14 problems

Let $X$ be a Calabi–Yau $3$-fold and let $L$ be a nef divisor on $X$. Nef abundance conjecture. The divisor $L$ is semiample. The surrounding discussion derives this assertion from…

Let $N>9$ be square-free, and let $d/m$ be the fundamental solution of the Pell equation $d^2-Nm^2=1$. Write $(d;m^{\times N})$ for the vector with first entry $d$ and $N$ further…

Assume the base field has positive characteristic. Let $\overline{\operatorname{M}}_{g,n}$ be the moduli space of stable $n$-pointed curves of genus $g$, and let $D$ be a nef divis…

Conjecture on asymptotically special nef divisors. The blow-up $X^n_r$ does not contain nef divisors which are asymptotically special.

Let $(X,B)$ be a klt pair with $K_X+B$ pseudoeffective, and let $L$ be a nef divisor on $X$ such that $K_X+B+L$ is also nef. Generalized abundance conjecture. The divisor $K_X+B+L$…

Non-vanishing conjecture. One has $\kappa(L)\geq 0$.

Boundedness conjecture. There is a natural number $m$ depending only on $d,epsilon$ and an integral divisor $L$ on $V$ such that

Let $X$ be a projective variety defined over a number field $k$, of dimension $n$. Let $E_1,\ldots,E_r$ be nef Cartier divisors on $X$ such that $\sum_{j=1}^rE_j$ is ample. Let…

Let $X$ be a normal projective klt variety with $H^1(X,\mathcal O_X)=0$ and $K_X\sim_{\mathbb Q}0$, and let $L$ be a nef divisor. Singular abundance conjecture. Then $L$ is semiamp…

Let $X$ be a projective manifold with $H^1(X,\mathcal O_X)=0$ and $K_X\sim 0$, and let $L$ be a nef divisor. Abundance conjecture for varieties with trivial canonical class. Then…

Let $X_\mathcal K$ be the blowup associated with the Klein line configuration, with $H$ the pullback of a line and $E_4,E_3$ the exceptional divisors over its quadruple and triple…

Let $V$ be a reduced $n\times(n+r)$ $F$-matrix and let $\gamma\in\mathcal{A}_{\Gamma}(V)$ be a non-singular chamber. A chamber is bordering if it lies on the boundary of the releva…

An $F$-nef divisor on $\overline{M}_{0,n}$ is a divisor having nonnegative intersection with every $F$-curve. The boundary divisors are the divisors parametrizing stable curves wit…

Exhaustiveness conjecture. The types for which $\Lambda(n,d,1,\gamma)$ is nef are exactly those constructed in Proposition 5.13.