40 problems

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

Let $X$ be a smooth projective variety over the complex numbers. Its holomorphic Kodaira dimension is denoted by $\kappa^h(X)$, and $X$ is uniruled if through a general point of…

Uniruledness conjecture. If $K_X$ is not $\mathbb Q$-effective, then $X$ is uniruled.

The affine uniruledness conjecture.

For a projective variety $Z$, let $\kappa(Z)$ denote its Kodaira dimension, and say that $Z$ is uniruled if through a general point of $Z$ there passes a rational curve. Uniruledne…

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 $f:X\to C$ 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 $X$ be a special manifold of dimension $n\geq 2$. A manifold is strongly irreducible special when every covering family with special general member is trivial in the sense defi…

Let $X$ be a smooth projective variety. Its $m$th plurigenus is $h^0(mK_X)=\dim_k H^0(X,\mathcal O(mK_X))$, where $K_X$ is the canonical class. A variety for which all plurigenera…

Non-uniruled sparsity conjecture. For every number field $E$ and every $0<\epsilon$, there is a subvariety $A\subsetneq X$ such that

The moduli space $\mathcal{M}_{{\operatorname{OG10}}}^{d,\gamma}$ parametrises polarised irreducible holomorphic symplectic manifolds of OG10 type, of degree $d$ and divisibility…

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 $(X,D)$ be a smooth projective simple normal crossing pair and set $U=X\backslash\operatorname{Supp}D$. A quasi-projective variety is $\mathbb{A}^1$-uniruled if it is covered b…

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…

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.

Uniruledness conjecture. If $M$ is a compact Kähler manifold, or even a general compact complex manifold, then uniruledness implies negative Kodaira dimension:

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…

Degree-bound conjecture. The set $S_f$ has degree of $\mathbb{K}$-uniruledness at most $d-1$.

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…