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Results tagged with birational-geometry
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user 4096
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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Explicit functor from Kuznetsov component to derived category of K3 for rational cubic fourf...
It depends on each case. In general there is an explicit birational map from $X$ to a rational variety (typically $P^4$, or a quadric, as it is the case for pfaffian cubics), whose indeterminacy locus …
- 3,779
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vote
2
answers
640
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Relative minimality for conic bundles
Due to difficulties in finding references I am not sure of what relative minimality is about for conic bundles. Suppose we're working on con. bun. over surfaces.
The definition is ok: the fiber over …
- 2,821
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Moduli spaces and conic bundles
here
https://arxiv.org/pdf/1409.5033.pdf
in section 5 there's an example of unirational, non-rational, 3fold moduli space. I am not sure whether it is a conic bundle, though. Probably not.
- 3,779
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1
answer
181
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Decomposition of a morphism with positive dimensional fibers
It is well known that any birational morphism between projective varieties is a sequence of blow ups. Suppose now that I have a morphism $f:X \to Y$ with positive dimensional fibers, that is a project …
- 4,111
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1
answer
745
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Is the Hasse principle a birational invariant?
Is the Hasse principle a birational invariant?
It is probably a very trivial question, but I am a beginner in arithmetics.
CommunityBot
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2
votes
1
answer
109
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Rational quadric bundles and group quotients
Suppose I have a rational projective variety $X$ and a quadric bundle $Q \to X$ such that the total space of $Q$ is rational. Assume now that I operate on $X$ with a finite group $G$ and that the quot …
- 4,111
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0
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116
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rationality of Fano 3fold $X_{18}$
I need a reference for an explicit proof of the rationality of the Fano 3-fold $X_{18}$. By explicit I mean by a sequence of explicit birational transformations.
Thank you!
- 3,779
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answers
115
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invariance of the dimension of severi varieties of surfaces
Suppose I have a smooth projective surface $S\subset P^n$ embedded by a very ample linear system $|L|$. Consider now the generalized Severi varieties that parametrize curves on $S$ belonging to $|L|$, …
- 3,779
2
votes
1
answer
300
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F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$
It is conjectured that a divisor on $\overline{\mathcal{M}}_{g,n}$ would be ample if $D\cdot C >0$ for all F-curves $C\subset \overline{\mathcal{M}}_{g,n}$. Does the intersection degree with all F-cur …
CommunityBot
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0
answers
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Fields over which cubic hypersurfaces are rational
All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold …
- 3,779
21
votes
1
answer
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Rationality of intersection of quadrics
Let $X \subset \mathbb{P}^n$ be a complete intersection of two quadrics. It is classical that, if $X$ contains a line, then it is rational. The proof is very simple and basically it is given by taking …
- 5,163
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1
answer
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do geometric fibers determine scheme-theoretic image?
Let $X,Y,Z$ be reduced algebraic varieties, and let $Y$ and $Z$ be normal. Let $f:X \to Y$ and $g:X \to Z$ two surjective projective morphisms of algebraic varieties such that the geometric fibers of …
- 19.6k