806 problems

Let $(\mathcal{M},g)$ be a $3+1$-dimensional Lorentzian spacetime solving the Einstein field equations, arising as the maximal future development of generic, regular, asymptoticall…

Let $V$ be a normal affine variety of complex dimension $n$ with an isolated singularity at $0$ that is numerically $\mathbb{Q}$-Gorenstein. Write $\operatorname{min.disc}(V,0)$ fo…

Eisenbud–Goto conjecture for selected classes. The conjecture may hold for smooth projective varieties, projectively normal varieties, projective toric varieties, and projective si…

Bridgeland stability conjecture. The category $\operatorname{\mathcal{K}\mkern-1mu\mathit{u}}(Y)$ admits a Bridgeland stability condition.

Let $p$ be an odd prime and let … which is a singular ring of characteristic $p$ and dimension $d-1$. For a Noetherian local ring $(R,\mathfrak{m})$ of characteristic $p$, write…

Let $\uppi\colon \EuScript{X}\to\mathop{\mathrm{Spec}}\EuScript{R}$ be a flopping contraction, and let $\Lambda_{\mathrm{con}}(\uppi)$ denote its contraction algebra, the stable en…

A sextic double solid is a double cover of $\mathbb{P}^3$ branched over a sextic surface; it is factorial when its local divisor class groups are generated by Cartier divisors, and…

Let $(R, \mathfrak{m}_R, k)$ be a one-dimensional complete local reduced $k$-algebra over a field of characteristic zero, and let $\Omega_R$ denote its universally finite module of…

Consider the mean curvature flow of embedded hypersurfaces, and let a blowup limit mean a limit obtained by parabolic rescaling near a singularity, counted with its varifold multip…

Let $X$ be a singularity over a field of characteristic zero, and let its tangent sheaf be free. Lipman–Zariski conjecture. The singularity $X$ is smooth. This conjecture asks whet…

Let $M_t$ be a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$, with $n dleq 6$, and suppose the flow develops a first finite singular time. Mean-curvature blow-up…

Ambro's lower semi-continuity conjecture. This function is lower semi-continuous. The conjecture concerns variation of minimal log discrepancies in families of singularities and, t…

Let $F$ be a field, and let $X$ be an affine finite-type $F$-scheme of dimension $d$. A scheme is regular if all of its local rings are regular, and it is $K_m$-regular when the ca…

Let $X$ be a projective algebraic variety with at worst Gorenstein canonical singularities. Assume that the stringy $E$-function $E_{\operatorname{st}}(X;u,v)$ is a polynomial. Bat…

Kaledin's conjecture. Every symplectic singularity is formally isomorphic to a conical symplectic singularity.

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…

Donaldson–Sun's metric-independence conjecture. Both $W$ and $\mathcal C$ depend only on the germ of the singularity $(Z,p)$, rather than on the metric.

Let $(M,g(t))_{t\in[0,T)}$ be a solution to Ricci flow on a closed smooth $n$-dimensional Riemannian manifold, with $T$ its maximal time. Hamilton's scalar-curvature blow-up conjec…

Let $J_{\bullet}$ be a graded sequence of ideals on $X$, let $\mathfrak{q}$ be a nonzero ideal on $X$, and suppose that the asymptotic jumping number…

Nash's conjecture. The number of irreducible components of $X_{\infty}^{\operatorname{Sing}(X)}$ is at most the number of essential irreducible components of the exceptional locus…

Let … be a $\mathbb{Q}$-Gorenstein flat family of klt singularities with a section $t\in T\mapsto x_t\in\mathcal{X}_t$. Constructibility conjecture. The function … is constructible…

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 $C$ be a rational cuspidal plane curve of degree $d$. The source identifies three specific series of tricuspidal curves in a table. Piontkowski's conjecture. If $d\geq 6$, then…

Let $X$ be a smooth projective variety. Minimal model conjecture. Either $X$ is uniruled, or there is a birational modification $X\dashrightarrow X'$ such that $X'$ has only termin…

Let $N\geq 3$ and $k$ be a positive integer. Set $l={N-1+k \choose N}$, and let $\mathfrak{m}$ be the maximal ideal in $\mathbb{C}[x_1,x_2,\dots,x_N]$. Consider the Hilbert scheme…