
## 4. Geometry and Topology of Moduli Spaces

1. #### Problem 4.05.

Is the moduli space $\mathcal{M}_d^N:={\rm{End}}^N_d/{\rm{PGL}}_{N+1}$ rational?
1. Remark. Known for $N=1$ [MR2741188].
• #### Problem 4.1.

Is $\mathcal{M}_d^N[n]$ (the moduli space of maps with marked points of period $n$) a variety of general type for $n\geq C(d,N)$?
1. Remark. $\mathcal{M}_2^1[6]$ is of general type [MR3431627].
• #### Problem 4.15.

[M. Manes and J. Silverman] Does the gonality of $\mathcal{M}_d^N[n]$ tend to infinity as $n\to\infty$? How about the moduli spaces $\mathcal{M}_d^N[\mathcal{P}]$ where $\mathcal{P}$ is a portrait?
1. Remark. The gonality of $X$ is defined to be the smallest degree of a dominant rational map $X\dashrightarrow\Bbb{P}^\ell$.
• #### Problem 4.2.

[K. Filom] Is the class of $\mathcal{M}_d^1$ the same as that of $\Bbb{A}^{2d-2}$ in the Grothendieck ring of varieties?
1. Remark. [D. Bejleri] Working in the closely related Grothendieck ring of stacks, the answer is recently shown to be positive for $d$ even [arXiv:2107.12231].
• Remark. $\mathcal{M}_d^1$ and $\Bbb{A}^{2d-2}$ are both rational varieties of dimension $2d-2$ [MR2741188], and they have the same point count over finite fields https://bit.ly/2WuJeAi.
• #### Problem 4.25.

[K. Filom] Compute Betti numbers and the fundamental group of $\mathcal{M}_d^N(\Bbb{C})$.
1. Remark. Known for $N=1$ [arXiv:1908.10792].
• #### Problem 4.3.

[H. Krieger] Understand “dynamically significant” “fibrations” $\mathcal{M}_d^N\rightarrow\Bbb{A}^{m}$ (singular fibers allowed), e.g. those defined by multipliers.
• #### Problem 4.35.

[J. Silverman, M. Weinreich] Obtain an dynamical understanding of $(\mathcal{M}_d^N)^{\rm{ss}}-\mathcal{M}_d^N$ and $(\mathcal{M}_d^N)^{\rm{unstable}}:=\Bbb{P}^M-(\mathcal{M}_d^N)^{\rm{ss}}\quad (M=\binom{N+d}{d}(N+1)-1)$.
• #### Problem 4.4.

[D. Bejleri] What is the equivariant cohomology of the semi-stable locus $(\mathcal{M}_d^N)^{\rm{ss}}$?
• #### Problem 4.45.

[D. Bejleri] What are some non-GIT compactifications of $\mathcal{M}_d^N$?
1. Remark. Measure-theoretical treatments of $\mathcal{M}_d^N$?
• Remark. Another compactification: [MR3711376].
• #### Problem 4.5.

[H. Moon] Understand different GIT quotients for $\mathcal{M}_d^N[n]$.
• #### Problem 4.55.

[J. Doyle] Are distinguished curves in $\mathcal{M}_d^N$ or $\mathcal{M}_d^N[n]$ (e.g. dynatomic curves) irreducible or smooth?

Cite this as: AimPL: Moduli spaces for algebraic dynamical systems, available at http://aimpl.org/modalgdyn.