37 problems

Let $\mathscr{H}_{k,g}$ be the Hurwitz stack parametrizing degree $k$ covers by genus $g$ curves of $\mathbb{P}^1$, up to automorphisms of the target $\mathbb{P}^1$, and let…

Let $\overline{\mathcal H}$ be the compactified Hurwitz space, and let its irreducible boundary components be the irreducible components of the complement of the locus parametrizin…

Tautological Picard group conjecture. The rational Picard group of $\widetilde{\mathcal H}^{\dagger}_{d,g}$ is

Weak Main Conjecture. There are no rational points at suitably high tower levels.

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 $1$.

No-Weigel-branch conjecture. There are no Weigel cusp branches on any infinite $K$ component branch of a modular tower.

g-$p'$ conjecture. Each $K$ component branch on a modular tower is defined by some g-$p'$ cusp branch.

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…

The isomonodromic tau-function is associated with semisimple Frobenius manifolds on Hurwitz spaces of branched coverings of genus $0$ and $1$. Existing formulae used auxiliary stru…

Let $G$ be a finite group, let $p$ be a prime divisor of $|G|$, let ${\underline C}_1$ be an unordered $r$-tuple of conjugacy classes of $G$ of order prime to $p$, and let $G_n$ an…

Let $c$ be a rack, let $G^c_c$ denote the group appearing in the definition of the associated Hurwitz space, and let $\Hur^c$ denote that Hurwitz space over $\mathbb C$. Moduli int…

Let $\mathscr{H}_{k,g}$ be the moduli space of simply-branched, degree-$k$, genus-$g$ covers of smooth genus-zero curves, and let $\overline{\mathscr{H}}_{k,g}$ be its admissible c…

Let $\operatorname{Hur}_{n,b}$ denote a Hurwitz scheme parametrizing branched degree $n$ covers of $\mathbb{P}^1$ with $b$ branch points, together with the relevant monodromy and r…

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…

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 $X$ be an elliptic curve, and let $\underline{\mathsf{Hur}}_{d,g}(X)$ denote the relevant Hurwitz stack of degree-$d$ covers of $X$ by genus-$g$ curves. A stack $Y$ has plenty…

Ellenberg–Venkatesh–Westerland's conjecture. For fixed $d$ and sufficiently large $g$, this map induces an isomorphism on $i$th homology. A slight variant in the cited source uses…

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…

Bujokas–Patel's conjecture. The multiset of splitting types of the syzygy bundles of a relative minimal resolution of a sufficiently general element of $\mathcal{H}_{d,g}$ is balan…

Eisenbud–Schreyer conjecture. The subscheme $\Sigma_{\vec{e}}$ is reduced.