25 problems

Cassels–Swinnerton-Dyer conjecture. If $X$ has an $L$-point for some field extension $L\supset K$ of degree $[L:K]$ coprime to $3$, then $X$ has a $K$-point.

Ciliberto–Ottaviani conjecture. The hessian map $h_{d,n}$ is birational onto its image for every $d\geq3$ and every $n\geq2$, except when $(d,n)=(3,2)$. This concerns the generic r…

A cubic hypersurface in $\mathbb{P}^n$ over $\mathbb{Q}$ is the vanishing locus of a cubic form in $n+1$ variables. The Hasse principle conjecture. When $n\geq 4$, every such cubic…

Let $K$ be a global field and let $X\subset \mathbb{P}_K^{n-1}$ be a cubic hypersurface defined over $K$. Folklore conjecture. The set $X(K)$ of $K$-rational points on $X$ should b…

Let $Y$ be a smooth cubic $n$-fold containing a $k$-plane $\varPi_0 \subseteq Y$. The quadric fibration … defines a sheaf of Clifford algebras with even part $\mathscr{C}_0$. Write…

Let $f\in\bK[x_0,\dots,x_n]$ define a smooth cubic hypersurface $X=V(f)\subset\bP^n$, and let $\cH_f$ be its Hessian hypersurface. Say that $f$ is of Thom–Sebastiani type if, after…

Wooley's conjecture. There exists a constant $c>0$ depending only on $X$ such that

Folklore local-to-global conjecture. If $X(\mathbf{Q}_p)\neq\emptyset$ for every prime $p$, then

Let $X=F_r(Y)$ be as above, with Voisin rational self-map $\Psi:X\dashrightarrow X$, and let $F\subset X$ be the closure of its fixed locus. A closed subvariety is constant-cycle w…

Let $Y$ be a cubic hypersurface and let $X=F_r(Y)$ be the variety of $r$-planes in $Y$, equipped with the Voisin rational self-map $\Psi:X\dashrightarrow X$. For $r\geq 2$ and…

For $k\geq 1$, let ${\mathcal K}_{2k+1}$ be the Klein cubic of odd dimension $d=2k+1$, defined by the cyclic cubic equation in projective space. Kollár's conjecture. The Klein cubi…

Let $[f]\in U$ be general, and let $Y$ be the associated surface. The surface $Y$ has two $2$-torsion line bundles: $[?]$ the line bundle $[?]$ defined by $K_Y=3H|_Y+\eta$, and…

Let $V_{[Z]}$ be the Hodge locus associated with $Z=r\mathbb{P}^{\frac n2}+\check r\check{\mathbb{P}}^{\frac n2}$, where the two linear cycles meet in $\mathbb{P}^m$. Generalized c…

Let $V_{[Z]}$ be the Hodge locus associated with $Z=r\mathbb{P}^{\frac n2}+\check r\check{\mathbb{P}}^{\frac n2}$ on a smooth cubic hypersurface, where the two linear cycles meet i…

Uniqueness conjecture. Every cubic hypersurface $X\subset {\mathbb P}^6$ with 35 nodes is projectively equivalent to $X_S$.

Let $\boldsymbol{\mathcal W}$ be a 6-web satisfying the five listed properties: it is linearizable and skew, has maximal 2-rank equal to $10$, has 1-rank at least $5$, and has a 5-…

Characteristic-number conjecture. For every $n$, the numbers given by the code for $n_p\geq n$ are the characteristic numbers.

Let $n\geq 3$ and let $X\subset \mathbb P^{n+1}$ be a very general cubic hypersurface. Stable irrationality conjecture. $X$ is not stably rational. The conjecture asks whether the…

Let $X_0$ be a cubic hypersurface in the full family of cubic hypersurfaces, and let $\mathbb{P}^{\frac{n}{2}}$ and $\check{\mathbb{P}}^{\frac{n}{2}}$ be linear cycles in $X_0$. Fo…

Let $X$ be a smooth cubic hypersurface in the complex projective space$\mathbb{P}^4$, and let $S_X$ be its hyperplane section. The complement $X\setminus S_X$ is an affine cubic hy…

Debarre–Manivel's conjecture. The inclusion $F\hookrightarrow G$ induces an isomorphism

Galkin–Kontsevich weak conjecture. If $\alpha$ is represented by a linear combination of varieties of dimension less than or equal to $2(d-2)$ and

Let $X\subset \mathbb{P}^5$ be a very general cubic hypersurface. Nonrationality conjecture. The variety $X$ is not rational. This is a classical rationality problem for cubic hype…

Let $X\subseteq\mathbb{P}^{n-1}$ be a cubic hypersurface defined over $\mathbb{Q}$ by a cubic form $C\in\mathbb{Z}[x_1,\ldots,x_n]$, with $n\geq 10$. Rational-points conjecture. On…

Let $X\subset\mathbb{P}^{n-1}$ be a cubic hypersurface over $\mathbb{Q}$, given as the zero locus of a cubic form $C\in\mathbb{Z}[x_1,\ldots,x_n]$. The ten-variable conjecture. One…