91 problems

Let $X\subset \mathbb{P}^n$ be a general hypersurface of degree $d\leq n$ and dimension at least $3$. Let $R_e(X)$ be the open locus in the Hilbert scheme of $X$ parameterizing smo…

Let $A$ be a smooth unbendable rational curve in a complex manifold $X$. The curve $A$ satisfies the formal principle: whenever a compact complex submanifold $\widetilde{A}$ in a c…

Let $\pi:\mathcal{X}\to B$ be a good Fano fibration. An intersection profile $\lambda$ specifies one generically smooth, geometrically integral irreducible component in each fiber,…

McKinnon's conjecture. There exists a rational curve $C$ on $X$ passing through $Q$ achieving the best approximation constant at $Q$ with respect to $L$.

Let $C$ be a rational cuspidal plane curve of degree $d$. The source identifies three specific series of tricuspidal curves in a table. Piontkowski's conjecture. If $d\geq 6$, then…

Generic non-jumping-line conjecture. The generic line is not jumping.

Let $(X,L)$ be a very general polarized $K3$ surface. A rational curve in $|L|$ is a curve of geometric genus $0$. Nodal rational curves conjecture. All rational curves in $|L|$ ar…

Let $\overline{M}_{0,n}$ be the moduli space of stable rational curves with $n$ marked points. A boundary divisor $\delta_I$ is associated to a partition of the markings into $I$ a…

Calabi–Yau rational curve conjecture in dimension $m$. The variety $X$ should contain a rational curve.

Let $X$ be a general hypersurface of degree $d$ in $\mathbb{P}^n$. Let $\overline{\mathcal{M}}_{0,0}(X,e)$ denote the Kontsevich space of degree $e$ stable rational maps to $X$. Di…

Let $S$ be a surface given in ${\bf{TK1}}$, ${\bf{TK2}}$, or ${\bf{TK3}}$. Let $F$ and $C$ denote the divisor classes used in these trigonal K3 surface constructions, and let $l$ b…

Let $S\subset {\mathbb P}^n$ be a primitive K3 surface, and let $d>0$. Consider the family of rational curves in the linear system $|{\mathcal O}_S(d)|$. Transitive monodromy conje…

Let $V$ be an algebraic variety defined over a field $k$, let $D$ be an ample divisor on $V$, and let $P\in V(k)$ be a $k$-rational point. Assume that there is a rational curve $C$…

Let ${\overline{\mathnormal{M}}}_{0,N}$ be the moduli space of stable $N$-pointed genus-zero curves. An $F$-divisor is a divisor that nonnegatively intersects every $F$-curve; let…

Let $X$ be a smooth projective variety over a finite field $k$. Assume that $X$ has trivial canonical class and that $X_{\bar{k}}$ has trivial algebraic fundamental group. An algeb…

Let $S$ be a K3 surface, and let $\Sigma=\bigcup D_\alpha$ be the union of all rational curves on $S$. Let $\overline{\Sigma}$ denote the closure of $\Sigma$ in the analytic topolo…

A maximally inflected plane quintic is a real rational curve of degree $5$ in $\mathbb{P}^2$ whose ramification occurs only at real points; a solitary point is an isolated real nod…

Let $d>r$ be integers and let $calpha^1,\ldots,\alpha^n$ be ramification data for degree $d$ rational curves in $\mathbb{P}^r$. Let $s_1,\ldots,s_n$ be distinct points of…

Mu-class closure conjecture. For every $\mu\leq\lfloor n/2\rfloor$,

Let $W$ be an arbitrary projective variety of dimension $n$. Assume that $W$ contains a rational curve and that every rational curve of $W$ moves in a family of dimension at least…

Cubic Calabi–Yau recursion conjecture. The recursive law is

Let $X\subset\mathbb{P}^4$ be a cubic threefold, let $Y\subset X$ be a hyperplane, and let $C\subset X$ be a smooth rational curve of degree $d$, with $X$, $Y$, and $C$ all general…

Let $X\subset\mathbb{P}^4$ be a generic quintic threefold. A rational curve means a curve on $X$ birational to $\mathbb{P}^1$, and its degree is measured with respect to the embedd…

Let $(X,\Delta)$ be a klt Fano pair over an algebraically closed field of characteristic $0$. A rational curve in $X$ is free when its pullback tangent bundle is nef. Free-curve co…

Let $k$ be a number field and let $X$ be a smooth projective geometrically irreducible variety over $k$. Let $Z_{rc}(X)\subseteq X$ be the Zariski closure of the union of all possi…