253 problems

Specialness criterion. The system $\mathcal L$ is special if and only if at least one of the following holds:

Let $X$ be a smooth complex projective surface, and let $C$ be a reduced curve on $X$. Bounded Negativity Conjecture. For each such $X$, there exists a number $B_X\geq0$, depending…

Let $P$ be a strictly convex polygon in $\mathbb{RP}^d$. Suppose that $P$ intersects a hyperplane with multiplicity $k$, where $k\geq d+2$. Call a vertex configuration a flattening…

Let $P$ be an embedded non-contractible closed polygon in $\mathbb{RP}^2$, and call a vertex configuration a flattening when it has the polygonal analogue of an inflection point. D…

Let $k\geq 2$, and consider the complex … Assume that $\beta$ is surjective and that $\alpha$ degenerates in codimension $2$; equivalently, the cokernel in degree $0$ is…

Let $X\subset \P^{2n-1}$ be an irreducible indecomposable Legendrian subvariety, and let $G<\P\operatorname{GL}_{2n}$ be a connected subgroup preserving $X$. Automorphism conjectur…

Generic injectivity conjecture. The map $\sigma$ is generically injective.

Projection conjecture. At most $k(n-1)$ points of $\phi_4(\Sigma)$ can be contained in a curve of degree $k$ in $\mathbb{P}^2$.

Let $A$ be a ring and let … be a bounded complex of finitely generated $A$-modules. Realization conjecture. Every such complex arises as the direct image complex of a family of she…

Vatne's conjecture. There is a degree-two Cohen–Macaulay subscheme $Z$ such that

Linear-configuration conjecture. The Hilbert function of $C_{\underline h}$ is $\underline h^{\min}$.

Minimal-Hilbert-function conjecture. Among these double schemes, there is a minimal Hilbert function, and it occurs when the support is $C_{t,r}$.

Let $L$ be a reversible Finsler metric on ${\mathbb R}P^2$. A smooth positive function $\rho$ on ${\mathbb R}P^2$ is a function whose product with $L$ defines the conformally resca…

Let $(r,d)$ be integers satisfying … and $(r,d)\notin\{(7,2),(8,2),(9,2),(8,3),(9,3)\}$. Let $l(d,\delta,\mu^r)$ denote the dimension of the space of homogeneous degree-$\delta$ po…

Let $(r,d)$ be integers satisfying … and $(r,d)\notin\{(7,2),(8,2),(9,3)\}$. Let $l(d,\delta,\mu^r)$ denote the dimension of the space of homogeneous degree-$\delta$ polynomials on…

Two-cyclic-factor conjecture. If $H$ can be realized by a 3-net in $\mathbb{P}^2$, then $H$ is a direct sum of at most two cyclic groups. If one class of the net is a pencil, then…

Let $X=\mathbb{P}^N\subset\mathbb{P}^{N+M}$ be a linear subspace of dimension $N\geq 6$. Let $W$ be a Cohen–Macaulay multiple monomial scheme with reduced subscheme…

Let $\mathbb{P}^N$ with $N\geq 6$ be a linear subspace of codimension two in $\mathbb{P}^{N+2}$. Let $M$ denote the integer occurring in the infinitesimal-neighbourhood notation, a…

Let $\mathbb{P}^N$ be a projective space of dimension at least six, and let $\mathcal{E}$ be a vector bundle of rank two on it. The rank two bundle conjecture. The bundle…

Let $C$ be a set of two skew lines in $\mathbb{P}^{4}$, spanning a hyperplane $H$, and let $C'$ be another set of two skew lines in $\mathbb{P}^{4}$, spanning a hyperplane $H'$. Tw…

A smooth hypersurface in $\mathbb{RP}^{n+1}$ is $(k,l)$-hyperbolic if its second quadratic form has signature $(k,l)$. Let $U\subset\mathbb{R}P^n$ be a domain bounded by a connecte…

Consider a domain $U\subset{\mathbb R}P^n$ bounded by a connected smooth hypersurface $B$. Suppose that the second fundamental form of $B$ is non-degenerate at every point and has…

Type conjecture. The type $\boldsymbol{t}$ of these pseudo-codewords has $t_0$ non-negative, $t_1$ positive, $t_2$ positive, and $t_\ell=0$ otherwise. If this is false, the smalles…

Structural conjecture. Every minimal pseudo-codeword is a sum of a few minimal pseudo-codewords, with one or two low-value components changed so that they become the large componen…

Open-domain projective categoricity conjecture. Two such sets $U$ and $V$ are $\mathcal{L}_{\mathrm{B}}$-elementarily equivalent if and only if they are projectively equivalent.