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Let $E$, $X$ be irreducible smooth algebraic varieties over the complex numbers and let $p \colon E \to X$ be a morphism which is locally trivial with respect to the Zariski topology. Since $p$ is a v …

I'm sorry for asking the question. There is an easy counter-example. Let $\mathbb{F}_1$ be the first Hirzebruch surface and let $\pi \colon \mathbb{F}_1 \to \mathbb{P}^1$ be a projection which makes $ …

Let $X$ be a smooth quasi-projective irreducible variety over the field of complex numbers $\mathbb{C}$. We denote by $\mathrm{Aut}(X)$ the group of algebraic automorphisms of $X$. Moreover, for a var …

Let $k$ be an algebraically closed field of characteristic zero and let $X$ be a toric variety over $k$, i.e. $X$ is a normal, irreducible $k$-variety and it admits an algebraic action of a torus with …

Let $k$ be an algebraically closed field and let $X$, $Y$ be varieties over $k$. Let us denote by $\mathcal{O}(X)$ and $\mathcal{O}(Y)$ the $k$-algebra of regular functions on $X$ and $Y$ respectivel …

Let $G$ be a linear algebraic group (not necessarily reductive) and let $X$ be an affine variety with a regular $G$-action (everything defined over the field of complex numbers $\mathbb{C}$). Denote …

Let $X$ be an affine spherical variety for some reductive algebraic group $G$. Let $X^0$ be the open orbit in $X$ under a fixed Borel subgroup $B \subseteq G$. Does there exists a function $f$ on $X$ …