Questions tagged [birational-geometry]
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.
675
questions
38
votes
5
answers
3k
views
Surfaces in $\mathbb{P}^3$ with isolated singularities
It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, ...
30
votes
3
answers
4k
views
Birational invariants and fundamental groups
In pondering this MO question and people's efforts to answer it, and recalling also something that I learned in my youth about using Morse theory ideas to prove some results of Lefschetz in the ...
21
votes
2
answers
2k
views
Applications of derived categories to "Traditional Algebraic Geometry"
I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical ...
21
votes
1
answer
2k
views
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 ...
19
votes
2
answers
2k
views
is the Hodge conjecture birationally invariant?
Let $X$ and $Y$ be two birational smooth projective varieties over the complex numbers. Assume $X$ satisfies the Hodge conjecture.
Is it known that the Hodge conjecture holds for $Y$?
18
votes
3
answers
1k
views
What is known about the birational involutions of P^3?
Describing the group of birational automorphisms of $\mathbb{P}^n$, $\mbox{Bir}(\mathbb{P}^n)$, for $n\ge 3$ is a fundamental open problem in birational geometry. For $n=2$, the classical theorem of ...
17
votes
2
answers
5k
views
Training towards research on birational geometry/minimal model program
Being a not yet enrolled independently supervised graduate student in mathematics, with prospects of applying to American graduate schools hopefully in a 1-2 years' time, I have a background of having ...
16
votes
7
answers
1k
views
When does $\operatorname{Aut}(X)=\operatorname{Bir}(X)$ hold?
Let $X$ be a projective complex manifold. Under what condition do we have the equality $\operatorname{Aut}(X)=\operatorname{Bir}(X)$? Here $\operatorname{Aut}(X)$ denotes the group of holomorphic ...
16
votes
3
answers
4k
views
Contracting divisors to a point
This is quite possibly a stupid question, but it is pretty far from what I normally do, so I wouldn't even know where to look it up.
If $X$ is a projective variety over an algebraically closed field ...
16
votes
1
answer
1k
views
The Order of Approximation of a Rational Map
Essentially, I am looking for a definition, which makes this a tricky question.
Consider a rational map $\phi: X \dashrightarrow \Bbb P^m$ of complex irreducible projective varieties. I want to ...
15
votes
0
answers
2k
views
Relative canonical divisors
Suppose that $X$ is a Gorenstein variety and that $\pi : Y \to X$ is a birational map of varieties with normal $Y$.
In this case the relative canonical divisor is defined to be $K_Y - \pi^*K_X$ (if ...
14
votes
4
answers
4k
views
Is the Euler characteristic a birational invariant
Suppose that $X$ and $Y$ are smooth projective varieties which are birationally equivalent. I would like to have that $$\textrm{deg} \ \textrm{td}(X) = \textrm{deg} \ \textrm{td}(Y).$$ Invoking the ...
14
votes
1
answer
690
views
If $X\times X$ is rational, must $X$ also be rational?
Is there an example of a smooth projective variety $X$ such that $X$ is irrational, but $X\times X$ is rational?
For instance, is $X\times X$ irrational for a smooth cubic threefold $X$?
14
votes
1
answer
491
views
Birational automorphisms of varieties of Picard number one
Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism.
Must $f$ necessarily contract a divisor?
13
votes
2
answers
988
views
Rationality of GIT quotients
I recently worked through most of the proof of the rationality of the moduli of genus 3 curves, which seemed to have the following structure:
Every nonhyperelliptic genus 3 curve is a smooth plane ...
13
votes
1
answer
780
views
Generalization of the rigidity lemma in birational geometry
Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected.
If there exists ...
12
votes
2
answers
1k
views
Blowups of Cohen-Macaulay varieties
Suppose that $X$ is a Cohen-Macaulay normal scheme/variety and $\pi : Y \to X$ is a proper birational map with $Y$ normal.
Question: Is $Y$ also Cohen-Macaulay? Are there common conditions which ...
12
votes
1
answer
560
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 $i>0$. ...
12
votes
2
answers
797
views
Minimal model which is necessarily singular
I was told during a summer school on the MMP a nice example (which I have also mentioned here on MO) that I'm not able to figure out anymore.
The example (due, I think, to Miles Reid) is a smooth ...
12
votes
1
answer
7k
views
Simple normal crossing divisors
I found the following definition.
A Weil divisor $D = \sum_{i}D_i \subset X$ on a smooth variety $X$ is simple
normal crossing if for every point $p \in X$ a local equation of $D$
is $x_1\cdot...\...
12
votes
1
answer
916
views
Schemes as a model category
I'm just learning some basics of model categories, so please forgive me if my question turns out to be trivial. I hope it does at least make sense.
A natural temptation is to relate this machinery to ...
12
votes
0
answers
277
views
birational geometry of moduli spaces: why work on the coarse space?
In studying the birational geometry of $\overline{\mathcal{M}}_g$, it seems standard to work with the coarse space $\overline{M}_g$ rather than the smooth stack $\overline{\mathcal{M}}_g$. Why is this?...
12
votes
0
answers
243
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 ...
11
votes
2
answers
2k
views
Motivation for birational geometry
I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...
11
votes
3
answers
716
views
Birational automorphisms of canonical models
Let $X$ be a variety with canonical singularities such that $K_X$ is ample.
Do you have a reference of the fact that every birational map from $X$ to itself is biregular?
Thank you
11
votes
1
answer
1k
views
Why is the standard flop a flop?
I have seen at least two ways to define flops (and similarly flips).
We start with $Y \to X$, a surjective birational morphism, contracting a locus of codimension at least 2, such that $K_Y$ is ...
11
votes
1
answer
857
views
Conditions for the contractibility of subvarieties
One often finds statements of the sort "and one can contract this subvariety $E\subset X$ to a point in the projective variety $W$," without any explanation of the reasons such a contraction exists, ...
11
votes
1
answer
471
views
A property of varieties between unirational and retract rational
EDIT: The vague question Q1 below is partially answered, while the concrete question Q2 seems to be still open.
Let $V$ be a geometrically integral variety over a field $K$.
I consider the following ...
11
votes
2
answers
746
views
Geometrically unirational varieties that are not unirational
By a variety over a field $k$, I mean a scheme that is separated and
of finite type over $k$. I indicate changes of the base ring by
subscripts.
Does there exist a smooth and projective variety $V$ ...
11
votes
0
answers
452
views
Is the $\hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?
The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ ...
10
votes
1
answer
736
views
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.
10
votes
3
answers
2k
views
Extending birational isomorphisms between planar curves to the P^2
Let $k$ be a field and let $C,D$ be two integral curves in $\mathbb{P}^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $\mathbb{P}^2_k$. To be precise, does there exist a ...
10
votes
2
answers
1k
views
Picard group of a cubic hypersurface
Consider the following cubic hypersurface in $\mathbb{P}^5$:
$$
X = \{z_0z_3z_5-z_1^2z_5-z_0z_4^2+2z_1z_2z_4-z_2^2z_3 = 0\}\subset\mathbb{P}^5
$$
The singular locus of $X$ is the Veronese surface $V\...
10
votes
1
answer
592
views
Are stably rational surfaces all rational?
Let $X$ be an irreducible surface such that $X \times \mathbb{P}^1$ is rational. Is it true that $X$ is rational?
If the field is not algebraically closed, the answer is no in general (see A. ...
10
votes
2
answers
919
views
Why is $\mathbb{Q}$-factoriality not local in the étale topology?
I was reading Kollár and Mori's book today and stumbled on the following passage:
"The $\mathbb{Q}$-factoriality assumption is a very natural one if we start with smooth varieties, and it makes many ...
10
votes
1
answer
513
views
Quadrics in the Grothendieck ring
Let $\mathcal{Q}$ be an irreducible quadric in $\mathbb{P}^n(k)$, with $n \geq 2$ and $k$ a finite field. Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. It is well known (it appears) that ...
10
votes
1
answer
631
views
$K_0$-equivalence of varieties
Let $k$ be an algebraically closed field of characteristic zero.
Let $A$ be the $\mathbf{Z}$-subalgebra of the Grothendieck ring of $k$-varieties $K_0(\text{Var}_k)$ generated by classes of semi-...
10
votes
0
answers
326
views
References about conic bundles
I'm interested in stable rationality of conic bundles by means of Brauer group/unramified cohomology non-triviality, and I was wondering if there are some references for the basic properties of conic ...
10
votes
0
answers
586
views
Letters of a bi-rationalist
V.V. Shokurov has written several papers over the course of about 10 years which are called "Letters of a bi-rationalist". Here are the ones that I could find:
Letters of a bi-rationalist. I. A ...
9
votes
3
answers
1k
views
Singular cohomology and birational equivalence
Let us remenber that we have the following proposition of Artin and Mumford (in "Some elementary examples unirational varieties which are not rational" proposition 1.):
"The torsion subgroup $T_2\...
9
votes
2
answers
821
views
Reference request: birational automorphism group is finite
I am interested in having a look at the proof of the following fact: If $X$ is a smooth variety of general type, then $\mathrm{Aut(X)}$ is finite.
I know that this is proved in "On algebraic groups ...
9
votes
2
answers
1k
views
Reference request on birational invariance of Chow group of zero cycles of degree zero
Let $CH_0(X)^0$ denote the group of zero cycles of degree zero modulo rational equivalence.
I am looking for a reference for the following fact:
If $X$ and $Y$ are smooth and projective varieties ...
9
votes
1
answer
520
views
Degree of secant varieties of Veronese varieties
Consider the degree two Veronese embedding $\mathbb{P}^n\rightarrow\mathbb{P}^N$ and let $V^n_{2}\subset\mathbb{P}^N$ be the corresponding Veronese variety.
Let $Sec_k(V^n_{2})\subseteq\mathbb{P}^N$ ...
9
votes
1
answer
235
views
Liftable rational varieties
Is there an example of a rational smooth projective variety over a perfect field of characteristic $p$, that is not liftable to characteristic zero?
9
votes
2
answers
1k
views
How much can small modifications change the nef cone?
First let me give a precise formulation of the question; I'll give some background/motivation at the end.
If X is a projective variety which is Q-factorial (meaning X is normal, and some sufficiently ...
9
votes
1
answer
298
views
Concerning $k \subset L \subset k(x,y)$
The following is a known result in algebraic geometry:
Let $k$ be an algebraically closed field of characteristic zero (for example, $k=\mathbb{C}$).
Let $L$ be a field such that $k \subset L \subset ...
9
votes
1
answer
377
views
Set theoretic equation for Veronese varieties
Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety ...
9
votes
1
answer
470
views
Is the number of minimal models finite
Let $X$ be a variety of general type.
Assume that $\dim X = 3$. In https://eudml.org/doc/164223 it is proven that $X$ has only finitely many minimal models (i.e., only $\mathbb Q$-factorial terminal ...
9
votes
1
answer
367
views
Dimension-specific phenomena in algebraic geometry
In differential topology, there are some funny phenomena that can only happen in dimension 4. For example, only in dimension 4 you can have a closed topological manifold admitting infinitely many ...
9
votes
1
answer
852
views
Why is proving fully-faithfulness of an integral functor locally analytically sufficient?
More than once I've come across a statement in a paper about derived categories in which it says something to the effect of "in order to prove that $\Phi:D^b(X)\rightarrow D^b(Y)$ is fully-faithful we ...