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Let $D$ be a pseudo-effective $\mathbb{R}$-divisor on a rational surface. Can we find an example that the numerical class of $D$ contains no effective divisor?

I would like to know if there is an example as follows: $X$ is a log canonical surface and $x \in X$ is an elliptic singularity such that The minimal resolution of $x$ is a circle of rational curves …

Let $A \subset B$ be a finite (finite type + integral) extension of integral domains and let $\mathfrak{p} \subset A, \mathfrak{q} \subset B$ be prime ideals such that $\mathfrak{q} \bigcap A =\mathfr …

Let $f:X \to Y$ be a surjective morphism of smooth projective varieties, $D$ be a simple normal crossings divisor on $X$ and $U_Y \subset Y$ be an open subset over which $(X,D)$ is log smooth (in the …

In Nakayama's book 2004, pg. 39, a formula is written: Let $f:X \to Y$ be a generically finite and proper surjective morphism, $D$ be a Cartier divisor on $Y$. Then, $f_*f^*D=(\deg f) D$. More preci …

Let $f:X \to Y$ be a finite surjective morphism from a $\mathbb{Q}$-factorial variety to a smooth variety. Let $D_Y$ be a prime divisor on $X$ and let $\bigcup D_i$ be the inverse image of $D_Y$. Do w …

Let $f:X \to Y$ be a finite map from a normal projective variety to a smooth projective variety, $D$ be a Cartier divisor on $X$. Do we have any relation between $\kappa(X,D)$ and $\kappa(Y,f_*D)$?

Q1: Let $(X,B)$ be a toric variety. There exists a toric resolution of singularities $f:(Y,E) \to (X,B)$. Here is my question: Is any lc center of $(X,B)$ an irreducible component of an intersection …

Given a log smooth pair $(X,B)$ with a reduced boundary divisor $B$, consider a birational model $\pi:X' \to X$ and a boundary divisor $B'$ which is given by $K_{X'}+B'=\pi^*(K_X+B)$. Here is my quest …

Let $(X,B)$ be a Calabi-Yau pair, that is, $(X,B)$ is lc (or klt for simplicity) and $K_X+B \sim_\mathbb{Q} 0$. Given a fibration $f:X \to Y$, there is an induced generalised pair $(Y,B_Y+M_Y)$ with $ …

Let $X$ be a projective normal variety, $D$ be a Cartier divisor on $X$ and $A$ be an ample divisor on $X$. Let $x \in X$ be a (not necessarily closed) point. If the asymptotic vanishing order of $D$ …

It is well-known that for any positive (1,1)-current $T$, there is a decomposition formula according to [Siu74]. That is, $T$ can be written as an infinite sum of prime divisors plus an extra part. In …

Let $\pi:X \longrightarrow C$ be a smooth projective family of varieties over a curve $C$. Fix a point $0\in C$ and assume the fiber $X_{0}$ has nonnegative Kodaira dimension. Is it possible to prove …

Let $\pi:X \longrightarrow C$ be a smooth projective morphism onto a smooth projective curve. If the general fibers are of nonnegative Kodaira dimension, is $\pi_{\ast} \mathcal{O}(k K_{X/C})$ nonzero …

Let $\pi:X \longrightarrow C$ be a smooth projective morphism onto a smooth projective curve, and $F$ be a central fiber. If the Kodaira dimension $\kappa (F)$ is nonnegative, is $\pi_{\ast} \mathcal …