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Results tagged with birational-geometry
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user 70593
Birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.
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Can Kummer surfaces coming from the same abelian surface be Cremona equivalent / isomorphic?
Assume we are given a simple abelian surface $A$ which has 2 non-equivalent principal polarizations $D_1$ and $D_2$ in $NS(A)$ (up to isomorphism), thus giving rise to two non-isomorphic smooth projec …
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Which blow ups in the base of a conic bundle preserve the "standard" condition?
Assume we are given a nontrivial standard conic bundle $\pi: X\rightarrow S$, that is $X$ and $S$ are smooth projective varieties (say over $\mathbb{C}$), $\pi$ is flat and furthermore we have $Pic(X) …
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Can one control the ramification of a Brauer class under birational morphisms?
Assume we are given a Brauer class $\xi\in Br(k(\mathbb{P}^n))$ ramified at some divisor $D\subset \mathbb{P}^n$, here $k=\mathbb{C}$.
If $f: \mathbb{P}^n\mathrel{-\,}\rightarrow \mathbb{P}^n$ is a b …
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How much information is encoded in the Jacobian-Kummer K3 surface of a curve of genus two?
Assume we work over $\mathbb{C}$.
Let $S\subset \mathbb{P}^3$ be a quartic surfaces with 16 nodes (ordinary double points). Then there is a simple principally polarized abelian surface $(A,\theta)$ …
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Are two conic bundles birational, if their bases are birational via a map preserving the ass...
Assume we have two standard conic bundles $\pi:C \rightarrow X$ and $\pi': C'\rightarrow X'$. That is $\pi$ and $\pi'$ are flat morphims of smooth varieties over $\mathbb{C}$ and both maps are relativ …
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vote
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How to test if these two threefolds are birationally equivalent?
Assume we have the projective plane $\mathbb{A}^2=Spec(\mathbb{C}[r,s])$. Now take the projective plane over this affine plane $\mathbb{P}^2_{\mathbb{A}^2}$ with homogenous coordinates $[u:v:w]$.
Def …
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Behaviour of (principal) polarizations of (singular) surfaces under birational maps
Assume we have two p.p. simple abelian surfaces $(A_i,D_i)$, i=1,2, over $\mathbb{C}$ with the following commutative diagram:
$\require{AMScd}
\begin{CD}
A_1 @>{birational}>> A_2\\
@V{2:1}VV @VV{2:1} …
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