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      Questions tagged [ag.algebraic-geometry]

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

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      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 ...
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      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 ...
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      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 ...
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      1answer
      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 ...
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      0answers
      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 (...
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      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 $...
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      0answers
      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 \...
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      1answer
      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 $b_1(X)=1$. ...
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      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
      1answer
      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 ...
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      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
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      1answer
      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
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      1answer
      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, \...
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      0answers
      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
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      0answers
      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-...

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