Mutations of blowups of toric surfaces and \(q\)-Painlevé systems
The same space of initial conditions can be constructed from different toric surfaces. Seeds record these constructions, and mutations connect them.
Seeds defining isomorphic \(q\)-Painlevé systems are mutation equivalent.
Discrete Painlevé systems are constructed as actions on families of rational surfaces. They fall into elliptic, multiplicative, and additive types. We consider the multiplicative type: \(q\)-Painlevé systems.
From seeds to spaces of initial conditions
A space of initial conditions for a \(q\)-Painlevé system can be constructed as a family of blowups at smooth points of the boundary of a toric surface. A seed records the boundary directions and the number of blowups.
A seed in a free abelian group \(N\) equipped with a skew-symmetric bilinear form is a multiset of elements of \(N\). We work with \(N=\mathbb Z^2\) and the wedge product
Assume that every vector in the seed is primitive and that the seed spans \(\mathbb Q^2\). A vector is primitive when the greatest common divisor of its coordinates is \(1\).
An example of a seed
Superscripts denote multiplicities in the multiset. For example, \((1,0)^3\) means that \((1,0)\) occurs three times.
Three choices in the construction
- Choose a complete smooth toric fan \(\Sigma\).
Each vector of the seed must generate a ray of the fan. Rays correspond to irreducible boundary components of the toric surface \(\operatorname{TV}(\Sigma)\). - Choose points \(p_v\) on the boundary.
For each occurrence of \(v\in s\), choose a point to blow up on the corresponding boundary component. - Order the seed and compose the blowups.\[\pi:Y\longrightarrow\operatorname{TV}(\Sigma).\]Let \(D\) be the strict transform of the toric boundary.
Allowing the points to vary gives a family of log Calabi–Yau surfaces. The construction concerns this family rather than one fixed choice of points.
A piecewise-linear map and one reversed vector
Choose one occurrence of \(v\in s\). Apply a piecewise-linear map to the seed, changing vectors on the side where the wedge product is positive. Then replace the chosen occurrence of \(v\) by \(-v\).
The operation \(\setminus\{v\}\) removes exactly one occurrence of \(v\) from the multiset. Other occurrences of \(v\) are retained.
Explore a mutation
Choose a direction to apply one mutation to the example seed above. Each choice starts from the same original seed.
Mutation induces an isomorphism \(Y\to Y'\). Geometrically, it corresponds to an elementary transformation combining a blowup with the contraction of a fiber (cf. Hartshorne, Example 5.7.1).
A formal composition of mutations and \(\mathrm{GL}(2,\mathbb Z)\) transformations is called a cluster transformation. Two seeds joined by a cluster transformation are mutation equivalent.
The boundary intersection matrix
In Sakai’s theory, spaces of initial conditions for \(q\)-Painlevé systems form families of generalized Halphen surfaces.
The surface \(Y\) defined by a seed is a generalized Halphen surface if and only if the intersection matrix of its boundary \(D\) is of affine type.
Self-intersections from the fan and multiplicities
Let \(u,w\) be the primitive generators of the rays adjacent to \(v\) in the smooth fan. The relation \(u+w+mv=0\) determines the self-intersection \(m\) of the toric boundary component. Subtract the number \(n\) of blowups, which is the multiplicity of \(v\) in the seed:
Example 1 · Two blowups on each component
For the four rays \((1,0),(0,1),(-1,0),(0,-1)\) of \(\mathbb P^1\times\mathbb P^1\), we have \(m=0\). With multiplicity two in each direction, the boundary intersection matrix satisfies
This example corresponds to a \(q\)-Painlevé system of type \(D_5^{(1)}\). The type of the boundary intersection matrix and the symmetry type of the system are distinct notions.
Example 2 · A different fan
Consider the rays \((1,0),(0,1),(-1,3),(0,-1)\) with multiplicities \(1,0,1,9\), respectively.
Infinitely many constructions, ten mutation classes
Seeds defining isomorphic \(q\)-Painlevé systems are mutation equivalent.
The proof reduces to the classification of a class of Fano polygons by Kasprzyk–Nill–Prince. Its outline has three steps.
- Define a size function on seeds giving \(q\)-Painlevé systems.
- Show that there are finitely many minimal seeds with respect to size: 35 in total.
- Determine mutation equivalence among these 35 seeds, obtaining ten classes.
Representative seeds
The type \(E_5^{(1)}\) is also denoted by \(D_5^{(1)}\).
Gluing the coordinate charts
For a seed \(s\), glue algebraic tori indexed by cluster transformations starting at \(s\). The resulting space is the cluster Poisson variety \(\mathcal X(s)\):
A cluster transformation \(\mu:s\to s'\) determines an isomorphism \(\mathcal X(\mu):\mathcal X(s)\to\mathcal X(s')\).
The isomorphism defined by a mutation agrees with an elementary transformation up to subsets of codimension two.
This correspondence connects cluster Poisson varieties with the interiors of spaces of initial conditions constructed by blowing up toric surfaces.
Realizing symmetries by cluster transformations
A cluster transformation \(\mu:s\to s\) gives an automorphism of \(\mathcal X(s)\). The group of automorphisms obtained in this way is called the cluster modular group.
The Cremona isometry group, describing the symmetries of the \(q\)-Painlevé system, is realized as a subgroup of the cluster modular group.
This realization leads to tropicalization through a definition over \(\mathbb N\), and to quantization.
An \(E_8^{(1)}\) action in coordinates
Consider a seed of 11 elements in which the directions \((0,1),(-1,0),(1,-1)\) have multiplicities \(3,2,6\), respectively.
Let \(W=\langle r_0,\ldots,r_8\rangle\) be the affine Weyl group of type \(E_8^{(1)}\), and let \(C=(C_{ij})\) be its Cartan matrix.
The action \(\iota:W^{\mathrm{op}}\to\operatorname{Aut}(\mathcal X(s))\) has the following coordinate expressions:
The variety \(\mathcal X(s)\) is a family of two-dimensional schemes over a nine-dimensional algebraic torus. The variables \(a_0,\ldots,a_8\) are coordinates on the base. The displayed formulas are obtained by composing the rational transformations associated with mutations.