155 problems

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface, and let $h\in A^1(X)$ denote the hyperplane class. Hartshorne's conjecture. … Apart from some easy results when…

Let $C_f$ be the hypersurface defined by a polynomial $f$, let $H_f$ be its Newton polytope, let $G(H)$ be the greatest common divisor of all coordinates of all elements of $H$, an…

Let $X\subset \mathbb{P}^N$ be a smooth Fano hypersurface of degree $d$, and let $F(X)$ denote its Fano scheme of lines. The expected dimension is $2N-d-3$. Debarre–de Jong conject…

Rational points conjecture. If $d\geq4$, then

Let $X\subset\mathbb P^n$ be a very general hypersurface of general type, and measure the degree of a rational curve on $X$ with respect to the hyperplane polarization. Voisin's co…

Let $M$ be a hypersurface in the Euclidean sphere. Its mean curvature and scalar curvature are the corresponding curvature functions of $M$, and it is isoparametric when its princi…

Cohomological special effect conjecture. The system ${\mathcal{L}}$ is special if and only if it is cohomologically special.

Cheltsov–Park cone conjecture. If $d\geq n$, then equality holds in this bound if and only if $V$ is a cone. This conjecture proposes a characterization of the equality case for th…

Let $X\subset \mathbb{P}^4$ be a generic hypersurface of degree $d$, and let $Y$ be a desingularization of a one-dimensional subvariety of $X$; write $g$ for the genus of $Y$. For…

Normal-index-one conjecture. The normal index of $\phi$ is always equal to one:

For a hypersurface with prescribed $k$-th elementary symmetric curvature, let $k$ denote the order of the higher-order mean curvature, with $k=1$ corresponding to mean curvature an…

Viehweg–Zuo's conjecture. Any smooth fiber of a non-rigid family of hypersurfaces in $\mathbb P^3$ of degree $d$ is defined, up to a projective transformation, by an equation

De Giorgi's nonsingularity conjecture. The evolving hypersurface does not develop singularities during the flow.

Let $X$ be a very general hypersurface of degree $d$ in $\mathbb{P}^n$, and let $p\in X$. Denote by $R_{\mathbb{P}^1,X,p}$ the space of points of $X$, other than $p$, that are rati…

Homma–Kim conjecture. If $n\geq 3$ is an odd integer, then equality in this bound with $k_{X^n}=\lfloor n/2\rfloor=(n-1)/2$ does not occur for $X^n$. Homma and Kim had observed thi…

De Jong–Debarre conjecture. If $d\leq n$, then

Let $X\subset\mathbb{P}^n$ be a smooth hypersurface of degree $d$, and let $F_k(X)\subset\mathbb{G}(k,n)$ be the Hilbert scheme of $k$-planes contained in $X$. Suppose that…

Abhyankar–Sathaye embedding conjecture. If $H$ is isomorphic to a coordinate hypersurface, then $H$ is equivalent to it under an automorphism of the ambient affine space.

Let $X\subset {\bf P}^n$ be a general hypersurface of degree $d$, and let $\overline{M}_{0,0}(X,e)$ be the coarse moduli space of stable maps from rational curves of degree $e$ to…

Given a hypersurface $X_d=f_d^*(0)$ in $\mathbb{P}^n$ of degree $d$, call a (very) general small deformation of $X_d$ hyperbolic if, for any (very) general degree-$d$ hypersurface…

Intersection-dimension conjecture. If $\dim R$ is even, then

Let $X$ be a projective algebraic variety of dimension $n$, let $W$ be a very ample linear system on $X$, and let $S_1,\dots,S_r$ be singularity types with Milnor numbers…

Even-dimensional vanishing conjecture. If $\dim R$ is even, then $\theta^R$ always vanishes. The source presents this as a conjecture motivated by results in dimensions $2$ and $4$…

Refined Iitaka–Severi conjecture. One should have

Iskovskikh–Manin conjecture. For every integer $n\geq 4$ there exists a non-unirational, smooth, degree-$n$ hypersurface in $\mathbb{P}^n$.