Looijenga Pairs in Surface Geometry

• Looijenga pairs are smooth rational surfaces paired with anticanonical cycles of rational curves arranged in a cycle, fundamental to mirror symmetry.
• They are classified by the number and self-intersections of the boundary curves, enabling precise control over deformations and automorphism groups.
• They provide a rich framework for exploring correspondences among Gromov–Witten, Donaldson–Thomas, and tropical invariants, offering concrete methods for enumerative geometry.

A Looijenga pair, also known as an anticanonical pair or log Calabi–Yau surface, is a pair $(Y, D)$ where $Y$ is a smooth rational projective surface and $D$ is a reduced nodal anticanonical divisor—typically a cycle of smooth rational curves—satisfying $D \in |-K_Y|$. Looijenga pairs are central objects in mirror symmetry, the birational geometry of surfaces, enumerative geometry, and the study of surface singularities. These pairs provide a fertile testing ground for correspondences among Gromov–Witten theory, Donaldson–Thomas invariants, open enumerative invariants, and tropical geometry.

1. Definition, Examples, and Classification

A Looijenga pair $(Y, D)$ is defined by the following properties:

• $Y$ is a smooth projective rational surface ($H^1(Y, \mathcal{O}_Y) = 0$).
• $D$ is a reduced, connected, nodal curve in the anticanonical linear system ($D \in |-K_Y|$), whose irreducible components $D_i \cong \mathbb{P}^1$ are arranged in a cycle: each $D_i$ meets $D_{i\pm1}$ transversely and does not intersect the other $D_j$.

The dual intersection graph of $D$ is a cycle. The length of the cycle, denoted by $l$, is the number of components of $D$.

Typical examples include:

• The pair $(\mathbb{P}^2, D)$ with $D$ being a triangle of lines, a line plus conic, or a nodal cubic curve; these correspond to the three Looijenga pairs obtainable from $\mathbb{P}^2$ (Garrel, 2023).
• Toric surfaces with the toric boundary forming a cycle give toric Looijenga pairs, characterized by their vanishing "charge," a numerical invariant defined as $Q(Y, D) = 12 + \sum_{i=1}^l (d_i-3)$, where $d_i = -D_i^2$ for $l>1$ (Engel, 2014).
• More generally, blowing up points along the toric boundary of a rational toric surface yields any Looijenga pair up to deformation (Brini, 2022).

There exist 18 deformation types of nef Looijenga pairs, classified according to the sequence of self-intersections $D_1^2,\ldots,D_l^2$ and the length $l$ (Bousseau et al., 2020). For negative-definite cycles of length $6 \leq n \leq 9$, the deformation type is uniquely determined except for certain alternating cycles of length 8 (where two types can occur) and at most three types for length 9 (Simonetti, 2023).

2. Intersection Theory, Root Systems, and Torelli Theorems

The Picard group $\operatorname{Pic}(Y)$ carries a symmetric bilinear intersection form of signature $(1, \rho-1)$, with the classes of boundary components spanning a negative semi-definite lattice. The sublattice orthogonal to the boundary, $$A(Y,D) = \{ \alpha \in H^2(Y, \mathbb{Z}) \mid \alpha\cdot[D_i]=0 \, \forall i \}$$, is of central importance.

One defines the Looijenga root lattice

$$R(Y, D) = \{ \delta \in H^2(Y, \mathbb{Z}) \mid \delta^2 = -2 ,~ \delta \cdot [D_i] = 0~\forall i \}.$$

Reflections in these roots generate the Weyl group acting on the Picard lattice, which controls much of the deformation and automorphism theory (Friedman, 2015).

The period map assigns to a Looijenga pair a point in $\operatorname{Hom}(A(Y,D), \mathbb{C}^*)$. The global Torelli theorem (Gross–Hacking–Keel) asserts that the period map, together with ample-cone data, determines the pair up to isomorphism modulo the Weyl group. The deformation space is smooth and unobstructed, with dimension equal to the rank of $A(Y,D)$ (Friedman, 2015).

3. Birational Models, Toric Degenerations, and Geometry

Every Looijenga pair is obtained from a toric Looijenga pair via a sequence of blowups at smooth points of the boundary (interior blowups) and nodes (corner blowups). The charge increases by one with each interior blowup and remains constant under corner operations. The toric model is thus a minimal birational contraction (Simonetti, 2023).

For understanding smoothability of cusp singularities, Looijenga's conjecture relates the existence of a Looijenga pair with a given self-intersection cycle to the smoothability of the dual cusp singularity, a statement proved using integral-affine geometry and tropical techniques (Engel, 2014).

Degenerations of Looijenga pairs to toric surfaces—together with surgeries in the integral-affine structure—yield explicit models of the associated dual complexes and play a key role in both classification and mirror symmetry constructions (Engel, 2014).

4. Cox Ring, Theta Functions, and Canonical Bases

The Cox ring $\operatorname{Cox}(X)$ of a Looijenga pair $(X,E)$ is defined as the direct sum of sections of all line bundles: $$\operatorname{Cox}(X) = \bigoplus_{L \in \operatorname{Pic}(X)} H^0(X, L),$$ with a natural $\operatorname{Pic}(X)$-grading. For positive Looijenga pairs (i.e., $E$ supports an ample combination; equivalently, the intersection matrix of $E$ is negative-definite), the Cox ring is finitely generated (Keel et al., 2024).

There exists a canonical basis of theta functions for $\operatorname{Cox}(X)$, indexed by integral points in the essential skeleton of the mirror, constructed via broken lines in a scattering diagram in the associated affine manifold. The product structure on the theta basis has structure constants given by naive counts of $k$-analytic disks in the universal deformation of the mirror, which in dimension two coincides with the analytic space $X \setminus E$. The theta basis is compatible with all deformations and respects birational modifications and automorphism actions (Keel et al., 2024).

5. Enumerative Invariants and "Web of Correspondences"

Enumerative geometry for Looijenga pairs is organized around a suite of invariants:

• Log Gromov–Witten invariants $N^{\log}_{g,\beta}(X, D)$ count genus $g$ stable log maps to $X$ in class $\beta$, maximally tangent to each component of $D$, with insertion of Hodge and point classes. These admit all-genus generating functions via tropical scattering (Brini, 2022, Bousseau et al., 2020).
• Local Gromov–Witten invariants for the total space of $\bigoplus_i \mathcal{O}_X(-D_i)$ are computed via mirror symmetry and linked to the log invariants by explicit formulas.
• Open Gromov–Witten invariants enumerate holomorphic disks bounded by Aganagic–Vafa branes in Calabi–Yau 3-folds associated to $(X,D)$, computed explicitly via the topological vertex.
• Donaldson–Thomas invariants for symmetric quivers determined by the combinatorics of $(X, D)$.
• BPS-type invariants (LMOV/KP/IP) extracted via plethystic–Möbius inversion from the above series.

The "web of correspondences" asserts equalities (after explicit multiplicative factors) among log GW, local GW, open GW, and DT/BPS invariants. In genus zero, log GW invariants are proportional (via products of intersection numbers and sign factors) to local GW and, in turn, to open GW invariants. These correspondences are established using tropical mirror symmetry, topological vertex techniques, and quiver methods and hold for nef or quasi-tame Looijenga pairs (Brini, 2022, Bousseau et al., 2020, Brini et al., 2022). Closed-form and hypergeometric formulas encapsulate these enumerative invariants for all basic deformation types (Bousseau et al., 2020, Brini et al., 2022).

6. Mirror Symmetry and Tropical/Scattering Techniques

Looijenga pairs furnish the two-dimensional case of the Gross–Hacking–Keel–Siebert mirror symmetry program. The mirror to $U = X \setminus D$ is constructed as a scheme over the base torus $T_{\operatorname{Pic}(X)}$, with the theta function basis of the Cox ring providing the coordinate functions of the mirror family. The essential skeleton $\operatorname{Sk}(U^\vee)$ forms a tropical affine manifold with singularities, and the canonical algebra $A_{\operatorname{Sk}(U^\vee)}$ is determined by broken lines (tropical analogues of holomorphic disks) and their counts (Keel et al., 2024).

In dimension two, positivity ensures that the mirror family is deformation-equivalent to the original pair, and all enumerative invariants (log GW, open GW, DT) can be recovered from the same scattering diagram/tropical data. Disk counts in the Berkovich analytic space correspond precisely to the theta function structure constants, and wall-crossing formulas capture recursive and associativity properties (Keel et al., 2024, Bousseau et al., 2020).

7. Automorphism Groups, Deformations, and Moduli

Automorphisms of a Looijenga pair act on $\operatorname{Pic}(Y)$ and, via the tropical skeleton, permute the theta basis. The automorphism group, up to its finite kernel, sits in an exact sequence involving the algebraic torus acting on the Cox ring and is further governed by the Weyl group generated by root reflections (Friedman, 2015). Deformations of $(Y, D)$ are unobstructed and parametrized by the period domain, with the canonical theta basis extending coherently over the universal family (Keel et al., 2024).

Recent results show that the diffeomorphism type of $(Y, D)$ (together with the labeling of $D$) determines its algebraic deformation type (Friedman, 2015). The root lattice is characterized intrinsically as those (−2)-classes becoming rational curves in a deformation, and the Weyl group action accounts for all nontrivial monodromy in families.

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References:

• (Keel et al., 2024) "Theta Function Basis of the Cox ring of Positive 2d Looijenga pairs"
• (Garrel, 2023) "Stable maps to Looijenga pairs built from the plane"
• (Brini, 2022) "Enumerative geometry of surfaces and topological strings"
• (Friedman, 2015) "On the geometry of anticanonical pairs"
• (Engel, 2014) "Looijenga's Conjecture via Integral-affine Geometry"
• (Simonetti, 2023) "Deformation types of Looijenga pairs of small length"
• (Brini et al., 2022) "On quasi-tame Looijenga pairs"
• (Bousseau et al., 2020, Bousseau et al., 2020) "Stable maps to Looijenga pairs" and its orbifold generalization