Let $T$ be a torus and $X$ a proper smooth curve over characteristic $0$ algebraically closed field $k$. What is $\text{Pic}(\text{Gr}_{T,X^n})$? Here $\text{Gr}_{T,X^n}$ is the BD Grassmannian over $X^n$, parametrising $n$-tuples of points in $X$, a $T$-bundle over $X$ and a trivialisation away ...

Definition A tree means a set-theoretic tree, that is a poset $(T,<)$ so that for each $x\in T$, the set $\{y\in T\mid y<x\}$ is well-ordered. Question: I would like to know if it is consistent with ZFC, the existence of a tree with height $\omega$ and each level of it has at most $\aleph_1$ elem...

Let $T$ be a torus and $X$ a proper smooth curve over characteristic $0$ algebraically closed field $k$. What is $\text{Pic}(\text{Gr}_{T,X^n})$? Here $\text{Gr}_{T,X^n}$ is the BD Grassmannian over $X^n$, parametrising $n$-tuples of points in $X$, a $T$-bundle over $X$ and a trivialisation away ...

If there is a smooth isometric embedding $f: (S^2, g)\rightarrow \mathbb{R}^n$, where $(S^2, g)$ is a sphere with Riemannian metric such that the corresponding sectional curvature is equal to $1$, and $n\geq 4$. Is $f(S^2)$ always a round $2$-dim unit sphere in $\mathbb{R}^3\subseteq \mathbb{R}^n$?