Resolution of Indeterminacies in Discrete Systems

• Resolution of Indeterminacies is the process of eliminating undefined points in rational maps using techniques such as blow-ups and root stack constructions.
• It employs precise geometric modifications to construct well-defined phase spaces that support the analysis of discrete integrable systems like Painlevé equations.
• This framework links algebraic regularization with dynamic properties such as controlled degree-growth and integrability, aiding cohomological and entropy computations.

Resolution of indeterminacies refers to the process, particularly in algebraic geometry and its applications to discrete integrable systems, of systematically eliminating points of non-definition in rational maps by geometric operations such as blow-ups, root stack constructions, or parametrized morphisms, thereby obtaining well-defined morphisms on modified spaces or stacks. In the context of discrete integrable systems, these geometric techniques also serve to clarify and control the complex behavior (entropic or otherwise) of iterated birational transformations, linking the analytic notion of integrability to explicit algebraic structural data.

1. Geometric Background: Rational Maps and Indeterminacy

A rational map between algebraic varieties or complex manifolds is a map defined by rational functions, typically not everywhere defined due to division by zero at specific points—these are the indeterminacy loci. For projective varieties, resolving these loci is crucial to extend the map to a regular morphism on a modified space. In classical surface theory, the indeterminacy of a rational map $\varphi: X \dashrightarrow Y$ is resolved by performing a finite sequence of blow-ups along the locus of indeterminacy, yielding a birational morphism

$\epsilon: \widetilde{X} \rightarrow X$

such that $\varphi \circ \epsilon$ becomes a morphism. In the setting of rational surfaces, this procedure is closely related to the construction and classification of exceptional divisors and the analysis of the structure of the Picard group.

For integrable discrete systems arising as birational maps on surfaces, the underlying algebraic surface is often modified iteratively to track the evolution of base points and ensure the regularity of the system's action.

2. Sakai Surfaces and Root Lattices

Within Sakai's framework for discrete Painlevé equations, the phase spaces are not fixed, but rather parametrized families of rational surfaces (Sakai surfaces), notable examples being generalized Halphen surfaces characterized by the presence of a unique effective anticanonical divisor $D$ of "canonical type" whose irreducible components have self-intersection $-2$.

For a concrete class, such as the D$_5^{(1)}$ type, the typical construction is via a sequence of eight blow-ups of $\mathbb{P}^1\times \mathbb{P}^1$ at carefully determined base points $b_1, \dots, b_8$ dependent holomorphically on parameters. The resulting surface $S_{\mathbf{a}}$ is equipped with a geometric basis for its Picard group: $$\mathrm{Pic}(S_{\mathbf{a}}) = \langle H_p, H_q, E_1, \dots, E_8 \rangle,$$ with $H_\bullet$ denoting the pullbacks of the respective rulings and $E_i$ the exceptional divisors of the blow-ups.

Embedded in the Picard group are two root sublattices:

• The surface (configuration) root lattice $Q(\mathcal{R})$ spanned by the classes $D_i$ of the irreducible components of $D$.
• The symmetry (complementary) root lattice $Q^{\perp}$, defined as those $F \in \mathrm{Pic}(S_{\mathbf{a}})$ with $F \cdot D_i = 0$ for all $i$. For $D_5^{(1)}$, $Q^{\perp}$ is of type $A^{(1)}_3$.

This configuration governs the intrinsic symmetries and the space of birational self-maps consistent with the surface blow-up structure.

3. Extended Affine Weyl Group Symmetry and Cremona Isometries

The action of the extended affine Weyl group associated with the symmetry root lattice is central to the structure of discrete integrable systems. Each simple root $\alpha_i$ in a chosen basis defines a reflection $r_i : F \mapsto F + (F \cdot \alpha_i)\alpha_i$ on the Picard group, and automorphisms of the associated Dynkin diagram extend this group further. The explicit permutation action on the geometric basis given in the literature allows for nontrivial transformations, such as

$$\sigma_1 = (H_q \ H_p)(E_1 \ E_7)(E_2 \ E_8)(E_3 \ E_5)(E_4 \ E_6),$$

which enact substantial changes on the parameters and exceptional divisor configuration.

Discrete Painlevé equations, within this paradigm, are defined by the action of infinite order elements (translations or quasi-translations) of the extended affine Weyl group on the family of Sakai surfaces, which, via a change of blowing-down (marking), induces a birational map on $\mathbb{P}^2$ that can represent the discrete evolution.

4. Parametrization, Period Maps, and Evolution of Root Variables

Parameter dependence in these systems is expressed through "root variables," typically denoted $a_i$, assigned via the period map as images of simple symmetry roots. The evolution of these parameters is dictated by the group action:

• In the additive case: $\bar{a_i} = \sum_j M_{ij}a_j$,
• In the multiplicative case: $\bar{a_i} = \prod_j a_j^{M_{ij}}$, where $M$ is the matrix of the corresponding Weyl group element.

This yields a parameter-dependent, non-autonomous, discrete evolution, characteristic of the higher Painlevé equations.

5. Resolution of Indeterminacies and Birational Dynamics

The iterative resolution of indeterminacy points by blow-ups underlies the construction of the phase space that supports the well-defined action of the entire symmetry group. Each step clarifies the structure of exceptional divisors, the anticanonical divisor, and the precise Picard lattice relations. For birational transformations governing discrete dynamical systems, the entropy—measured by the degree-growth rate of the iterates—becomes computable via cohomological tools afforded by this resolution process, linking algebraic entropy to integrability.

The Cremona isometries governing the system’s dynamics act as automorphisms of the Picard lattice and translate directly into explicit birational maps on the underlying rational surfaces. This connection provides not only a geometric origin for the equations but also precise control over the associated dynamical properties.

6. Significance for Discrete Integrable Systems

This algebro-geometric framework reveals how discrete Painlevé equations and similar integrable systems possess controlled degree growth and satisfy conditions such as singularity confinement. The entire structure—root lattices, symmetry groups, parameter evolution, and blow-up configurations—serves to explain and enforce the "integrability" of such discrete models. The resolution of indeterminacies ensures that each iterate of the mapping remains within a well-behaved geometric context, avoids undefined operations, and supports a rich symmetry structure that parallels the continuous Painlevé theory.

The explicit construction of the phase space and symmetries, as for the D₅^1 Sakai surface, illustrates the deep relation between classical algebraic geometry and the modern theory of discrete integrable systems, substantiating integrability via the existence of large symmetry groups and geometric regularization techniques.

7. Cohomological Computations and Algebraic Entropy

The cohomological computation of algebraic entropy for birational transformations is directly facilitated by the explicit tracking of divisors, Picard groups, and the effect of the map on these data through the resolution of indeterminacies. Sub-exponential (often quadratic) degree growth is a hallmark of integrable systems and serves as a testable criterion for integrability, cementing the critical importance of these geometric constructions in both classifying and analyzing the complexity of discrete dynamical systems.

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Thus, the resolution of indeterminacies—realized through techniques such as blow-ups, root stack constructions, and affine Weyl group actions on the Picard lattices of rational surfaces—forms a foundational methodology for both regularizing rational dynamics and uncovering the algebraic structures underpinning discrete integrable systems, particularly in the context of discrete Painlevé equations (Alecci et al., 14 Oct 2025).