All Questions

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Let $\pi\colon X\rightarrow \text{Spec}(R)$ be a smooth projective morphism with geometrically integral fibers. In ...

Following up on this question which received a negative answer, I wonder if something weaker is true. We work in the same set-up as the previous question. Let $B$ be a smooth quasi-projective variety ...

Let $S$ be a connected smooth complex projective surface. Let $Sym^{d}(S)$, $d\in \mathbb{Z}^+_0$, be the $d$-th symmetric product of $S$ parametrizing $0$-cycles of degree $d$. Let $Sym^{d,d}(S)=...

Let $B$ be a smooth quasi-projective variety over a field of characteristic zero. Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $...

Let $Sym^m(X)$ be the $m$th symmetric product of a smooth projective variety $X$, $n=\dim(X)$, $Y_1$ an ample hypersurface of $X$, and $CH_0(X)_{hom}$ the Chow groups of $0$-cycles of degree $0$....