38 problems

Colliot-Thélène–Sansuc conjecture. The Brauer–Manin obstruction is the only obstruction to the local-to-global principle for $Y$.

Let $V$ be a smooth cubic surface over a number field $k$. A point of degree prime to $3$ on $V$ means a closed point whose degree over $k$ is coprime to $3$. The prime-to-3 degree…

Let $z_1,z_2,z_3$ be coordinates on a three-dimensional complex vector space, let the Hessian groups act on the polynomial ring ${\mathbb C}[z_1,z_2,z_3]$, and let $C_6$, $C_9$, an…

Kleppe's conjecture. If

Kleppe–Ellia's conjecture. If $C$ is linearly normal, then every small deformation $C'$ of $C$ in $\mathbb P^3$ is contained in a cubic surface $S'\subset\mathbb P^3$; equivalently…

Topological entropy conjecture. The topological entropy of $c$ should agree with the logarithm of its first dynamical degree:

Generalized Eckardt point conjecture. If the total log canonical threshold of $X$ is $\frac{n-1}{n}$, then a wild tiger $S$ for $X$ is a cone in $\mathbb{P}^{n-1}$ over a smooth hy…

Let $S$ be a cubic surface with $27$ marked lines $D_1,\ldots,D_{27}$, and set … where $\varepsilon\in\mathbb{Q}$ and $0<\varepsilon\ll 1$. Hacking–Keel–Tevelev's conjecture. The c…

Let $S_0$ be the cubic surface introduced earlier in the paper, and consider its integral points with coordinates satisfying $1\leq x,y\leq N$. A growth conjecture asserts that the…

Let $S$ be a smooth, affine, cubic surface whose automorphism group is infinite and discrete. A classification conjecture asserts that $S$ is either a Markov-type surface, or the c…

Brauer-adjusted heuristic conjecture. The main heuristic holds with

Oblong-surface conjecture. The main heuristic holds with

The logarithmic-bound conjecture. As $B\to\infty$,

For a degree $d$ form $f\in\mathbb{Z}[x_0,\ldots,x_n]$, let $X_f\subset\mathbb{P}^n$ be the hypersurface defined by $f=0$, and order forms by the Euclidean height of their coeffici…

Dolgachev–Duncan's conjecture. In characteristic $2$, every cubic surface can be written in Emch normal form. This conjecture concerns the existence of a uniform normal form in the…

Let $S'$ be a cubic surface over $\mathbb{Q}$, let $U'$ be the relevant open subset, let $H$ be its height function, and let $rho'$ denote the corresponding Picard rank. Write…

Let $S$ be a cubic surface, and let $L$ be the certain Lagrangian on $S$ referred to in the source. A cubic surface has $21$ terms in its mirror superpotential, corresponding to th…

Let $V \subset \mathbb{P}^3_L$ be a smooth cubic surface defined over a number field $L$. Let $\nu_1$ and $\nu_2$ be two different places of $L$. Suppose that there exists a non-as…

Let $S$ be a smooth cubic surface in $\mathbb P^3$. Gizatullin's automorphism conjecture. Every automorphism of the affine variety $\mathbb P^3\setminus S$ is induced by an automor…

Consider the cubic surface … With respect to the coordinate hyperplanes $x_i=0$, the covariance-matrix conjecture. An analogue of Theorem should hold with covariance matrix … This…

Divisibility conjecture. Restricted to the $3$-planes coming from eigenschemes, the eigendiscriminant divides the Hurwitz form.

Let $M$ be the matrix whose determinantal rank loci are denoted by $V_k$, and let $N_k$ denote the locus of cubic surfaces with $k$ singularities. For each $k$, consider the minors…

Let $V_k$ be the locus of cubic surfaces for which the matrix $M$ has rank at most $20-k$, and let $N_2$ denote the locus of binodal cubic surfaces. The $19\times 19$ minors of $M$…

Kristin Shaw's conjecture. If the surface is tropically smooth, then its $27$ lines have distinct tropicalizations. In particular, all these lines and points are defined over the…

Let $\mathcal{H}_{3,3}$ denote the moduli space of smooth cubic surfaces, and let $\mathcal{H}_{3,3}(r)$ denote the moduli space of smooth cubic surfaces together with $r$ specifie…