11 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…

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

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

Semiampleness conjecture. Then $D$ is semiample.

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…

Generalized abundance conjecture.

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…