127 problems

Let $X\subseteq\mathbb{P}^{4}$ be a threefold hypersurface of degree $d\geq 6$, and let $C\subseteq X$ be any curve. Griffiths–Harris conjecture. One expects … The strongest form o…

Let $(A,\theta)$ be a polarized abelian variety, and let $C\subset A$ be a smooth curve of genus $h$. Write $c$ for the gonality of $C$ and … for its $\theta$-degree. Gonality–genu…

Abramovich–Harris conjecture. All potentially dense sets of points of degree $d$ on $C$ should be geometrically explained by a morphism of degree $d$ from $C$ to a curve of genus…

Let $g\geq 2$ be an integer. A smooth curve is a curve defined over $\mathbb{Q}$, and its rational points are its points over $\mathbb{Q}$. Uniformity Conjecture over $\mathbb{Q}$.…

Grothendieck's section conjecture. The section map

Grothendieck's section conjecture. The natural map $C(k)\to\Pi_{C/k}(k)$ is bijective.

Generalized Sato–Tate conjecture. Curves of fixed genus $g$ are classified into certain families, and the quantities $a_k(m;g)$ are the same for all curves in each family. Moreover…

Let $C$ be a pointless curve of genus at least $2$ over a number field $k$, let $J$ be its Jacobian, and let $C({\mathbb A}_k)^{\operatorname{PS}}$ denote the Poonen–Scharaschkin s…

Let $f:S\to\mathbb P^1$ be a nontrivial fibration of semistable curves of genus $g\geq 1$, and let $s$ be the number of its singular fibers. Beauville's conjecture. If $g\geq 2$, t…

A closed smooth curve is a smooth embedded curve in three-dimensional space. Steinhaus's conjecture. Every closed smooth curve in three-dimensional space has a pair of parallel tan…

Let $X\subset \mathbb{P}^4$ be a generic hypersurface of degree $d$, and let $Y$ be a desingularization of a one-dimensional subvariety of $X$; write $g$ for the genus of $Y$. For…

Mistretta–Stoppino's conjecture. If

Let $k$ be a number field, and let $C/k$ be a curve of positive genus with no $k$-rational points. A potential Hasse principle failure (PHPF) is a variety for which there exists an…

Let $C$ be a smooth projective curve embedded by a sufficiently positive line bundle $L$, and let $\Sigma_k$ denote its $k^{\text{th}}$ secant variety. Property $N_{k+2,p}$ means t…

Sander's conjecture. Nontrivial rational points exist only when $(k,\ell)=(2,2)$.

Let $F$ be a field, let $B$ be a quaternion algebra over $F$, and let $g>1$. A curve $C/F$ is $B$-modular if there is a Shimura curve $X_K$ associated with a suitable compact open…

Let $X$ be a smooth projective variety of general type with canonical line bundle $K_X$. Let $C$ be a curve in $X$ of geometric genus $g$, and let $\operatorname{deg}_C(K_X)$ denot…

Diagonal dimension conjecture. One has

Bogomolov conjecture. There should exist an $\epsilon>0$ such that $C(\epsilon)$ is finite.

Let $k$ be an algebraically closed field of characteristic $p$, and let $k[[t]]$ be the formal power series ring. Local Oort conjecture. Every cyclic extension of $k[[t]]$ lifts to…

Clifford-dimension conjecture. One has

Boundary correspondence conjecture. A one-dimensional boundary component $B$ of $\overline{\mathcal M}$ of type a, b, c, or d corresponds to a boundary component of type IIa, IIb,…

Let $\gamma$ be a closed curve in $\mathbb{R}^3$ of length $L$, with curvature $\kappa$. Let $\mu_1$ denote the first eigenvalue of the periodic Sturm–Liouville operator … on the i…

Local-field curve-index conjecture. (a) For all $g$ and $I\mid g-1$, if $K$ admits a totally ramified cyclic extension of degree $I$, then there exists a genus-$g$ curve $C_{/K}$ w…

SNC index specialization conjecture. Under these assumptions,