gamma ) \in \Lambda _{\tau }$ , $c(\gamma )\in \mathbb {C}[P]$ by demanding that

\begin{equation*} \mathrm {Mono}(\gamma )= c(\gamma ) \cdot z^{\varphi _{\tau }(s(\gamma ))}.\end{equation*}

Here, $\mathrm {Mono}(\gamma )$ denotes the monomial attached to the last domain of linearity of $\gamma$ .

Write $\mathrm {Limits}(\gamma )=(q,Q)$ if $\gamma$ is a broken line for $q$ with endpoint $Q \in B_0$ . We say $Q$ is general if the ray starting at $0$ and passing through $Q$ has irrational slope.

Theorem 3.8 [Reference Gross, Hacking and KeelGHK15a, Theorem 2.34 and Corollary 6.11]. Let $(Y,D)$ be positive. Consider the free $R$ -module

\begin{equation*}A= \bigoplus _{q\in B(\mathbb {Z})} R\cdot \vartheta _q.\end{equation*}

For $p_1,p_2,q \in B(\mathbb {Z})$ and $z \in B_0$ a general point sufficiently close to $q$ , there can only be finite many pairs of broken lines $(\gamma _1,\gamma _2)$ such that $\textrm {Limits}(\gamma _i)=(p_i,z)$ and $s(\gamma _1)+s(\gamma _2)=q$ . Define

\begin{equation*} \alpha _z(p_1,p_2,q)=\sum _{\substack {(\gamma _1, \gamma _2)\\ \mathrm {Limits}(\gamma _i) = (p_i, z) \\ s(\gamma _1)+s(\gamma _2)=q }} c(\gamma _1)c(\gamma _2).\end{equation*}

In particular, $\alpha _z(p_1,p_2,q)$ is independent of the choice of $z$ near $q$ and we define

\begin{equation*}\alpha (p_1, p_2, q) := \alpha _z(p_1,p_2,q) \end{equation*}

for $z$ chosen sufficiently close to $q$ . Then we can define a multiplication rule on $A$ given by

\begin{equation*} \vartheta _{p_1}\vartheta _{p_2}=\sum _{q\in B(\mathbb {Z})} \alpha (p_1,p_2,q)\vartheta _r.\end{equation*}

This gives a commutative and associative $R$ -algebra structure on $A$ . The induced map

\begin{equation*} f:\textrm {Spec}(A)\rightarrow \textrm {Spec}(R) \end{equation*}

is a flat affine family of Gorenstein semi-log canonical surfaces with smooth generic fiber.

For the rest of the paper, we assume that the pair $(Y,D)$ is positive. As mentioned before, this means that the intersection matrix for $D$ is not negative semi-definite. In [Reference Gross, Hacking and KeelGHK15a, Lemma 6.9], it is stated that the following are equivalent.

1. (i) The pair $(Y,D)$ is positive.

2. (ii) The surface $Y\setminus D$ is the minimal resolution of an affine surface with, at worst, du Val singularities.

3. (iii) There exist positive integers $a_1,\ldots ,a_n$ such that for all $j$ , $\left (\sum a_i D_i\right )\cdot D_j\gt 0$ .

There are several important implications of this assumption, also from the same lemma. These include the following.

1. (i) The cone ${\textrm {NE}}(Y)_{{\mathbb {R}}}$ is rational polyhedral, generated by only finitely many classes of rational curves. Furthermore, every nef line bundle on $Y$ is semi-ample.

2. (ii) The subgroup $G\subset {\textrm {Aut}}(\textrm {Pic}(Y))$ that fixes $[D_i]$ is finite.

3. (iii) The union $R\subset Y$ of all curves disjoint from $D$ is contractible.

The key takeaway is that the mirror construction gives an actual algebraic family over $\textrm {Spec} (\mathbb {C}[{\textrm {NE}}(Y)])$ with smooth generic fibers.

Definition 3.10. Two types of weights will come into play associated with the action of $T^{D}$ on the canonical algebra generated by theta functions. One comes from the map $w:A_1(Y)\rightarrow \chi (T^D)$ given by

\begin{equation*} C\mapsto \sum _i (C\cdot D_i) e_{D_i}\end{equation*}

and the other comes from $w:B\rightarrow \chi (T^D)\otimes {\mathbb {R}}$ , where $w$ is the unique piecewise linear map with $w(0)=0$ and $w(v_i)=e_{D_i}$ for $v_i$ the primitive generator of $\rho _{i}$ . Together, we get a weight map

\begin{align*} w: B(\mathbb {Z}) \times \mathrm {NE}(Y) & \rightarrow \chi (T^{D}) \\ (q,C) & \mapsto w(q) + w(C). \end{align*}

The following theorem is a special case of [Reference Gross, Hacking and KeelGHK15a, Theorem 5.2] when $(Y,D)$ is positive.

4. Compactifications via polygons

4.1 Generalized half-spaces and polygons in $B$

In this section, the convex geometry of $B$ is developed. Namely, the notion of a generalized half-space is defined and we show how convex polytopes come from intersections of generalized half-spaces. Most importantly, we construct the polytopes that will be used to compactify the mirror family. First, we set up the notion of convex polygons in $B$ .

Definition 4.1. A closed, connected subset $F$ in $B$ with $0$ in the interior is a polygon in $B$ if for each maximal cell $\sigma$ in $\Sigma$ , $F\cap \sigma$ is a polygon. Given a polygon $F$ on $B$ , we say $F$ is convex if any immersed line segment on $B_0$ with endpoints in $F$ is contained in $F$ . We say a polygon is rational if for every vertex $v$ , we have $v \in B(\mathbb {Q})$ and each edge has rational slope. We say a convex rational polygon is nonsingular if for any vertex $v$ as the intersection of two edges $E_1$ and $E_2$ , the primitive generators of $E_1$ and $E_2$ form a basis of $\Lambda _{v}$ .

Remark 4.2. We emphasize that by our definition, a polygon in $B$ will always have $0$ in its interior.

Recall the notion of the developing map on $B_0$ .

Definition 4.3. Let $q:\tilde {B}\rightarrow B_0$ be the universal cover. Then the universal cover inherits an integral affine structure pulled back from $B_0$ . If we patch together integral affine charts on $B_{0}$ , then up to integral linear functions, we obtain a canonical map called the developing map $\delta :\tilde {B} \rightarrow \mathbb {R}^2$ . The fan $\Sigma$ induces a decomposition of $\tilde {B}$ into cones, which we denote by $\tilde {\Sigma }$ .

Lemma 4.4. The developing map is one-to-one if $(Y,D)$ is negative semi-definite. In this case, the closure of the image is a convex cone and is strictly convex if $(Y,D)$ is negative definite. If the developing map is not one-to-one, then it surjects onto $\mathbb {R}^{2}\setminus \{0\}$ .

Proof. Since developing maps are unique up to $\textrm { SL}_{2}(\mathbb {Z})$ -transformations, it suffices to prove the lemma for any particular choice of the developing map. We can assume that $D$ has no $-1$ curves by contracting irreducible $(-1)$ -curves in $D$ . Then $(Y,D)$ is negative semi-definite if $D_{i}^{2}\leqslant -2$ for all $i$ and negative definite if, in addition, $D_{i}^{2}\lt -2$ for at least one $i$ . If $D_{i}^{2}=-2$ for all $i$ , the developing map that takes a lift of $\sigma _{i,i+1}$ to the cone generated by $(i+1,1)$ and $(i,1)$ identifies $\tilde {B}$ with the upper half plane of $\mathbb {R}^{2}$ . If $D_{i}^{2}\lt -2$ for some $i$ , it follows that the developing map identifies $\tilde {B}$ with a strictly convex cone in $\mathbb {R}^{2}$ .

If $(Y,D)$ is positive and contains no $(-1)$ -curves, $D_{i}^{2}\geqslant 0$ for some $i$ . Then the image under the developing map, restricted to a lift of $\sigma _{i-1,i}\cup \sigma _{i,i+1}$ , will contain a half-space in $\mathbb {R}^{2}$ . It follows that the developing map cannot be injective and is surjective onto $\mathbb {R}^{2}\setminus \{0\}$ .

Corollary 4.5. A Looijenga pair $(Y,D)$ is positive if and only if every immersed, directed ray on $B$ not passing through $0$ goes to infinity.

Proof. We use the developing map $\delta$ . Let $R$ be an immersed directed ray on $B$ not passing through $0$ and choose a lift $\tilde {R}$ of $R$ to the universal cover $\tilde {B}$ . By Lemma 4.4, if $(Y,D)$ is positive, then $\delta$ is surjective and therefore there exists a cone $\sigma$ in $\Sigma$ such that $\tilde {R}$ goes to infinity in a connected component of $\delta ^{-1}(\sigma )$ . Conversely, if $(Y,D)$ is negative semi-definite, again by Lemma 4.4, $\delta$ is injective and the closure of its image is a convex cone $\mathcal {C}$ . Let $\tilde {\Sigma }$ be the decomposition of $\tilde {B} \setminus \{0\}$ into cones induced by $\Sigma$ . Let $\tilde {R}'$ be a ray in $\mathbb {R}^2$ not passing through the origin such that its endpoint is in the interior of $\mathcal {C}$ but it escapes $\mathcal {C}$ . Let $R'$ be the projection of $\tilde {R}'$ onto $B$ . Notice that $\tilde {R}'$ intersects rays in $\tilde {\Sigma }$ infinitely many times. Therefore, $R'$ wraps around $0$ infinitely many times and does not escape to infinity.

Let $\bar {L}\subset B_0$ be a line segment with rational slope and integral unit tangent vector $v$ . For $p\in B_0$ , we can view the vector $\overrightarrow {0p}$ as an element in $T_{p,B_0}$ , and if $p\in B(\mathbb {Z})$ , then, furthermore, we can view it as an element of $\Lambda _p$ . It is important to note that $\overrightarrow {0p} \wedge v$ is the same as long as we choose $p\in \bar {L}$ . If $\overrightarrow {0p}\wedge v=0$ , then $\bar {L}$ must be on the ray spanned by $\overrightarrow {0p}$ . Then, if $\overrightarrow {0p}\wedge v\ne 0$ for all $p \in \overline {L}$ , it follows that $\bar {L}$ can be extended to an immersed line $L$ which does not pass through $0\in B$ . The distance of $L$ to $0$ is denoted by

\begin{equation*} d(L):= |\overrightarrow {0p}\wedge v|, \end{equation*}

for any $p\in L$ . Note that this is well defined even when we cross charts since the transition functions are in $ \rm {SL}_2(\mathbb {Z})$ . Given a line $L$ , there are two sides to it. On one side, there is $0$ and on the other side, there is $\infty$ . We always orient $L$ in a way such that $0$ will be on the left of $L$ . In particular, the distance of $L$ from $0\in B$ will always be positive.

Throughout the paper, assume that an immersed line has rational slope.

Definition 4.6. By Corollary 4.5, if $(Y,D)$ is positive, any immersed line $L$ not passing through $0$ has two asymptotic directions. Parametrize $L$ so that the orientation of $L$ agrees with that of $\mathbb {R}$ oriented from $-\infty$ to $+\infty$ . We say the line $L$ escapes or goes to infinity parallel to $q$ (respectively, comes from infinity parallel to $q$ ) if for a maximal cone $\sigma \ni q$ on $B$ , there exists $t_{\sigma }$ such that for any $t\gt t_{\sigma }$ (respectively, $t\lt t_{\sigma }$ ), $l(t)$ is contained in $\sigma$ and $l'(t)=q$ (respectively, $-l'(t)=q$ ) via parallel transport within $\sigma$ . We denote by $L_{\infty }$ (respectively, $L_{-\infty }$ ) the direction in which $L$ escapes to infinity (respectively, comes from infinity).

Given $x\in B_{0}$ , consider all immersed straight segments $I_{px} \subset B_{0}$ between $p$ and $x$ with $p\in L$ . Then the lemma below follows from Corollary 4.5.

Lemma 4.7. Suppose $(Y,D)$ is positive. Let $L$ be an immersed line in $B_0$ . Then given a fixed $x\in B_{0}$ , there are only finitely many values for $v\wedge {\overrightarrow{px}}$ , where $\overrightarrow{px}\in T_{p}B$ is the displacement vector along $I_{px}$ as we vary immersed straight segments $I_{px} \subset B_0$ between $p$ and $x$ with $p \in L$ .

Definition 4.8. The generalized half-space $Z\left (L\right )$ , or simply the half-space, of an immersed line $L$ in $B_0$ with $d(L)\gt 0$ is the closure of all $x$ in $B_{0}$ such that $v\wedge \overrightarrow{px}\geqslant 0$ for all straight immersed line segments $I_{px}$ , as in Lemma 4.7.

The definition of $Z(L)$ is meant to emulate regular half-spaces in usual Euclidean geometry. However, we now give an important example to depict why $Z(L)$ can be more complicated than initially imagined.

Example 4.9. Let $(Y,D)$ be the pair where $Y$ is a del Pezzo surface of degree $1$ and $D\in |\!-\!K_Y|$ is an irreducible nodal curve where $D^2=1$ . We show what $Z(L)$ looks like in $B_{(Y,D)}$ . Let $L$ be a line coming from infinity parallel to $\rho _1$ . We work on a toric blowup of $(Y,D)$ . Since a toric blowup $(Y',D')\rightarrow (Y,D)$ does not change the integral affine structure, we draw the half-space in $B_{(Y',D')}$ . We perform one toric blowup to get a pair $(Y',D')$ where $D'=\tilde {D}+E$ , $\tilde {D}$ is the strict transform of $D$ , and $E$ is the exceptional curve from the blowup. The self-intersection numbers are: $\tilde {D}^2=-3$ , $E^2=-1$ . Figure 1 shows the developing map where $\rho _1$ and $\rho _2$ correspond to $\tilde {D}$ and $E$ , respectively.

Let $L$ be an immersed line coming from infinity parallel to $\rho _{1}$ . As we can see in Figure 1, $L$ also goes to infinity parallel to $\rho _{1}$ . As shown in Figure 2, $L$ intersects itself twice and $ Z(L)$ is a bounded polygon in $B_{(Y',D')}=B_{(Y,D)}$ .

The following construction will be used to prove facts about half-spaces and polygons.

Construction 4.10. Suppose that $(Y,D)$ is positive. Let $L\subset B$ be a line that does not pass through zero with the orientation such that $0$ is on the left. For $L$ , there are two (not necessarily distinct) asymptotic rays, $\rho _{\infty }$ and $\rho _{-\infty }$ , parallel to $L_{\infty }$ and $L_{-\infty }$ , respectively. We refine $\Sigma$ by adding these asymptotic rays if they are not already there. Now let $\bar {U}\subset B$ be the closure of the cones in $\Sigma$ that meet $L$ . Denote the distance from $0$ to $L$ by $d(L)$ . Denote by $\sigma$ the cone containing the $-\infty$ asymptotic end of $L$ . Note that one of the generating rays of $\sigma$ is $\rho _{-\infty }$ . Denote the other generating ray of $\sigma$ by $\rho$ . Now consider the developing map $\delta$ that takes a lift of $\rho _{-\infty }$ and $\rho$ to the negative $y$ -axis and the positive $x$ -axis, respectively. Then $\delta$ gives a canonical, multivalued, orientation-preserving immersion

\begin{equation*} h:\bar {U}\rightarrow {\mathbb {R}}^2,\end{equation*}

which takes $L$ to the line $\{(x_1, x_2) \mid x_1=d(L)\}$ with upward orientation and $p$ to $(d(L),0)$ . Here $p$ is the point when $L$ first intersects $\rho$ . We can see that

\begin{equation*} h(\bar {U})=\{(x_1,x_2) \mid x_1 \ge 0\}.\end{equation*}

We denote $\bar {H}:=h(\bar {U})$ . Then, when restricted to $\bar {H}$ , the inverse of $h$ is a single-valued immersion, which we call $g:\bar {H}\rightarrow B_0$ . Under $g$ , $L$ is the image of an affine immersion $l:{\mathbb {R}}\rightarrow B_0$ . Then we can get a lift $\tilde {l}:{\mathbb {R}}\rightarrow \tilde {B}$ by choosing a lift of $p$ on $\tilde {B}$ . Then $\delta \circ \tilde {l}:{\mathbb {R}}\rightarrow \tilde {B}$ gives a parametrized line. Now let $\tilde {H}$ be the union of cones in $\tilde {\Sigma }$ that meet the image of $\tilde {l}$ . Then we get a linear isomorphism $\delta :\tilde {H}\rightarrow \bar {H}$ . Now if we pullback $\Sigma$ along $g$ , we get a fan $\bar {\Sigma }$ whose support is $\bar {H}$ . Then for each maximal cone $\bar {\sigma }\in \bar {\Sigma }$ , $g:\bar {\sigma }\rightarrow \sigma$ is an isomorphism onto a cone of $\Sigma$ .

Figure 1.

The developing map for the degree $1$ del Pezzo.

Figure 2.

The half-space $Z(L)$ is a bounded polygon.

Lemma 4.11. If $(Y,D)$ is positive, then given an immersed line $L\subset B_0$ , the generalized half-space $Z(L)$ is a convex polygon.

Proof. Given $x\notin Z(L)$ , we want to show that the ray $\rho _x$ (generated by $\overrightarrow {0x}$ ) intersects $L$ . Let $I_{px}$ be a line segment from $p\in L$ to $x$ that points to the right of $L$ . Then $d(I_{px},0)$ is a non-constant affine function as we vary $p\in L$ . That means there exists $p$ such that $d(I_{px},0)=0$ , meaning that the immersed line spanned by $I_{px}$ goes through $0$ . However, the immersed ray starting at $p$ in the $\overrightarrow {px}$ direction cannot contain zero since it would imply that $x$ is on the left-hand side of $L$ . Thus, we see that $p$ is on the ray $\rho _x$ .

For the rest of the proof, we use Construction 4.10 and the notation there. Now we wish to show that

(4.1) \begin{equation} Z\left (L\right )^{c}=g\left (\left \{ (x_1,x_2)\mid x_1\gt d\left (L\right )\right \} \right ). \end{equation}

As we have shown, if $x \notin Z(L)$ , then $x$ is in the image of $g$ . Let $I_{px}$ be a line segment that points to the right of $L$ . Then because $I_{px}$ is in the image of $g$ , we can lift it to a line segment $\tilde {I}_{px}\subset \bar {H}$ from a point $\bar {p}\in \{(x_1,x_2)\mid x_1=d(L)\}$ to a point $\bar {x}$ such that $\overrightarrow {\bar {p}\bar {x}}$ points to the right of $\{(x_1,x_2)\mid x_1=d(L)\}$ . Thus, we get that $Z(L)^c\subset g(\{(x_1,x_2) \mid x_1\gt d(L)\})$ . The other inclusion is immediate.

Now given an immersed line segment $I$ with endpoints $x$ and $y$ in $Z(L)$ , suppose that $I$ is not contained in $Z(L)$ . Then $I$ intersects $L$ at a point $p$ such that either $v\wedge \overrightarrow {px} \lt 0$ or $v\wedge \overrightarrow {py} \lt 0$ , in contradiction with the assumption that both $x,y$ are in $Z(L)$ . Hence the assumption is false and every immersed line segment with endpoints in $Z(L)$ is contained in $Z(L)$ , i.e. $Z(L)$ is convex.

For the rest of this subsection, we assume that $(Y,D)$ is positive.

Lemma 4.12. Let $F\subset B$ be a convex polygon. If $E$ is an edge of $F$ and $L$ is the immersed line generated by $E$ , then $F\subset Z(L)$ and $F\cap L=E$ .

Proof. By our definition of polygons, $0$ is in the interior of $F$ . Let $g:H \rightarrow B_0$ be as defined in Construction 4.10. We claim that $g^{-1}(F)$ is convex. Indeed, since $F$ is convex, $g^{-1}(F)$ is locally convex at any vertex $v$ . Therefore $g^{-1}(F)$ is convex. Since $0$ is contained in the interior of $F$ , we have

\begin{equation*} g^{-1}(F)\subset \{(x_1,x_2) \mid 0\le x_1 \le d(L)\}.\end{equation*}

By Equation (4.1),

\begin{equation*} F\cap Z(L)^c = g ( g^{-1}(F) \cap \{ (x_1,x_2) \mid x_1\gt d(L) \}) = \emptyset . \end{equation*}

Hence, $F \subset Z(L)$ .

To show that $F\cap L=E$ , it remains to show that $F\cap L\subset E$ . Let $q\in F\cap L$ and $p\in E$ . Let $I_{pq}\subset L$ be the immersed line segment from $p$ to $q$ . We want to show that $I_{pq}$ is contained in $E$ . If not, then $I_{pq}$ will cross an edge, $E'$ , of $F$ at a vertex of $F$ . Now let $L'$ be the immersed line generated by $E'$ . We know that $F\subset Z(L')$ and that $I_{pq}\subset Z(L')$ . Notice though that $I_{pq}$ must cross $E'$ from left to right, implying that $q\notin Z(L')$ , which is in contradiction with that $I_{pq}\subset Z(L')$ . Thus, it must be that $I_{pq}\subset E$ . This implies that $F\cap L=E$ , as desired.

Proposition 4.13. Let $F\subset B$ be a convex polygon. Then $F$ is equal to the intersection of the half-spaces spanned by the edges.

Proof. Let $E\subset F$ be an edge and $Z_E$ the corresponding half-space obtained by the corresponding immersed line. Then define $P:=\bigcap _E Z_E$ . By Lemma 4.12, $F\subset P$ and $\partial (F)\subset \partial (P)$ , which implies that $F=P$ .

By Lemma 4.11 and Proposition 4.13, we have the following corollary.

Corollary 4.14. A polygon $F$ is convex if and only if it is the intersection of generalized half-spaces.