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Results tagged with birational-geometry
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user 3377
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.
3
votes
Blowing up of a singular subvariety
Just blow up the singular
point in a variety $X_1$ which is obtained from
a smooth, irreducible manifold $X$ by identifying
points $x$ and $y$. The blow-up divisor
is ${\Bbb P} T_xX\coprod {\Bbb P}T_y …
- 9,237
1
vote
K3 surfaces and density of rational curves
Rational curves are dense on K3 for all K3 outside of a Baire set
(countable union of closed, nowhere dense). Here is the reference:
https://arxiv.org/abs/1004.5167, Density of Rational Curves on K3 S …
- 9,237
5
votes
2
answers
383
views
fibers of birational contraction for complex manifolds - are they Moishezon?
Let $X$ be a smooth complex manifold and
$\phi:\; X \mapsto Y$ a proper holomorphic
map which is birational ("birational contraction"),
and $Z= \phi^{-1}(y)$ its fiber in a point $y$.
The variety $Y$ …
- 9,237
4
votes
0
answers
98
views
Existence of a rational curve in the center of a birational contraction for symplectic singu...
Let $M$ be a holomorphically symplectic
complex manifold, and $f: M \to X$
a holomorphic, birational contraction to a Stein
variety $X$, contracting a subvariety $E$
to a point, and bijective outside …
- 9,237
13
votes
1
answer
642
views
Does a resolution of a rational singularity have rationally connected fibers?
A rational singularity is a singularity of a
complex variety $X$ such that for any
resolution $\pi:\; \tilde X\rightarrow X$ the
higher direct images $R^i\pi_*(O_{\tilde X})$
vanish for all $i>0$. Sup …
- 9,237