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.

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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, ...
Francesco Polizzi's user avatar
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 ...
Tom Goodwillie's user avatar
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 ...
IMeasy's user avatar
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19 votes
2 answers
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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 $Y$?
hughes's user avatar
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18 votes
3 answers
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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 ...
J.C. Ottem's user avatar
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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 ...
Javier Álvarez's user avatar
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 ...
Koopa's user avatar
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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 ...
Lars's user avatar
  • 4,400
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 ...
Jesko Hüttenhain's user avatar
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 ...
Karl Schwede's user avatar
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14 votes
4 answers
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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 ...
Ariyan Javanpeykar's user avatar
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 $X$?
byu's user avatar
  • 666
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?
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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 ...
Charles Siegel's user avatar
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 ...
user avatar
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 ...
Karl Schwede's user avatar
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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 $i>0$. ...
Misha Verbitsky's user avatar
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 ...
Andrea Ferretti's user avatar
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...\...
user avatar
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 ...
Andrea Ferretti's user avatar
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?...
Hans Sachs's user avatar
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 ...
dhy's user avatar
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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 ...
roymend's user avatar
  • 221
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
Gianni Bello's user avatar
  • 1,150
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 ...
bananastack's user avatar
  • 1,250
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, ...
HNuer's user avatar
  • 2,098
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 ...
Arno Fehm's user avatar
  • 1,989
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 $V$ ...
R.P.'s user avatar
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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 $Y$ ...
JME's user avatar
  • 2,972
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.
IMeasy's user avatar
  • 3,707
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\...
user avatar
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. ...
Jérémy Blanc's user avatar
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 ...
Morgan Brown's user avatar
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 ...
THC's user avatar
  • 4,313
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-...
user avatar
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 ...
Caligula's user avatar
  • 375
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 ...
YangMills's user avatar
  • 6,636
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\...
David C's user avatar
  • 9,792
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 ...
Stefano's user avatar
  • 625
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 ...
Joachim's user avatar
  • 449
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 $Sec_k(V^n_{2})\subseteq\mathbb{P}^N$ ...
user avatar
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?
user avatar
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 ...
user avatar
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 ...
user237522's user avatar
  • 2,783
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 ...
user avatar
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 ...
Ste3an's user avatar
  • 133
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 ...
user avatar
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 ...
HNuer's user avatar
  • 2,098

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