27 problems

Castelnuovo's conjecture. The linear system $\mathcal{L}_{\bullet}$ has dimension $r-10$, and one of the following conditions occurs: (A) a general element…

Let $f\colon X\to Z\ni o$ be a contraction from a threefold with only terminal singularities such that $-K_X$ is $f$-nef and $f$-big. An $n$-complement of $K_X$ is a complement wit…

MNOP dimension-zero Donaldson–Thomas conjecture. For any smooth projective threefold $X$, the dimension-zero Donaldson–Thomas series has the form

Let $X$ be a $\mathbb Q$-Fano threefold, and let $H\in|-K_X|$ be a generic section of the anticanonical divisor. Reid's general elephant conjecture. The surface $H$ has at worst ca…

Stringy flip conjecture. One has

Let r,d\begin{in}\mathbb{N}\end{in} be two positive integers and let $X$ be an irreducible smooth threefold. For a point $[\mathcal F]$ of the Quot scheme…

Let $X$ be a quasi-smooth Fano hypersurface, and let $\iota_X$ denote its Fano index. A Fano variety is solid if it admits no birational map to a Mori fibre space over a positive-d…

Noether inequality. Then

Let $X$ be a threefold satisfying the stated properties, including a decomposition of its intermediate Jacobian … where the $S_i$ are smooth projective curves of positive genus and…

Fix real numbers $\alpha_0>0$ and $\beta_0$, and let $E\in\operatorname{Coh}^{\alpha_0\omega,B+\beta_0\omega}(X)$ be a $\nu_{\alpha_0\omega,B+\beta_0\omega}$-semistable object. Def…

Threefold numerical-effectivity conjecture. Then $L$ is numerically equivalent to an effective divisor.

Let $P\in X$ be the germ of a smooth threefold, let $\mathfrak{m}$ be the maximal ideal in $\mathscr{O}_X$ defining $P$, and let … be $\mathbf{R}$-ideals on $X$. Fix a finite subse…

Let $P\in X$ be the germ of a smooth threefold, and let $\mathfrak{a}$ and $\mathfrak{b}$ be $\mathbf{R}$-ideals on $X$. For a product of $\mathbf{R}$-ideals, membership in $I$ mea…

Flip F-bound conjecture. If $F(Y)\leq 24$, then

Pseudo-effectivity conjecture. The class $c_2(X)$ is pseudo-effective.

Bayer–Macrì–Stellari quadratic Bogomolov–Gieseker inequality conjecture. For every $\nu_{\omega,B}$-semistable object $E\in\mathcal{B}_{\omega,B}$,

Let $X$ be a smooth projective threefold. For an object $E\in D^b(X)$, let $\overline{\beta}(E)$ be the limiting value of the parameter $\beta$ associated with the kernel of the re…

Let $X$ be a smooth projective threefold and let $H\in\operatorname{NS}(X)$ be an ample class. For a real number $\beta$, write $\operatorname{ch}_i^\beta$ for the $\beta H$-twiste…

Let $X$ be a smooth projective threefold. Let $u{\omega,B}$ be the tilt slope on the tilted heart $\operatorname{Coh}^{\omega,B}(X)$, where $\omega$ is ample and $B$ is a real divisor…

Let $X=Z(f_1,\dots,f_{n-3})\subset\mathbb{P}^n$ be a complete intersection threefold, where $f_i$ has degree $k_i$, and let $C\subset X$ be a curve of degree $d$ and genus $g(C)$.…

Let $X=Z(f_1,\dots,f_{n-3})\subset\mathbb{P}^n$ be a complete intersection threefold, where $f_i$ has degree $k_i$, and let $C\subset X$ be a curve of degree $d$ and genus $g$. The…

Suppose that $Y\to Y_{\mathrm{con}}$ and $Z\to Z_{\mathrm{con}}$ are flopping contractions of a single curve in smooth projective $3$-folds, to points $p$ and $q$ respectively. Ass…

Basepoint free conjecture. The linear system $|b(K_X+S+A)|$ is basepoint free for every $b\gg 0$.

Asymptotic purity conjecture. $X$ is AP if and only if the big and ample cones are equal.

Stable-sheaf inequality conjecture. Then