15 problems

Fubini–Study conjecture. The two-plane $\Omega^\ast$ agrees with the two-plane in equation $$ . Equivalently, $\mathcal{C}_{a,b,c}$ has a geometric interpretation on $X_{a,b,c}…

Natural hyperkähler structure conjecture. The theory $\mathcal{C}_X$ has a geometric interpretation whose full hyperkähler structure is the natural one induced on $X$ by its embedd…

Let $\mathbb{Q}(\sqrt{-3})$ act on abelian fourfolds with signature $(3,1)$, equipped with a level $\sqrt{-3}$ structure and a $(1,1,1,3)$-polarisation. Let…

Let $Z$ be a set of six points in linearly general position in $\mathbb{P}^3$, and let $\mathcal W(Z)$ denote its Weddle surface. Upper-bound conjecture for lines on Weddle surface…

Consider the four families of quartic surfaces in $\mathbb P^3$ defined by homogeneous polynomials that are, respectively, symmetric in all four variables; symmetric in three varia…

Uniqueness conjecture. The quartic surface discovered in the cited work is the only quartic that attains the bound of $52$ lines.

Sharp upper-bound conjecture.

Let $K$ be an algebraically closed field of characteristic $2$, and let $X\subset\mathbb{P}^3_K$ be a normal quartic surface. Write $\mu(4)$ for the maximal number of singular poin…

800-conic conjecture. One has

Let $f \in \mathbb{C}[w,x,y,z]_4$ define a smooth quartic surface $X_f$ in $\mathbb{P}^3(\mathbb{C})$. Its periods form a rank-$21$ subgroup of $\mathbb{C}$, and proving that numer…

Generalized Degtyarev–Itenberg conjecture. The polynomial $f$ exists if and only if

Burnside's conjecture. One has

Burnside's conjecture. The most symmetric nonsingular quartic surface is projectively equivalent to $V(h)$, and

Lines on singular quartic surfaces conjecture. (a) If a non-$K3$ quartic surface $S\subset\mathbb P^3(\mathbb C)$ contains more than $31$ lines, then it is ruled by lines. (b) Let…

Quotient conjecture. For any hyperplane $L\subset\mathbb{P}^{4}$, $L\cap I_4$ is the quotient of $X_L$ by the Heisenberg group, and