27 problems

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $\lambda$ be an intersection profile, and let $\mathsf{S}_{\lambda,\mathbb Z}$ be the corresponding integral section classes…

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $\lambda$ be an intersection profile, and let $c(\mathcal X,\lambda)$ be the leading constant in the all-height Manin conjec…

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $\lambda$ be an intersection profile, let $M_{\alpha}$ be the unique Manin component for sufficiently large…

Let $\pi:\mathcal X\to B$ be a good Fano fibration and let $\mathsf Z\subset\mathcal X_{\eta}(K(B))$ be the exceptional set. A properly constructible thin set is a finite union of…

Let $\pi:\mathcal X\to B$ be a good Fano fibration, let $q$ be the cardinality of the constant field, and suppose that $\mathcal X_{\eta}(K(B))$ is not thin. With…

Let $\pi:\mathcal{X}\to B$ be a good Fano fibration, let $\mathcal{X}_{\eta}$ be its generic fiber, and let $\mathsf Z$ be the union of the images of all breaking thin maps into…

Let $\pi:\mathcal{X}\to B$ be a good Fano fibration. An intersection profile $\lambda$ specifies one generically smooth, geometrically integral irreducible component in each fiber,…

Sun–Zhang's conjecture. A polarized Fano fibration $(\pi:X\to Y,\xi)$ admits a Kähler–Ricci shrinker $(\omega,\xi)$, unique up to automorphisms of $X$ preserving $\xi$, if and only…

Let $\mathcal{M}^o$ be a finite-type algebraic $\mathbbm{k}$-stack underlying a $\mathbb{Q}$-Gorenstein faithfully flat affine family of $\mathbb{Q}$-Fano fibrations, equipped with…

Consider $T$-equivariant Fano fibrations $\pi\colon X\to Y$, fixing the multi-Hilbert function of $T\curvearrowright Y$ and the Hilbert polynomial of $-K_X$ on the fibers of $\pi$.…

Let $T$ be a torus and let $c>0$ be a real number. Consider isomorphism classes of $T$-equivariant $(\mathbb{Q})$-Fano fibrations $T\curvearrowright(X\xrightarrow{\pi}Y)$, with wei…

Tangent flows at infinity conjecture. Given a non-collapsed ancient Kähler-Ricci flow $(X,\omega(t))$, $t\in(-\infty,0]$: 1. $X$ admits a non-collapsing weak Fano fibration structu…

Finite-time two-step degeneration conjecture. For any $p\in Y$ and $q\in\pi^{-1}(p)$: 1. The Kähler-Ricci flow $\omega(t)$ naturally induces a weighted-volume-minimizing, hence K-s…

K-moduli conjecture for polarized Fano fibrations. Given a positive integer $n$ and $\epsilon>0$, there exists a projective moduli space parametrizing K-polystable polarized Fano f…

Two-step degeneration conjecture for weighted volume. 1. There is a unique minimizer $v_*$ of $\mathbb W$ on $\operatorname{Val}_{\pi,p}^*$, which is quasi-monomial. 2. The graded…

Let $d$ be a natural number and let $\epsilon$ be a positive real number. Assume $(X,B)$ is an $\epsilon$-lc pair of dimension $d$, $f:X\to Z$ is a contraction with $\dim Z>0$,…

Bounded relative-global klt complement conjecture. For every positive integer $d$ and positive real number $\epsilon$, there exists a positive integer $N$, depending only on $d$ an…

Let $\pi:\mathcal{X}\to B$ be a Fano fibration over an algebraically closed field $\kappa$ of characteristic $0$. Let $A$ be the affine space of section classes and let…

Let $\pi:\mathcal{X}\to B$ be a Fano fibration. Let $A$ be the affine space of curve classes having intersection $1$ with a general fiber, and let $\mathcal{P}$ be the convex hull…

Let $\pi:\mathcal{X}\to B$ be a Fano fibration, and let $\operatorname{Sec}(\mathcal{X}/B)$ be its moduli space of sections. A component is accumulating when it is governed by the…

Let $d$ be a positive integer and $\epsilon$ a positive real number. Let $X$ be a normal variety of dimension $d$ and let $\pi:X\to C$ be a contraction to a curve $C$. Assume that…

Let $d$ be a positive integer and $\epsilon$ a positive real number. Let $(X,B)$ be a log pair of dimension $d$ and let $\pi:X\to C$ be a contraction, where $X$ is of Fano type ove…

Movable Bend-and-Break conjecture. There is a constant $Q=Q(\mathcal{X})$ such that every movable section $C$ satisfying

Shokurov's conjecture. There is $\delta\in\mathbb R^{>0}$ such that one can write

Let $n>0$ be an integer and let $0<\epsilon<1$. A normal projective variety is of $\epsilon$-Fano type if there exists an effective $\mathbb{Q}$-divisor $B$ such that $(X,B)$ is an…