28 problems

Tenenbaum and I recently asked the following question: let $n_1 < n_2 < \ldots$ be an infinite sequence of positive integers. Is it then true that there always is a positive intege…

I proved long ago that every $m < n!$ is the distinct sum of $n - 1$ or fewer divisors of $n!$. Let $h(m)$ be the smallest integer, if it exists, for which every integer less than…

Semi-ampleness conjecture. Any nef integral divisor on a SCY3 is semi-ample.

Let $X$ be a smooth projective variety of dimension $d$, and let $\textnormal{Big}(X)$ denote its big cone. An open subset $U\subseteq\textnormal{Big}(X)$ is called “large” in the…

Existence conjecture. Without imposing any condition on $N_1(F/\mathbb F_q)$, there exists a non-special divisor of degree $g$ for every function field with

Divisibility conjecture. If $q(M)=0$ and $h^0(X,\mathcal{O}_X(M))\geq n+2$, then $M$ is divisible in $\operatorname{Pic}(X)$.

Let $n\geq 1$ be an integer, let $\mathcal{D}_n$ be the set of positive divisors of $n$, and define … Fix real numbers $\theta$ and $\epsilon$ satisfying $0<\epsilon<\theta<1$. Sho…

Let $X$ be a smooth irreducible complex projective variety, let $(E_i)_{i\in S}$ be a finite set of mutually distinct prime divisors on $X$, and let $\bm a=(a_i)_{i\in S}$ with…

Let $X$ be a smooth projective variety over $\mathbb{C}$, and let $D$ be a pseudoeffective $\mathbb{R}$-divisor on $X$. Fujino's conjecture. There exist $m_0\geq 1$, $c>0$, and an…

Let $\alpha$ and $\beta$ satisfy … For a positive integer $m$, consider its divisors in the interval $[m^\alpha,m^\alpha+m^\beta]$. Divisor clustering conjecture. There is a consta…

Let $S$ be a compact tropical manifold, let $\mathcal{L}$ be a line bundle on $S$, and let $[s]$ denote the line bundle associated with a permissible $C^\infty$-divisor $s$. Existe…

Borman–Sheridan class compatibility conjecture. Under this equivalence, the action of $t$ on $\mathcal{W}_{S^1}(X)_{-1}$ coincides with the action of $\tau$ on…

Let $\Gamma\subseteq[0,1]$ be a finite set and let $X\ni x$ be a germ. Local stability conjecture. There exists a positive real number $\tau$ depending only on $\Gamma$ and…

Oguiso's nef divisor conjecture. The divisor $D$ is $\mathbb{Q}$-linearly equivalent to an effective divisor:

Let $\varepsilon(y,z)$ and $\varepsilon_1(y,z)$ be the densities of integers having at least one, respectively precisely one, divisor in $(y,z]$, and put…

Erdős's conjecture. When $z=2y$ and $y\to\infty$, the quantity $\varrho_1(y,z)$ tends to a positive limit. The source reports subsequent estimates, including results showing relate…

Erdős's conjecture. The quantity $G(n)$ tends to infinity for all integers $n$ outside a set of density zero. This is described as an unconventional problem, and the source gives n…

Let $\tau(n)$ be the number of divisors of $n$, and let $\tau^+(n)$ be the number of integers $k$ for which $n$ has a divisor $d$ satisfying $2^k<d\leqslant2^{k+1}$. Erdős's conjec…

Erdős's conjecture. There exists a strictly decreasing sequence $\\{\alpha_r\\}_{r=1}^{\infty}$ such that

Let $X$ be a normal projective variety, and let $D$ be a pseudoeffective $d53d$-Cartier divisor on $X$. The Boucksom–Broustet–Pacienza conjecture. The restricted base locus and the…

Let $X$ be a normal projective variety over an algebraically closed field and let $D$ be an ${\mathbb{R}}$-Cartier ${\mathbb{R}}$-divisor on $X$. The non-nef locus…

Let $\Delta$ be a tropical surface, let $D$ be a Cartier divisor on $\Delta$, and let $K_\Delta$ be the divisor obtained by summing $(\operatorname{deg} r-2)[r]$ over the ridges…

Let $\Delta$ be a tropical complex and let $D$ be a Weil divisor on $\Delta$. Define $h^0(\Delta,D)$ as the smallest cardinality of a set of rational points that is not contained i…

Let $X$ be a complex normal projective variety and $D$ a Weil divisor on $X$. Global generation conjecture. There is an ample Cartier divisor $H$ on $X$ such that for any…

Ruzsa's conjecture. There is a constant $K_\epsilon$ such that, for every positive integer $n$, the number of such divisors is at most $K_\epsilon$.