15 problems

A projective uniruled $3$-fold is a projective uniruled three-dimensional variety. A trivial $\mathbb P^1$-bundle is a product with fiber $\mathbb P^1$, and a conic bundle is a con…

Let $X$ be a smooth projective variety. A variety is uniruled if there are a variety $B$ of dimension $\dim X-1$ and a dominant rational map…

Let $X$ be a smooth complex projective variety. The numerical characterization conjecture. $X$ is uniruled if and only if … for every $m\geq 1$. This conjecture is presented as a r…

Kollár's conjectures. 1. If $M$ is an orientable Seifert fibered manifold, then there is a uniruled algebraic variety $X$ such that $M$ is diffeomorphic to a connected component of…

Let $X$ be a smooth projective variety. Negative Kodaira dimension conjecture. If $X$ has negative Kodaira dimension, then $X$ is uniruled. The source says that a positive answer i…

Let $X$ be a projective variety with at most terminal $\mathbb Q$-factorial singularities. Minimal model conjecture. There exists a minimal model $X'$ birational to $X$ if and only…

Let $K$ be a complete discrete valued field with equal characteristic $0$, and let $\mathcal{X}$ be a uniruled variety over $K$. Say that a proper geometrically integral smooth $K$…

Let $X$ be a smooth, complex projective variety. Its Kodaira dimension is denoted by $\kappa(X)$, and $X$ is uniruled if through a general point of $X$ there passes a rational curv…

Let $Y$ be a variety, and let $\kappa(Y)$ denote its Kodaira dimension. A variety is uniruled if rational curves cover it. Non-vanishing conjecture. If $Y$ is not uniruled, then ……

Let $X$ be a smooth complex projective manifold. If $K_X$ is pseudoeffective, write $\nu_1(X):=\nu(X,\Omega_X^1)$. If $X$ is uniruled, let $r:X\to R_X$ be its maximal rationally co…

MRC-fibration pseudoeffectivity conjecture. One has

Weak abundance conjecture. $X$ is uniruled if and only if it has negative Kodaira dimension:

Uniruled-variety conjecture. Every uniruled variety admits a Kähler metric with

The converse to separable uniruledness implying negative Kodaira dimension. If $X$ has negative Kodaira dimension, then $X$ is separably uniruled, at least in characteristic zero.

Let $X$ be a smooth projective variety over a field of characteristic zero. The plurigenus characterization conjecture. $X$ is uniruled if and only if all its plurigenera vanish: ……