21 problems

Let $k$ be a field and let $\mathcal{M}_\sim(k)$ be the category of motives over $k$ for an adequate equivalence relation $\sim$. Let $\operatorname{Pic}(\mathcal{M}_\sim(k))$ deno…

Let $f:\mathcal{X}\rightarrow S$ be a toric degeneration, and write $X=\mathcal{X}_0$. Let $Z$ be the singular set of the degeneration, and let $\mathcal{M}_X$ denote the induced l…

Let $X$ be an $n$-dimensional toric Fano variety, and let $\rho$ denote its Picard number. The preceding discussion suggests that varieties with maximal Picard number are $S_3$-bun…

Let $X$ be a smooth Fano variety of Picard number $\rho(X)=1$ over an algebraically closed field of characteristic zero. An endomorphism of degree greater than $1$ is a self-map of…

Let $d>0$ and let $n$ be even with $n\geq 2d-2$. Let $c\subset S_d$ be the conjugacy class of transpositions, and let…

Picard-rank finiteness conjecture. The Picard number of $\mathcal{Y}_s$ is greater than or equal to $2$ for at most finitely many values of $s\in\mathbb{P}^1(\mathbb{C})$.

Fausk–Lewis–May conjecture. The canonical map

Reduced Homological Vanishing Conjecture. For the parameters under consideration, this map is an isomorphism

Picard group conjecture.

Let $C$ be a smooth projective curve over $\mathbb F_q$, let $S\subset C$ be a reduced effective divisor, and consider smooth projective double covers $D\to C$ unramified over $S$.…

Let $X=D53F^1\timesD53F^1$ and let $M_{H_m}({\bf v})$ be the moduli space associated with a character ${\bf v}=(r,\nu,\Delta)\in K(X)$, where $r\geq 2$ and $\Delta\geq\frac{1}{2}$.…

Strengthened independence conjecture. For “most” sections, the following maps are injections:

Hopkins's exotic Picard-group conjecture. The group $\kappa_n$ is a finite $p$-group. The source notes that $\kappa_n=0$ when $2(p-1)\geq n^2$, while it is nontrivial in many other…

Work over an algebraically closed field of characteristic zero. Let $\widetilde H_{d,g}$ be the Hurwitz space of degree-$d$ covers of smooth genus-$0$ curves by irreducible genus-…

Cartwright's Picard-group conjecture.

Let $K$) be the function field of a hyperelliptic curve $X$ of odd degree. Write … For a point $\mathfrak{p}\in X$, let $\deg\mathfrak{p}$ denote its degree and let its class in th…

Non-splittable extremal conjecture. One has

Non-$S_3$-splittable extremal conjecture. One has

Assarf–Joswig–Paffenholz conjecture. If $d>3k$, then $X$ is isomorphic to $(S_3)^m\times Y$, where $m$ is a positive integer and $Y$ is a toric Fano manifold of dimension at most…

Let $(R,\mathfrak{m})$ be a local complete intersection ring of dimension $3$. Let … be the punctured spectrum of $R$. Gabber's conjecture. The group $\operatorname{Pic}(U_R)$ is t…

Domination conjecture. The variety ${\mathbb C}^*$ is dominated by one of the components of ${\mathcal M}^\perp\cap {\mathcal N}^\perp$ under the map $h$.