19 problems

Uniform non-modularity conjecture. The same conclusion holds for every $p\neq 3$.

Mazur–Merel version of the Strong Main Conjecture. Assuming the hypotheses of the Strong Main Conjecture, for all $p\notin P_{\mathbf C}$, only finitely many $V\in\mathcal V_p(J)$…

Strong Main Conjecture. For all $p\notin P_{\mathbf C}$, only finitely many $V\in\mathcal V_p(J)$ give a space $\mathcal H(V\times^s J,\mathbf C)^{\mathrm{in},\mathrm{rd}}$ having…

High-level genus conjecture. For large $k$, all components of $\overline{\mathcal H}_k$ have genus exceeding $1$.

Let ${\mathbb P}{\cal H}_n$ be the projectivized Hurwitz space. Let $\Psi_n,\Delta_n\in H^2({\mathbb P}{\cal H}_n)$, where $\Psi_n$ is the distinguished class used for intersection…

Let ${\mathbb P}{\cal H}_n$ be the projectivized Hurwitz space, and let $C_n$ and $M_n$ be the cohomology classes of the caustic and Maxwell stratum, respectively. Consider the sub…

Let $g\geq2$ be an integer and let $G$ be a finite group. Let $\operatorname{CHur}_g^G$ be the algebraic stack over $\mathbb C$ whose objects are geometrically connected finite éta…

Let $d>0$ and let $n$ be even with $n\geq 2d-2$. Let $c\subset S_d$ be the conjugacy class of transpositions, and let…

Stable Chow groups conjecture. The $i$th rational Chow groups of $\operatorname{CHur}^{G,c_1,\ldots,c_k}_{n_1,\ldots,n_k}$ stabilize, in the sense that they are independent of…

Stable homology conjecture. The only stable homology of these Hurwitz spaces should be the obvious stable homology coming from the configuration space. The conjecture is known for…

Let $k$ be a field of characteristic not dividing $6$, let $g$ be the genus, and let $\mathrm{Hur}_{3,g,k}$ be the degree-$3$ Hurwitz stack. Write $\mathbb L$ for the class of the…

Fix an integer $g\ge2$. Let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$, and for each $d\ge1$ let $\phi_{g/1,d}$ be the morphism from the compactif…

Let $\alpha$ and $\beta$ be permutations in $S_n$. Write $[\alpha,\beta]=\alpha\beta\alpha^{-1}\beta^{-1}$ for their commutator, and let $\alpha^\gamma=\gamma^{-1}\alpha\gamma$ den…

Let $\mathscr{H}_{g,k}$ denote the Hurwitz space of degree-$k$ covers $C\to\mathbb{P}^1$, and let $\rho(g,k,1)$ be the Brill–Noether number. For a general cover, let $N_1$ be the f…

Work over an algebraically closed field of characteristic zero. Let $\widetilde H_{d,g}$ be the Hurwitz space of degree-$d$ covers of smooth genus-$0$ curves by irreducible genus-…

Fix $n\geq 1$. Let $N(k)$ be the number of isomorphism classes of marked degree-$n$ extensions of $\mathbb{F}_q(t)$ with squarefree discriminant of degree $k$, totally split above…

Let $G$ be a symmetric group on more than two letters, and let $c$ be the conjugacy class of transpositions. For each integer $i\geq 2$, consider the rational homology of the conne…

Let $\bar O_k$ be a component at level $k$, and let ${{\mathcal H}}^{{\text{rd}}}_{\bar O_k}$ denote the corresponding reduced Hurwitz-space component. Orbit shortening is the redu…

Assume the Main Conjecture hypotheses, and let $\bar O_k$ be any component at level $k$. Elliptic ramification from $\bar O_k$ to a component at level $k+1$ means the ramification…