16 problems

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…

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 $S_0$ 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…

Oblong-surface conjecture. The main heuristic holds with

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 $V \subset \mathbb{P}^3_L$ 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…

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 $M$…

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…

Distribution conjecture. The number of Hasse failures for $0\leq k\leq K$ satisfies

Let $\overline{\mathcal{M}_3}^{ALG}:=\mathbb{P}^{19}//SL(4,\mathbb{C})$ be the algebraic compactification of the moduli space of cubic surfaces, and let…

Isotropicity conjecture. The five vector spaces of dimension $2$ canonically defined in the relevant construction are maximal isotropic subspaces for the symplectic form on…

A smooth tropical cubic surface is a smooth tropical surface of degree $3$. The 27-line conjecture. A general smooth tropical cubic surface contains exactly $27$ tropical lines. Th…