79 problems

Let $R$ be an admissible hypersurface, meaning that $\widehat{R}$ is a quotient of an unramified or equicharacteristic regular local ring by a nonzero element, with an isolated sin…

Zero-dimensional intersection conjecture. Given $d,m\geq 1$ and $m\leq r\leq \binom{d+m}{m}$, we have

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…

Let $V\subset\mathbb{P}^{4}$ be a hypersurface of degree $d$ with at most isolated ordinary double points, and let $\mathrm{Sing}(V)$ denote its singular locus. Factoriality conjec…

Let $n\geq 5$, let $X\subset\mathbb{P}^n$ be a non-singular hypersurface of degree $d\geq 3$, and set … An $m$-plane means a linear projective subspace of dimension $m$ contained i…

Let $n\geqslant3$, let $F\in\mathbb{Z}[X_0,\ldots,X_n]$ be a non-singular form of degree $d\geqslant2$, and define … For $B\geqslant1$,…

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

Let $X\subset \mathbb{P}^n$ be a general hypersurface of degree $d$, and let $\overline{\mathcal{M}}_{0,0}(X,e)$ be the Kontsevich moduli space of degree-$e$ stable maps to $X$. Ca…

Cayley hypersurface characterisation conjecture. Then, in a suitable affine coordinate system, $\Sigma$ is given by the Cayley hypersurface equation.

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface, let $h\in A^1(X)$ denote the hyperplane class, and let $i_1,i_2,i_3>0$. Triple-intersection conjecture. … The s…

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface. Assume that either the dimension $n$ is odd, or the degree of $X$ is $n+2$, so that $X$ is Calabi–Yau. Intersec…

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 product conjecture. … This is presented as a co…

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 $(n,d)$ be a pair with … Let $M$ be the GIT moduli space of degree $d$ hypersurfaces in $\mathbb{P}^n$ with $m$-rational singularities, and let $M^{\rm BBH}$ denote its Hodge-t…

Low-degree curve plane-containment conjecture. Every curve $C$ of degree $\delta \leq d$ must be contained in a $2$-plane. This extends the stated minimality theorem for plane curv…

Let $C$ be a smooth curve of genus $g$, let $G(r,n)$ be a Grassmannian, and let $X_{\ell}$ be a general hypersurface of degree $\ell$ in $G(r,n)$ with $\ell<n$. Denote by…

Let $X\subset\operatorname{\mathbb{P}}^4$ be a nodal hypersurface, meaning a hypersurface of degree $d$ whose singularities are all ordinary double points. Assume that $X$ has at m…

Dimension growth conjecture. For every $\varepsilon>0$,

Let $k$ be an algebraically closed field and let $i\colon X=V(f)\hookrightarrow\mathbb{P}^n_k$ be a closed embedding of a smooth irreducible hypersurface of degree $d$. For a bound…

Let $X$ be a smooth hypersurface and suppose that $n=\dim(X)$ is odd. Let $T_v$ be the connected component of the analytic covering space of the smooth hyperplane-section parameter…

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

Let $k$ be a $C_1$ field, and let $V$ be a rational hypersurface over $k$. Manin's conjecture. The hypersurface $V$ has a $k$-rational point. This is a weaker variant of Lang's con…

Additive-action multiplicity conjecture. There are at least two non-equivalent induced additive actions on $X$.