60 problems

Bayer–Macrì–Toda conjecture. The heart obtained by the double-tilt construction on $\operatorname{Coh}(X)$, together with the central charge $Z^X_{a,b}$, defines a Bridgeland stabi…

Let $f:X\to Z$ be a $K$-negative contraction of terminal threefolds. A general member of the anticanonical system should satisfy an analog of the result that, for a three-dimension…

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…

Degree-zero DT conjecture. The degree-zero Donaldson–Thomas partition function is

Bayer–Macrì–Toda's conjecture. If $E$ is $\sigma_{s,q}$-semistable for a certain choice of $(s,q)\in\Delta$, then

Let $X$ be a terminal or canonical weak $Q$-Fano $3$-fold, and let $(-K_X)^3$ denote its anti-canonical volume. Prokhorov's conjecture. The upper bound should be … At the time of t…

Noether inequality. Then

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…

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 $f\colon X\to Z\ni o$ 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…

Let $X$ be a smooth quasi-projective $3$-fold over $\mathbb{C}$ equipped with an action of an algebraic torus $T$, and suppose that the fixed locus $X^T$ is compact. Let…

Let $X$ be a smooth projective $3$-fold over $\mathbb{C}$ and let $S\subset X$ be a nonsingular divisor. Let $I_0(X/S,n)$ be the moduli space of relative ideal sheaves, define … an…

Donaldson–Thomas degree-zero partition-function conjecture.

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

Let $X_r^3$ be the blow-up of $\mathbb{P}^3$ at $r$ points in general position. For a special effective divisor $D$ on $X_r^3$, let $B$ be an irreducible and reduced positive-dimen…

Log abundance conjecture. The linear system $|m(K_X+\Delta)|$ is base point free for some $m\in\mathbb{N}$.

Let an OADP variety be a variety with one apparent double point, and let FRPT denote the property that the general tangential projection is birational. Consider smooth nondegenerat…

Strong general elephant conjecture. The type of a Du Val singularity $Q\in S$ for any point $Q$ is exactly the type of a general Du Val section of the germ $Q\in Y$.

Beltrametti–Schneider–Sommese conjecture. There is an integer $d_0$ such that every threefold on ${\cal Q}^{5}$ of degree $d\geq d_0$ is a minimal model.

Accurate Resolution Conjecture. For an arbitrary Q-Gorenstein threefold $X$ there exists a resolution of singularities $\pi:Y\longrightarrow X,$ such that for every Q-Cartier divis…

Hunt's Chern-ratio conjecture. There are no smooth threefolds whose Chern ratios lie above the curve

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…

Completeness conjecture. All threefolds on the Noether line are in this list for $p_g$ sufficiently large.

Terminal extraction conjecture. There exists a prime divisor $E$ over $X\ni x$ and a divisorial contraction $f:Y\to X$ extracting $E$, such that $Y$ is terminal and

The generalized Bogomolov–Gieseker conjecture. Every polarized threefold satisfies this inequality for all $(\alpha,\beta)\in U$ and $\gamma=e^{-B}$. This was the original conjectu…