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Let $\pi\colon X\to Y$ be a morphism between irreducible varieties with terminal singularities (let us say smooth if you want). I suppose that I have an open subset $U$ of $Y$ over which the fibres ...

Fix a field $k$. Let $X,Y$ be normal integral $k$-schemes of finite type and $h: Y \to X$ a proper birational $k$-morphism. Moreover, assume that $X$ is Gorenstein, i.e. it admits a canonical divisor $...

Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that $$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$ is an isomorphism, where $i:S\...