Moduli Spaces of Degree Two Rational Maps with Portraits Up to Six Points

Abstract: We consider the moduli space, $\mathcal{M}_d<sup>N[\mathcal{P}]$, of degree $d$ rational self maps of $\mathbb{P}<sup>N$ with prescribed pre-periodic structure $\mathcal{P}$ which were introduced by Doyle and Silverman. It was shown in arXiv:1305.1054 that, $\mathcal{M}_2<sup>1[\mathcal{P}_6]$, the moduli space of degree two rational maps with a $6$-cycle is a surface of general type. Here we compute the Kodaira dimension of all the moduli spaces with up to six pre-periodic points and show that $κ= -\infty, 0, 1,2$ are all realized for some $\mathcal{P}$.

• The paper provides a comprehensive classification of moduli spaces for degree two rational maps by analyzing pre-periodic portraits up to six points.
• The methodology involves explicit coordinate reductions, blowups, and branch locus analysis to resolve singularities and determine the Kodaira dimension.
• The results demonstrate that moduli spaces cover the full Kodaira spectrum, impacting the predicted arithmetic density of rational points in dynamical systems.

Moduli Spaces of Degree Two Rational Maps with Pre-periodic Portraits: Classification and Kodaira Dimension

Introduction and Background

The paper "Moduli Spaces of Degree Two Rational Maps with Portraits Up to Six Points" (2606.20982) investigates the birational geometry of moduli spaces parameterizing degree two rational self-maps of $\mathbb{P}^1$ with prescribed pre-periodic structure ("portraits") involving up to six points. This research sits within the framework of arithmetic dynamics, engaging with deep conjectures such as the Uniform Boundedness Conjecture of Morton and Silverman and questions about the distribution of rational pre-periodic points for dynamical systems defined over number fields.

Of particular relevance is Doyle and Silverman's construction of moduli spaces $M_{d,N,\mathcal{P}}$, which encode the pre-periodic portrait structure for maps of degree $d$ on $\mathbb{P}^N$ [Doyle_2020]. The Kodaira dimension of these moduli spaces is a vital invariant, reflecting their geometric complexity and having consequences for the arithmetic density of rational points. This focus is motivated by conjectures suggesting that an increase in the number of prescribed pre-periodic points may force the moduli space into the category of varieties of general type, whose rational points are expected to be sparse.

Methodology: Portrait Classification and Moduli Construction

The approach in the paper is to classify the moduli spaces $M_{2,1,\mathcal{P}}$ according to the portrait $\mathcal{P}$, which is a pre-periodic structure determined by partitions of up to six points into cycles. The construction of the moduli space uses explicit algebraic conditions: for a tuple $(f, p_1, \ldots, p_n)$, $f(p_i) = p_{\mathcal{P}(i)}$ with all points distinct and the resultant of $f$ non-zero. The geometric quotient is taken under the action of the automorphism group $Aut(\mathbb{P}^1)$.

For $M_{d,N,\mathcal{P}}$0, the moduli spaces are shown to be rational surfaces or empty, utilizing coordinate reductions, explicit equations in affine and projective space, and resultant calculations. In the case $M_{d,N,\mathcal{P}}$1, the complexity increases substantially, requiring models as hypersurfaces in $M_{d,N,\mathcal{P}}$2 and, for more structured portraits, as covers branched over discriminant loci, leveraging blowup constructions to resolve singularities.

The algebraic geometry is rigorous: the singular loci are identified, and resolutions are constructed via blowups. The canonical class computations, adjunction formula applications, and nef/bigness arguments establish the Kodaira dimension for each case. For model surfaces, elliptic fibrations and double covers are constructed and analyzed, and the explicit calculation of branch loci and discriminants is performed.

Main Results: Kodaira Dimension Spectrum

The principal achievement is the comprehensive classification of moduli spaces $M_{d,N,\mathcal{P}}$3 for portraits involving up to six points:

• For $M_{d,N,\mathcal{P}}$4 points: All non-empty moduli spaces are rational surfaces ($M_{d,N,\mathcal{P}}$5).
• For $M_{d,N,\mathcal{P}}$6 points (excluding pure cycles previously classified):
  • $M_{d,N,\mathcal{P}}$7: For portraits $M_{d,N,\mathcal{P}}$8, corresponding to cubic weak del Pezzo surfaces.
  • $M_{d,N,\mathcal{P}}$9: For $d$0, shown to be a $d$1 surface via explicit resolution and canonical divisor calculation.
  • $d$2: For $d$3, $d$4, and $d$5, established as elliptic surfaces with elliptic fibrations arising from pencil systems.
  • $d$6: For $d$7, where blowups yield general type surfaces with $d$8.

These results demonstrate that all possible values of Kodaira dimension for surfaces ($d$9) are attained among moduli spaces for degree two maps with six-point portraits.

The paper further provides strong corollaries: if a portrait includes a six-cycle or a $\mathbb{P}^N$0 subportrait, the associated moduli space is of general type. Additionally, assuming general type emerges for sufficiently large portraits, moduli spaces with $\mathbb{P}^N$1 points are guaranteed to be of general type due to partition restrictions.

Implications and Theoretical Significance

The explicit construction and resolution of moduli spaces with portrait data up to six points give critical insight into the structure and arithmetic of dynamical systems. The realization that six points are insufficient for a universal general type result, contrary to conjectural expectations, forces a revision in thresholds for the transition to general type in the dynamical moduli context.

The geometric classification has arithmetic consequences: varieties of general type are expected to have finitely many rational points, and so the density of dynamical systems with prescribed portrait structure among rational maps is tightly constrained by the geometric type of the associated moduli spaces. These findings reinforce the link between birational geometry and arithmetic dynamics, emphasizing how explicit moduli constructions inform uniformity conjectures.

The analytical techniques—explicit surface equations, resolution via blowups, branch locus analysis, and canonical divisor calculations—set a technical framework for similar computations in higher-degree or higher-dimensional cases. The correspondence between partitions and geometric complexity suggests avenues for systematizing moduli space geometry as a function of portrait structure.

Future Directions

The classification achieved for $\mathbb{P}^N$2, $\mathbb{P}^N$3, and $\mathbb{P}^N$4 portraits opens questions for larger $\mathbb{P}^N$5, higher degree $\mathbb{P}^N$6, and higher-dimensional projective varieties $\mathbb{P}^N$7. The conjecture that sufficiently large portraits always yield general type moduli spaces needs resolution for explicit $\mathbb{P}^N$8, $\mathbb{P}^N$9, and $M_{2,1,\mathcal{P}}$0, with the transition value for $M_{2,1,\mathcal{P}}$1 remaining an important open problem.

Further work could extend these resolutions and classifications to families of polynomial maps, higher-dimensional endomorphisms, and weighted portraits. The interaction between the combinatorics of portraits and the geometry of moduli spaces, especially via covering maps and discriminant loci, is ripe for exploration with modern computational algebraic geometry tools.

Technically, advancing the understanding of branch loci, singularity resolutions, and fibration structures in moduli spaces of dynamical systems is crucial for unifying geometric invariant theory with computational dynamical systems.

Conclusion

This paper achieves a comprehensive algebraic-geometric classification of moduli spaces for degree two rational self-maps of $M_{2,1,\mathcal{P}}$2 with prescribed pre-periodic portraits up to six points. The geometric complexity, measured by the Kodaira dimension, is shown to range across the full spectrum for surfaces, with explicit criteria and models constructed for each portrait structure. These results substantially clarify the landscape of dynamical moduli spaces and their arithmetic implications, and they provide a technical foundation for future work in the interplay between algebraic geometry and dynamical systems in arithmetic contexts.