---
created: 2024-05-03T00:14
updated: 2024-05-04T21:37
---

These notes are an expanded version of the talk I gave for my oral exams at the University of Georgia in December 2022, which is an exposition on Phil Engel's thesis work [Looijenga's Conjecture via Integral-affine Geometry](https://arxiv.org/abs/1409.7676).

## Intro

The essential content of Looijenga's conjecture is the following: there are particular complex algebraic surfaces which have cusp singularities that come in "dual" pairs $D$ and $\hat D$. Is is a theorem that if $D$ is smoothable then there exists a certain anticanonical pair (ACP) of the form $(\hat Y, \hat D)$; Looijenga conjectured in 1981 that if there exists an ACP of the form $(Y, D)$ then $\hat D$ is smoothable. More precisely, for the forward direction, we have the following:


Theorem (Looijenga, 1981)
If a cusp singularity $D$ is smoothable, then the minimal resolution of the dual cusp $\hat D$ is the anticanonical divisor of a smooth rational surface.

In the converse direction, resolving the conjecture, we have

Theorem (Gross-Hacking-Keel 2015, Engel 2018)
If $D$ is the anticanonical divisor of a rational surface, then the cusp associated to $\hat{D}$ is smoothable.

## Cusps and Resolutions

We first recall some basic facts and examples about resolving singularities of curves embedded in surfaces by means of successive blowups. Consider $X \da V(y^2-x^3) \subseteq \AA^2$, which defines a cuspidal cubic plane curve. The singular locus is precisely one point $p$ consisting of the origin, as can be checked using the Jacobian criterion. Blowing up the origin once produces a parabola $V(y^2-x)$ and an exceptional divisor along the $y\dash$axis, and a second blowup at the origin results a configuration $X$ of three intersecting lines: the strict transform of the original curve, the strict transform of the exceptional divisor in the first blowup, and a new exceptional curve. Taking the dual complex $\Gamma(X; p)$ of this configuration yields the Dynkin diagram $A_3$, which has 3 nodes and 2 edges. This motivates the following

Definition
A surface singularity $p$ is a **cusp** if and only iff the dual complex of a minimal resolution is an affine Dynkin diagram of type $\tilde A_n$.

For illustrative purposes, we will often draw the following "cartoon" picture of a cusp singularity on a surface:

\<pic>

Cusp singularities arise in the classification of complex analytic surfaces, and in our case of interest, the study of hyperbolic Inoue surfaces. For experts, these are surfaces $V$ of Kodaira type $\rm{VII}_0$ with first Betti number $\beta_1 = 1$ and Kodaira dimension $\kappa(X) = -\infty$. The only curves on such a surface $V$ are components of divisors $D$ and $\hat D$, which are comprised of cycles of rational curves intersecting transversally. They satisfy $D + \hat {D} \in \abs{-K_V}$, and blow down to a surface $\bar V$ with (?) two corresponding cusps $p$ and $\hat p$. Our cartoon picture is the following:

\<pic>

One can construct $\bar V$ as $\dcosetl{ \Gamma} {\overline{\HH \times \CC}}$ where $\Gamma$ is a discrete group.

It is interesting to compare this to the notion of "cusps" in the theory of modular forms and the moduli space of elliptic curves. One requires a classical modular form for $\SL_2(\ZZ)$ to have certain growth rates or holomorphicity properties at "cusps" of a fundamental domain for the modular curve $X(0) \da \dcosetl{ \SL_2(\ZZ) }{\HH}$. To obtain the compactified modular curve $\bar{X(0)}$, one regards $X(0) \subset \CP^1$ as a subset of the Riemann sphere, adds the cusp $\ts{\infty}$, and then adds the orbit $\SL_2(\ZZ) . \ts{\infty} = \PP^1(\QQ) = \QQ \union \ts{\infty}$, which is a dense set of rational points on the equator of $\CP^1$.

We now proceed with a more precise definition:

Definition
A cusp singularity $(\bar V, p)$ is the germ of a minimally elliptic surface singularity such that the minimal resolution $\pi: V\to \bar V$ has exceptional divisor $\pi\inv(p) = D = \sum_{i=1}^n D_i \in \abs{-K_V}$ where the $D_i$ are smooth rational transverse curves.

Remark:
The germ of $p$ is uniquely determined up to symmetry. Letting $d_i\da -D_i^2$ be the negative self-intersection number of the $i$th curve in $D$, we consider the vector $\vec d \da (d_1,\cdots, d_n)$, which is the diagonal of the intersection matrix $M_{ij} \da D_i \cdot D_j$. Note that $D$ is contractible, and by Artin's contractibility criterion $M$ is negative definite, and one can show that this forces all $d_i\geq 2$ and at least one $d_j \geq 3$.

If we write $\vec d = (a_1 + 3,2,2,\cdots, 2, a_2+3, 2, 2,\cdots, 2, \cdots, 2, a_{k}+3, 2, 2,\cdots, 2)$, using the $a_i$ to single out all of the $d_j$ with $d_j\geq 3$, then there is a formula for the corresponding vector $\hat{\vec d}$ for $\hat D$:
$$
\hat{\vec d} = (\hat d_1, \cdots, \hat d_n) = (b_1 + 3, 2,\cdots, 2, b_2 +3, 2,\cdots, 2, b_k + 3, 2\cdots, 2)
$$
More transparently, one simply swaps the roles of $a_i$ and $b_i$ in the original vector $\vec d$ to obtain $\hat{\vec d}$.

Definition:
The charge of a cycle $D$ is given by the formula
$$
Q(D) \da 12 + \sum_{i=1}^n (d_i - 3)
.$$

Remark
One can show that $Q(D) = 12 + \sum_{i=1}^k (a_i - b_i)$, and moreover $Q(D) + Q(\hat D) = 24$. The latter follows immediately from the duality formula:
$$
Q(D) + Q(\hat D) = \qty{ 12 + \sum (a_i - b_i) } + \qty{ 12 + \sum (b_i - a_i)} = 12 + 12 = 24
.$$
This formula holds more generally for Type III anticanonical pairs $(Y, D)$ and in a sense measures how far the pair is from being toric, i.e. a toric variety with its toric boundary. One can show that a necessary condition for $(Y, D)$ to be toric is that $Q(Y, D) = 0$, and that if $Q(Y, D) \geq 1$ then the pair is not toric.

Definition:
An anticanonical pair $(Y, D)$ is a rational surface $Y\birationaliso \PP^2$ with $D = \sum D_i \in \abs{-K_Y}$ a cycle of transversally intersecting rational curves.

We give some examples of toric pairs: $\PP^1\times \PP^1, \PP^2, \FF_n$. For an example a non-toric pair, we can take the following: $(\PP^2, \text{two curves with two intersections})$. This is non-toric since $D$ is not the toric boundary of $\PP^2$, and moreover one can compute that $Q(\PP^2, D) = 4 \neq 0$ as a verification of nontoricity.