All Questions
7 questions
3
votes
0
answers
171
views
Nice blowups are universal algebraic fiber spaces?
We say that a proper (maybe projective) morphism $f:X \to Y$ is a universal algebraic fiber space if $f_* O_X = O_Y$ holds universally. (This means: for any morphism $Y' \to Y$, if $X' = Y' \times_Y X$...
- 339
2
votes
1
answer
185
views
Finite étale cover of factorial ring
Let $A$ be a regular factorial ring.
Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?
- 6,262
9
votes
1
answer
1k
views
Picard group and reduced schemes
$\DeclareMathOperator\Pic{Pic}$If $A$ is a ring, then we know that $\Pic(A)=\Pic(A_\text{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $\Pic(X)=H^{1}_{et}(X,\...
- 12.9k
3
votes
0
answers
547
views
Finite Picard group
Does there exist a connected scheme, smooth, proper, and positive-dimensional over $\mathbb{C}$ with finite Picard group? Note that Picard group has cardinality$>1$. Also note that this can not ...
- 9,501
3
votes
1
answer
600
views
Why is it useful for the (relative) Picard functor to be representable?
I have been studying Chapter 8 of Neron models by Bosch et al. The first part deals with the relative Picard functor. A lot of work is done to make it representable. My question would be why this work ...
- 16.2k
8
votes
2
answers
814
views
Are there varieties with non finitely generated Picard group and vanishing irregularity?
Let $X$ be a smooth projective variety over an algebraically closed field $k$.
Can it happen that $q(X) := \dim H^1(X,\mathcal O_X) =0$ and $\textrm{Pic} \,X$ is not finitely generated?
Certainly, ...
- 66.3k
21
votes
1
answer
1k
views
Does every relative curve have a Picard scheme?
More precisely:
Let $X \to S$ be a smooth proper morphism of schemes such that the geometric fibers
are integral curves of genus $g$. Must the fppf relative Picard functor
$\operatorname{\bf ...
- 5,019