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Results tagged with birational-geometry
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user 51424
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
Accepted
geometric genus of curves and generically finite morphism of surfaces
Does this work?
Take both $X$ and $Y$ to be $\mathbb{P}^1\times\mathbb{P}^1$, with the map $X\rightarrow Y$ corresponding to the product of a degree $2$ map $\mathbb{P}^1\rightarrow\mathbb{P}^1$ and …
- 5,958
12
votes
0
answers
254
views
Curves on rational surfaces and Lang's conjecture for M_g
There are a group of related conjectures associated to Lang's name - for this question I'll consider only the weakest one, namely that rational curves in a projective variety of general type are not Z …
- 5,958
1
vote
Accepted
Lifting of automorphism of rational surface to that on abelian variety
Denote $X\backslash\text{Sing}(X)$ by $X_0$ and its preimage in $Y$ as $Y_0$. Note that $Y_0$ is the Galois cover of $X_0$ corresponding to the normal subgroup $\mathbb{Z}[i]\times\mathbb{Z}[i]$ insid …
- 5,958
5
votes
Accepted
Current progress on rationality problem for complex hypersurfaces
For upper bounds, there is the paper https://arxiv.org/abs/1801.05397 of Schreieder, which shows (over any uncountable field of characteristic not equal to two) that for $N>2$, a very general $N$-dime …
- 5,958
6
votes
Accepted
$K_0$-equivalence of varieties
(Expanding my comments into an answer for more visibility.)
By Larsen-Lunts $K_0(\operatorname{Var}_k)/[\mathbb{A}^1]$ is the free abelian group on stable birational equivalence classes. It thus suff …
- 5,958