19 problems

Let $X$ be a $Z_n$ curve described by the family in the source. For each positive integer $c$, let $x_c$ be the number of indices $i$ with $c_i=c$, and for each positive integer…

Let $X$ be the $Z_n$ curve associated to the family in the source, with parameters $p$ and $q$, and let $g$ be its genus. A divisor of degree $g$ is non-special when it is not spec…

Urbano's conjecture. There exists a very ample divisor $A$ such that $\mathcal{O}_X(mD)\otimes \mathcal{O}_X(A)^{\otimes m}$ is globally generated for every $m\geq 1$.

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…

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…

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

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…

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 $K_\epsilon$.

Erdős–Rosenfeld conjecture. There is an absolute constant $K$ such that, for every $c$, the number of such divisors is at most $K$ whenever $n>n_0(c)$.

Let $q\geq 1$ and $r\geq q+2$ be integers. Let $X\subseteq \mathbb{P}^\ell$ be a nonsingular projective variety of dimension $q$, defined over $K$. Let $D=\sum_{i=1}^r D_i$ be an e…

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…