180 problems

For every connected complex reductive algebraic group $G$ and every involutive automorphism $\theta:G\to G$, let $H=G^\theta$ and let $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{s}$…

Let $A$ be a nonsingular projective $g$-dimensional variety over $\mathbb C$. For each prime $\ell$, let $G_{A,\ell}$ be the Zariski closure of the image of the $\ell$-adic represe…

Let $X$ be an affine variety of dimension at least $2$, and let … be generated by finitely many ${\mathbb G}_a$-subgroups. Suppose that $G$ acts $2$-transitively on one of its orbi…

Let $G$ be a reductive algebraic group over a field $k$ of characteristic $p$, let $G_rT$ be the corresponding Frobenius kernel together with a maximal torus, and let $X_r$ denote…

Let $G$ be a semisimple algebraic group over $k$, let $r\geq 1$, let $M$ be a rational $G$-module, and let $\operatorname{St}_r$ denote the $r$-th Steinberg representation. A good…

Let $X$ be an affine algebraic variety defined over an algebraically closed field, and let $G$ be a subgroup of $\operatorname{Aut}(X)$ generated by a finite collection of…

Let $R$ be a regular local ring containing a field $k$, and let $G$ be a smooth reductive algebraic group over $k$. Write $K$ for the field of fractions of $R$. Grothendieck–Serre…

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p>0$. For a positive integer $m$, let $j_m(G)$ be the dimension of the subvariety of eleme…

Let $K$ be a finitely generated field, let $X$ be a smooth projective variety over $K$, and consider a $\mathds{Q}$-compatible system of Galois representations … Let…

Let $k$ be the base field, and let $G$ be a simply connected, almost $k$-simple, $k$-isotropic algebraic group over $k$. Let $G(k)^+$ denote the abstract subgroup of $G(k)$ generat…

Let $S$ be a Noetherian scheme and let $G$ be an isotrivial reductive group over $S$, meaning that $G$ becomes split after a finite étale cover of $S$. A $G$-equivariant $S$-fiberw…

Let $G$ be an algebraic group, let $x,y\in G$, and let $\rho$ range over the irreducible rational representations of $G$. Steinberg's conjecture. The elements $x$ and $y$ are conju…

Let $S$ be a global excellent Dedekind scheme with function field $K$, and let $G$ be a smooth algebraic group over $K$. A Néron lft-model of $G$ is a smooth separated $S$-group al…

Dense-orbit and finite-orbit equivalence. Unless

Let $G=\operatorname{SL}_n(\mathbb{C})$, let $v=\operatorname{id}$ and $w=w_0$, and let $\twist_v^w$ be the twist automorphism of the open Richardson variety, preserving its totall…

Let $R$ be a regular local ring, let $K$ be its field of fractions, and set $U:=\operatorname{Spec} R$. Let $\mathbf G$ be a reductive group scheme over $U$, and let $\mathcal G$ b…

Let $X$ be an irreducible affine variety. For locally nilpotent derivations $\partial_1,\ldots,\partial_k$ of ${\mathbb K}[X]$, let … A Lie-algebra closure conjecture asserts that…

Let $R$ be a finitely generated commutative ring with unity that is infinitely generated as a $\b7\mathbb{Z}$-module. For $n\b7\geq 2$, let $\b7\mathbf{A}_n(R)$ be the corresponding…

Finite generation conjecture. The group $\Gamma$ is never finitely generated.

Let $G$ be a split semisimple algebraic group with Weyl group $W$, and let $L^{u,v}=N\overline{u}N\cap B_-vB_-$ be the reduced double Bruhat cell for $u,v\in W$. For a reduced word…

Katsylo's conjecture. The following properties are equivalent: $\dim V=\dim W$; and there exists a $G$-equivariant birational map

Let $H$ be an algebraic group over the ground field, and call it easy if every $h$ lies in the neutral connected component of its centralizer. The easy-group characterization conje…

Let $V$ be a symplectic vector space, let $X'\subset \P(V)$ be an irreducible Legendrian subvariety, let $g\in\mathfrak{sp}(V)$ be a nilpotent endomorphism, and let $m$ be an integ…

Let $G$ be a reductive group over a field $k$ of good characteristic, with Lie algebra ${\mathfrak g}$. Let ${\cal N}_1$ be the restricted nilpotent cone, let $e\in{\cal N}_1$, and…

Let $C$ be a smooth projective geometrically connected curve over $k$, of genus $g$, and let $G$ be a split semisimple connected algebraic group over $k$. Denote by…