152 problems

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…

Coolidge–Nagata conjecture. Every rational cuspidal curve can be mapped onto a line by a Cremona transformation of the plane.

Hassett–Tschinkel's conjecture. These are all the extremal rays of the cone of curves.

Oguiso's conjecture. $X$ contains a rational curve.

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 $X_{cSigma}$ be a smooth toric variety satisfying conditions $({\rm homogenous}.1)$ and $({\rm homogenous}.2)$, and let $D=(d_1,\ldots,d_r)$ be a multidegree. Define … where…

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…

Let $X\subset\mathbb P^n$ be a very general hypersurface of general type, and measure the degree of a rational curve on $X$ with respect to the hyperplane polarization. Voisin's co…

Rational-curve family conjecture. If $d\leq n-1$ and $n>3$, then the family of rational curves of degree $e$ on $X$ is irreducible and has the expected dimension.

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 $M_{\alpha}$ be an irreducible component of the moduli space of maps from a curve $B$ to a variety $X$, and let $\operatorname{Top}(B,X)_{\alpha}$ be the corresponding space of…

Batyrev's conjecture. There exists a polynomial $P(d)$ in $d$ such that the number of components of $\operatorname{Mor}(\mathbb P^1,X)$ parametrizing rational curves of $L$-degree…

Movable Bend and Break conjecture. There exists a constant $Q(\mathcal X)$ such that, whenever

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

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 $X\subset {\bf P}^n$ be a general hypersurface of degree $d$, and let $\overline{M}_{0,0}(X,e)$ be the coarse moduli space of stable maps from rational curves of degree $e$ to…

Let $F$ be a holomorphic symplectic fourfold deformation equivalent to the Hilbert square of a K3 surface, and suppose its Picard lattice contains two nodal classes $\rho_1$ and…

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…

A variety is unirational if it admits a dominant rational map from some projective space, and rationally connected if two general points can be joined by a rational curve. Uniratio…

Let $\gamma:\mathbb{CP}^1\to\mathbb{CP}^n$ be a rational curve. A point $t\in\mathbb{CP}^1$ is a flattening point of $\gamma$ if its osculating frame formed by…

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 $X$ be a smooth projective variety. A variety is uniruled if through a general point of it there passes a rational curve, and its Kodaira dimension is denoted by $\kappa(X)$. U…

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…