13 problems

Ambro–Kawamata conjecture. One has

Let $X$ be a quasismooth Fano threefold weighted hypersurface with Fano index $iota_X=1$, and let $X$ be any quasismooth member of its deformation family. Corti–Pukhlikov–Reid conj…

A quasismooth and well-formed Fano threefold weighted complete intersection is a Fano threefold embedded as a quasismooth and well-formed complete intersection in a weighted projec…

Generalized Frobenius conjecture. Then

Weighted-complete-intersection semigroup conjecture. For every positive integer $k$ such that $h\mid k$ and $k>\delta(d;a)$, there exist natural numbers $x_0,\ldots,x_n$ satisfying

Pizzato–Sano–Tasin conjecture. Then

Let $X \subset \mathbb{P}(a_0,\ldots,a_N)$ be a smooth well-formed Fano weighted complete intersection of multidegree $(d_1,\ldots,d_c)$. A nef-partition for $X$ is a splitting … I…

Let $(\overline{d},\overline{a}) \in \mathcal{CS}$ be a regular pair, and let $(\mathcal{D},\mathcal{A})$ be the associated pair of weighted simplicial complexes. Write…

Let $X \subset \mathbb{P}$ be a smooth well formed Fano weighted complete intersection of dimension at least $2$ which is not an intersection with a linear cone. Let $s_2(X)$ denot…

Nef-partition conjecture. The variety $X$ has a good nef-partition and a toric Landau–Ginzburg model.

Let $(d;a)=(d_1,\ldots,d_c;a_0,\ldots,a_n) \in \mathbb N^c \times \mathbb N^{n+1}$ be a regular pair such that $a_i \ne 1$ and $d_j \ne a_i$ for any $i,j$. Assume $c \le n$ and…

Let $X$ be a smooth well formed Fano weighted complete intersection. A nice nef partition conjecture asserts that there exists a nice nef partition for $X$, and that $X$ has a tori…

Let $m$ and $\epsilon$ be fixed positive numbers, let $\alpha$ be a negative integer, and consider quasismooth weighted complete intersections of dimension $m$ and amplitude…