19 problems

Fujita's spectrum conjecture. For each fixed dimension $d$, the set of pseudo-effective thresholds of smooth polarized $d$-folds should be finite away from $0$.

Let $X$ be a normal variety, let $B$ be a log divisor on $X$, and let $W\subset X$ be a closed subvariety with $a(\eta_W;X,B)=0$. Let $W^\nu$ be the normalization of $W$, and let…

The precise inverse of adjunction conjecture.

Let $(X,B)$ be a log variety and let $j:W\to X$ be the normalization of an irreducible component of $LCS(X,B)$ such that $K_X+B$ is log canonical at the generic point of $j(W)$. Th…

Let $X$ be a smooth projective variety of dimension $n$ over $\a0\mathbb{C}$, and let $K_X$ be its canonical divisor. A nef divisor is 1-big when $L^{n-1}\cdot H>0$ for some ample…

Let $X$ be a smooth projective variety of dimension $n$ over $\a0\mathbb{C}$, with canonical divisor $K_X$, and let $L$ be an effective divisor. Say that adjunction terminates in t…

Let $X$ be a smooth projective variety of dimension $n$ over $\a0\mathbb{C}$, and let $K_X$ be its canonical divisor. Ambro's conjecture. If $D$ is an ample divisor on $X$ and…

Let $(X,L)$ be a quasi polarized manifold of dimension $n$, meaning that $X$ is a projective manifold and $L$ is a nef and big Cartier divisor. Let $s>0$. Effective adjunction thre…

Let $(X,L)$ be a projective variety of dimension $n$ with at most terminal $\mathbb{Q}$-factorial singularities, where $L$ is a nef and big Cartier divisor; such a pair $(X,L)$ is…

Let $X$ be a normal projective variety and let $H$ be an effective Cartier divisor. Adjunction termination conjecture. If there exists an integer $m_0\geq 1$ such that … for every…

Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth ample divisor. Suppose that $p:Y\to Z$ exhibits $Y$ as a $\mathbb{P}^d$-bundle over a $b$-dimensional smooth…

Let $(X,\mathcal L)$ be an $n$-dimensional toric polarized variety, not necessarily smooth or Gorenstein, and let $\mu(\mathcal L)$ and $\tau(\mathcal L)$ denote the corresponding…

Let $(X,\mathcal L)$ be a nonsingular polarized variety of dimension $n$, with effective log threshold $\mu(\mathcal L)$ and nef value $\tau(\mathcal L)$. Beltrametti–Sommese–Wisni…

Let $(X,\Delta)$ be a log canonical pair and let $M$ be a big divisor, and define the Kodaira energy by … Suppose that the set of Kodaira energies has limit points. Fujita's one-si…

Let $(X,\Delta)$ be a log canonical pair and let $M$ be a big divisor. The Kodaira energy is … In the case $\Delta=0$ and $M$ ample, consider the set of negative Kodaira energies.…

Let $(X,L)$ be a polarised projective manifold of dimension $n$, and let $\varphi\colon X\rightarrow Y$ be an adjunction-theoretic scroll over a normal variety $Y$ of dimension $m$…

Fujita–Sommese conjecture. If

Dickenstein–Nill conjecture. If

Let $(X,\mathcal{L})$ be a polarized manifold of dimension $n\geq 3$. For each integer $i$ with $2\leq i\leq n-1$, let $K_X$ denote the canonical divisor, let $g_i(X,\mathcal{L})$…