24 problems

Grothendieck's section conjecture. Every section $s_X\in\mathcal{S}(\pi_1(X)/\pi_1(k))$ arises from a point $x\in X(k)$, and

Grothendieck's section conjecture. The section map

Extension conjecture. Every $\rho\in{\sf Fib}_k(U)$ extends to a functor in ${\sf Fib}_k(V)$.

Section conjecture reformulated. The map

Grothendieck's section conjecture. If $k$ is of finite type over $\mathbb{Q}$, then: (SC1) the map

General Section Conjecture. Both inclusions

Toric $p$-adic section conjecture. This map is bijective. The statement is supported by the paper's proof of the corresponding result for torsors under abelian varieties over finit…

Toric section conjecture. This map is bijective. The conjecture is proposed as an intermediate step toward Grothendieck's section conjecture; the paper proves injectivity of the na…

Grothendieck's section conjecture. This map is bijective. Injectivity is known, while surjectivity remains widely open.

Existence conjecture. There is a Kodaira fibration with no topological section. The source motivates this by the universal family of curves, which has no topological sections, but…

Weak topological section conjecture. The fibration $f$ admits an algebraic section if and only if its short exact sequence of fundamental groups splits. This is introduced as a con…

Let $X$ be a smooth projective curve of genus at least $2$ over a number field. A section of the fundamental exact sequence of $X$ is locally geometric if, at every place of the nu…

Let $g>2$ and let $k$ be a field. For each boundary stratum of the Deligne–Mumford compactification $\overline{\mathscr{M}_g}$ indexed by a stable graph $\Gamma$, let…

Let $X$ be a hyperbolic curve over a field $k$ finitely generated over $\mathbb{Q}$, let $U\subseteq X$ be an open subset, and let $s$ be a Galois section of $X$. Cuspidalization c…

Let $k$ be a field finitely generated over $\mathbb{Q}$, let $X$ be a smooth, geometrically connected hyperbolic affine curve, and let $\widehat{X}$ be its infinite root stack, obt…

Let $k$ be a field finitely generated over $Q$, let $X$ be a smooth, projective, hyperbolic curve over $k$, and let $L$ be a field finitely generated over $k$. Write…

Let $K$ be a field of characteristic $0$, let $X$ be a hyperbolic curve over $K$, and choose a geometric point $\ast$ of $X_{\overline{K}}$. Write…

Let $k$ be finitely generated over $\mathbb{Q}$, let $X$ be a smooth geometrically connected curve with smooth completion, and let $\Pi_{X/k}(k)$ denote the set of sections over…

Let $k$ be finitely generated over $\mathbb{Q}$. Let $T/k$ be a smooth variety, and let $X/k$ be a smooth, proper, geometrically connected hyperbolic curve. Grothendieck's Hom conj…

Let $X/k$ be a smooth, geometrically connected hyperbolic curve, let $K$ be its function field, and let $\overline X$ be its smooth compactification. A section of $\pi_1(X/k)$ is c…

Let $k$ be a number field or a finite extension of $\mathbb Q_p$, and let $X/k$ be a smooth, projective and geometrically connected curve of genus at least $2$, with function field…

Let $k/\mathbb Q_p$ be a finite extension, and let $X/k$ be a smooth, projective and geometrically connected curve of genus at least $2$. A rational point $a\in X(k)$ determines a…

Let $k$ be a number field, let $X$ be a smooth projective curve over $k$ of genus at least $2$, and fix a geometric point $\bar x$ of $X$. Consider the natural exact sequence … A c…

Let $X$ be a smooth, projective, geometrically connected curve over a field $k$ with algebraic closure $k^{\rm alg}$, and let … be its étale fundamental-group sequence. Write…