# Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

15,472
questions

**1**

vote

**0**answers

64 views

### Which endomorphisms of the Tate algebra are “algebraic”?

For an abelian variety $A$ over a field $k$ with characteristic different from $\ell$ and Galois group $G = Gal(\overline k/k)$, there is always an injective map of the form:
$$\mathbb Q_\ell\otimes ...

**4**

votes

**0**answers

84 views

### Convolution in K-Theory via an Example (From StackExchange)

I've spent lots of time in Chriss and Ginzburg's "Complex Geometry and Representation Theory" and despite convolution (in Borel-Moore homology or K-theory) being very central, I feel like I'm still ...

**0**

votes

**0**answers

114 views

### Bertini's theorem for singular varieties

I have extended the Bertini's theorem (Hartshorne II.8.18) for singular projective varieties, in order to show this, this or this.
(I think that they use Bertini for a singular veriety, but Hartshorne ...

**4**

votes

**1**answer

110 views

### Simple vector bundle isomorphic to one of its twistings

Let $V$ be a vector bundle over an algebraic curve $C$, ad assume that that $V \cong V \otimes L$ for some line bundle $L$.
If $V$ is decomposable this is clearly possible, for example take $V \cong ...

**2**

votes

**0**answers

49 views

### Understanding the universal sheaf locally

Suppose I have a projective, flat morphism $\pi : X \to S$ between smooth projective varieties over $\mathbb{C}$. By the work of Simpson, there exists a moduli space $M = M(X/S, v)$ of torsion-free (...

**1**

vote

**0**answers

45 views

### Definitions of hypercovers by generalized cover

Let $f: E\to B$ be a map between presheaves of sets. One says that $f$ is a generalized cover if given any map $rX\to B$, there is a covering sieve $R\hookrightarrow rX$ such that for every element $...

**2**

votes

**0**answers

62 views

### Base Change Theorem in the inverse other

I suspect this may be an error or an typing error, but I cannot be sure.
In https://edoc.ub.uni-muenchen.de/19890/1/ModulGeneralHiggs.pdf (line 2, page 9) it is claimed that $f_{X,*}\pi_Y^* \simeq \...

**4**

votes

**1**answer

153 views

### Cobordism modelling fibration over $S^1$

Let $X$ be a closed oriented manifold which is a fibration over $S^1$ whose fiber $F$ is connected, i.e. $X\cong F\times[0,1]/\sim h$, for an $h\in \mathrm{Diff}(F)$.
Suppose that $b_1(X)=1$. ...

**1**

vote

**0**answers

72 views

### Is nefness preserved under base change

Let $f:X \rightarrow Y$ be a morphism between (geometrically normal) varieties over a field $k$, $\bar{k}$ be the algebraic closure of $k$ and $B$ be a Cartier divisor on $X$ which is $f$-nef, that is ...

**2**

votes

**1**answer

174 views

### Failure of $H^1(X, \mathcal{T}_X)$ to act freely on the isomorphism classes of liftings of a deformation

It is well know that the isomorphism classes of first order deformations of a nonsingular variety $X$ are in $1$ to $1$ correspondence with $H^1(X,\mathcal{T}_X)$.
It is also known that given any ...

**-1**

votes

**0**answers

84 views

### Classification of quartic and quintic algebraic curves [on hold]

Do you have idea of the classification of quartic and quintic reducible and irreducible algebraic curve?

**3**

votes

**1**answer

180 views

### Is $B\mathbb{G}_m$ strongly $A^1$-invariant?

I have just seen the definition of strongly ${A}_1$ invariance:
A sheaf of group $G$ is strongly $A_1$ invariance , if both $H^0(-;G)$ and $H^1(-;G)$ is $A_1$-invariant.
I haven't got too much ...

**4**

votes

**1**answer

200 views

### Question about an implication of Thomason's étale descent spectral sequence

On page 5 of this paper by Dwyer and Mitchell, it is said that Thomason's étale descent spectral sequence from his paper Algebraic K-theory and étale cohomology, which reads
$$H^p_{\acute{e}t}(X, \...

**4**

votes

**0**answers

107 views

### Weak homotopy equivalence of sites

There are several notions of weak homotopy equivalence for topological spaces. The standard one can be formulated as follows: a map of spaces $X\to Y$ is a homotopy equivalence if the map of ...

**13**

votes

**0**answers

284 views

### When simple cohomological computations predict ingenious algebro-geometric constructions?

Classical algebraic geometry is full of ingenious constructions and miraculous coincidences: 27 lines on a cubic surface are related to Weyl lattice of type $E_6,$ lines on an intersection of four-...