22 problems

Let $f:X\to C$ be a fibration over a function field, let $r_K:X\to R(X_K)$ be its rational quotient, and call $X_K$ dominated when it admits the dominant product-type rational map…

Let $(X,\Delta_\epsilon)$ be a klt Campana orbifold over a field of characteristic $0$. It is strongly Campana uniruled when there exists a free Campana curve $f:C\to X$ such that…

Let $(M,\omega)$ be a compact Kähler manifold of arbitrary dimension. A compact complex manifold $M$ is uniruled if every point of $M$ lies on a rational curve, meaning a smooth co…

BDPP conjecture. The canonical class $K_X$ is pseudoeffective if and only if $X$ is not uniruled.

Let $X$ be a smooth projective variety, and let $f:X\to\mathbb{P}^n$ be a surjective morphism. Let $\Delta(f)$ denote the discriminant locus of $f$. Higher-dimensional uniruledness…

Let $X$ be a smooth projective variety, and let $f:X\to\mathbb{P}^1$ be a surjective morphism with at most $2$ singular fibers. Two-singular-fiber uniruledness conjecture. Then $X$…

Let $X$ be a smooth projective variety of dimension $n$ over $\a0\mathbb{C}$, with canonical divisor $K_X$, and let $L$ be an effective divisor. Say that adjunction terminates in t…

Uniruledness conjecture. The Kodaira dimension of $S$ is $-\infty$. In particular, if $S$ is a projective manifold, then $S$ is uniruled.

The affine uniruledness conjecture.

Kollár's conjecture. A general member $H\in |H|$ is not uniruled.

Hamiltonian no-torsion uniruledness conjecture. Each closed symplectic manifold with non-trivial Hamiltonian torsion must be uniruled.

Let $X_d\bsubseteq \mathbb{P}^{n+1}$ be a very general hypersurface of degree $d\geq n+3$, and let $Y$ be a smooth projective $n$-fold that is non-uniruled. Let $H_d:=X_d\cap H$, w…

Let $X$ be a projective manifold. Mumford's equivalent-conditions conjecture. If … for every $m\geq 1$, then each of the following holds: $X$ is uniruled; $K_X$ is not pseudo-effec…

Let $X$ be a projective manifold, let $K_X$ be its canonical bundle, let $K_X^{-1}$ be its anticanonical bundle, and let $\kappa(X)$ denote its Kodaira dimension. Uniruledness conj…

Let $X$ be a compact Kähler manifold. Say that $X$ is uniruled if it is covered by rational curves, and let a smooth Kähler metric mean a smooth positive Kähler form on $X$. Kähler…

Pseudoeffectivity–uniruledness conjecture. The canonical class $K_X$ is pseudoeffective if and only if $X$ is not uniruled.

Uniruledness conjecture. $X$ is uniruled, meaning covered by rational curves, if and only if

Let $X$ be a compact Kähler manifold and let $[\alpha]$ be a nef and big $(1,1)$ class which is not Kähler, with $[\alpha]+\lambda c_1(X)$ a Kähler class for some $\lambda>0$. The…

Pseudoeffectivity and uniruledness conjecture. The canonical class $K_X$ is pseudoeffective if and only if $X$ is not uniruled.

Let $X$ be a non-uniruled projective manifold. A vector bundle $E$ on $X$ is sufficiently nef if, for every $x \in X$, there is a family of curves through $x$ covering $X$ such tha…

Let $X$ be a projective manifold, let $L$ be a numerically trivial line bundle, and let $v \in H^0(X,\bigwedge^q T_X \otimes L)$ be a non-trivial section. A zero of $v$ is a point…

Let $Y$ be a smooth projective variety. A variety is uniruled if through a general point of $Y$ there passes a rational curve. Abundance conjecture. If $\kappa(Y)=-\infty$, then…