10 problems

An unmixed Beauville structure is a pair of generating triples satisfying the required Beauville disjointness condition. Beauville's conjecture. Every finite non-abelian simple gro…

A strongly real Beauville group is a group admitting a Beauville structure satisfying the strong reality condition: there are an automorphism and conjugating elements that simultan…

A characteristically simple group is a finite group isomorphic to $H^k$, where $H$ is finite simple and $k$ is a positive integer. The characteristically simple Beauville group con…

Let $\mathfrak M$ be the disjoint union of the coarse moduli spaces $\mathfrak M_{x,y}$ of canonical models of surfaces of general type with $\chi(\mathcal{O}_X)=x$ and $K_X^2=y$,…

Alternating-group Beauville structure conjecture. Let $(r,s,t)$ and $(r',s',t')$ be two hyperbolic types. Then almost all alternating groups ${\mathcal{A}}_n$ have an unmixed Beauv…

Call $(r,s,t)\in{\mathbb N}^3$ hyperbolic when … For an unmixed Beauville structure $v=(a_1,c_1;a_2,c_2)$, its two generating systems have types $(r,s,t)$ and $(r',s',t')$, respect…

A non-abelian finite simple group is a finite simple group that is not abelian. The finite simple Beauville group conjecture. Every non-abelian finite simple group other than the a…

Let $c\tau_1=(r_1,s_1,t_1)$ and $c\tau_2=(r_2,s_2,t_2)$ be two hyperbolic types. Let $G$ be a finite simple classical group of Lie type, with Lie rank sufficiently large. Fixed-typ…

Let $q$ be a prime with $q\equiv -1$ modulo $4$ and $q\geq 11$. A primitive root $d$ of $F_q$ is called suitable when $d-1+d^{-1}$ is a square in $F_q^*$ (equivalently, $d^2-d+1$ i…

Classical-group Beauville structure conjecture. If the Lie rank of $G$ is large enough, then $G$ admits an unmixed Beauville structure $(x_1,y_1;x_2,y_2)$ such that