6 problems

Let $(K,v)$ be an algebraically maximal valued field with divisible value group and perfect residue field $k$. A field is called geometrically $C_1$ if every smooth projective sepa…

Let $(K,v)$ be an algebraically maximal valued field with divisible value group and perfect residue field $k$. A field is geometrically $C_1$ if every smooth projective separably r…

A field $k$ is called $C_1$ if every non-constant homogeneous polynomial of degree $d$ in $n+1$ variables over $k$ with $d\leq n$ has a non-trivial zero over $k$. A field $k$ is ge…

A field $k$ is $C_1$ if every hypersurface of degree $d$ in ${\mathbb P}^n$ over $k$ has a rational point whenever $d\leq n$. Let ${\mathbb Q}^{\rm ab}$ be the maximal abelian exte…

A field $L$ is $C_1$ if every hypersurface in $\mathbf{P}^{n}_{L}$ of degree $d\leq n$ has an $L$-rational point. A variety is separably rationally connected if two general points…

Let $k$ be a field of characteristic $0$ that is $C_1$, meaning that every hypersurface of degree at most $n$ in $\mathbb{P}^n_k$ has a $k$-rational point. Let $X$ be a terminal…