All Questions
15 questions
9
votes
1
answer
1k
views
Reference request: tangent space to moduli space of coherent sheaves is $\operatorname{Ext}^1(E, E)$
Is there a standard reference for the fact that, in an appropriate algebraic-geometrical context, the tangent space at the point $[E]$ to the moduli space $\mathcal M$ is something like $\operatorname{...
- 1,980
8
votes
0
answers
284
views
Def-Obs theory of sheaves with fixed determinant on CY3.
Let $\mathcal{E}$ be a stable sheaf on a smooth complex projective threefold $X$ and $Ext^k_0(\mathcal{E},\mathcal{E})$ be the traceless Ext groups, defined by the kernel of the trace map
$$
Ext^k(\...
- 81
6
votes
1
answer
480
views
Is the Quot-scheme over non-singular curve reduced
Let $k$ be an algebraically closed field, $C$ a non-singular projective curve over $k$ of genus at least $2$ and $\mathcal{F}$ a locally free sheaf on $C$. Let $r,d$ be two integers satisfying $\...
- 1,981
3
votes
1
answer
506
views
Torsion free sheaves in flat families
Let $R$ be a dvr, $X$ a flat, projective, integral, normal $R$-scheme such every closed fiber is again integral, normal. Let $F$ be a torsion-free coherent sheaf on $X$, flat over $R$. Is it true that ...
- 1,981
3
votes
0
answers
132
views
Obstruction to the existence of a deformation of a subvariety compatible with the given deformation of a variety
Let $X$ be a smooth projective variety over a field $k$ of characteristic 0,
and let $A$ be a local Artinian $k$-algebra, say, $A=k\oplus I$
where $I$ is an ideal such that $I^2=0$.
Let $\frak X$ be a ...
- 14.1k
3
votes
0
answers
233
views
Obstruction to lifting coherent sheaves on discrete valuation ring
Let $R$ be a discrete valuation ring with algebraically closed residue field $k$. Let $K:=\mathrm{Frac}(R)$ the fraction field of $R$. Suppose $K$ is of characteristic zero. Denote by $\overline{K}$ ...
- 1,593
3
votes
0
answers
309
views
Examples of varieties with every stable sheaf simple
Are there examples of projective varieties over a non-algebraically closed field such that every geometrically stable sheaf on the variety is simple? I see, for example in Huybrechts-Lehn and in some ...
- 1,981
2
votes
1
answer
753
views
Push-forward of flat module under a finite, flat morphism
Let $f:X \to Y$ be a finite, faithfully flat morphism of noetherian, affine $\mathbb{C}$-schemes. One can assume $Y$ is non-singular. Let $A$ be a local artinian $\mathbb{C}$-algebra and $f_A:X_A \to ...
- 2,323
2
votes
0
answers
186
views
Base change, descent theory and coherent sheaves
Let $k$ be a field of characteristic zero and $X$ a smooth, projective $k$-variety. Let $E_{\overline{k}}$ be a coherent sheaf on $X_{\overline{k}}$ ($\overline{k}$ denotes the algebraic closure of $k$...
- 2,126
2
votes
0
answers
167
views
Open nature of $\mathcal{H}om$ functor/upper semi-continuity of $\operatorname{Ext}^i$
Let $k$ be an algebraically closed field, $T$ a $k$-scheme (can assume connected) and $X$ a projective variety over $k$. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k T$ flat over $T$. ...
- 1,981
1
vote
1
answer
280
views
Strong form of Grothendieck's algebrization theorem
Let $R$ be a Henselian discrete valuation ring with algebraically closed residue field $k$ ($R$ is not necessarily complete), $X$ a regular surface over $\mathrm{Spec}(R)$ and a sequence of locally ...
- 2,022
1
vote
1
answer
295
views
Isomorphism of sheaves in families of projective varieties
Let $\pi:\mathcal{X} \to S$ be a flat, family of projective varieties (here $\mathcal{X}$ and $S$ are noetherian). Let $E$ and $F$ be two locally free sheaves on $\mathcal{X}$ such that for all $s \in ...
- 2,022
1
vote
1
answer
219
views
Extending locally free sheaves and compatibility with fibers
Let $X$ be a smooth, projective variety over an algebraically closed field $k$ (of characteristic zero), $B$ a connected, noetherian scheme (possibly non-reduced) and $U$ an open subscheme of $X \...
- 2,323
1
vote
0
answers
381
views
On tangent space of relative quot schemes in positive characteristic
Let $k$ be an algebraically closed field of positive characteristic and $f:X \to S$ be a smooth, flat, projective morphism between noetherian $k$-schemes. Assume that $S$ is a non-singular quasi-...
- 503
0
votes
1
answer
918
views
Euler characteristic on flat families of quasi-projective schemes
Let $A$ be a noetherian integral domain (may be regular). Let $\pi:X \to \mathrm{Spec}(A)$ be a flat morphism. Suppose that each fiber of $\pi$ are quasi-projective. Let $\mathcal{F}$ be a coherent ...
- 2,323