8 problems

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…

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 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 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…

Let $X$ be a smooth projective variety, and let $\kappa^h(X)$ denote its holomorphic Kodaira dimension. Mumford's conjecture. If … then $X$ is uniruled. This is the converse to the…

MRC-fibration pseudoeffectivity conjecture. One has

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