
## 1. Automorphism Loci

1. #### Problem 1.1.

[L. DeMarco and J. Silverman] Given a subgroup $G$ of ${\rm{PGL}}_{N+1}$, we define $\mathcal{M}_d^N[G]$ to be the Zariski closed subset defined by maps whose automorphism group contains an isomorphic copy of $G$. Investigate these loci.
1. Remark. The automorphism locus and its stratification is thoroughly studied in the case of $N=1$ and an algebraically closed field of characteristic 0 [MR3709645]. There are also results in positive characteristic [arXiv:2003.12113].
• Remark. If the finite group $G$ can be embedded in ${\rm{PGL}}_{N+1}$, then $\mathcal{M}_d^N[G]$ becomes non-empty for some $d$ [MR3846410].
• #### Problem 1.2.

[J. Silverman] What is the class determined by $\mathcal{M}_d^N[G]$ in the homology?
• #### Problem 1.3.

[M. Weinreich] What are possible automorphism groups for maps at the boundary of $(\mathcal{M}_d^N)^{\rm{ss}}$?
1. Remark. Can be infinite [MR4045425].
• #### Problem 1.4.

[D. Bejleri] What is the essential dimension of $\mathcal{M}_d^N[G]$?
1. Remark. The essential dimension is the least transcendence degree of a field of definition.
• #### Problem 1.5.

[K. Filom] Relate the singular locus of $\mathcal{M}_d^N$ to the automorphism locus $$\{\langle f\rangle\in \mathcal{M}_d^N\mid {\rm{Aut}}(f) \text{ is non-trivial}\}$$ which is where the conjugation action fails to be free.
1. Remark. For $N=1$ and $d>2$, the automorphism locus of $\mathcal{M}_d^N(\Bbb{C})$ (i.e. the orbifold locus) coincides with its singular locus (i.e. the subset of non-manifold points) [MR3709645].

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