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.
679
questions
5
votes
1
answer
281
views
Isomorphisms of complete intersections
Let $X, Y\in \mathbb{P}^n$ be two singular Fano complete intersections of the same multidegree $(d_1,…,d_r)$.
If we assume there is an isomorphism $f\colon X\rightarrow Y$ are there any assumptions so ...
5
votes
1
answer
957
views
Which singularities of log pairs do not depend on the resolution?
Let $(X,\Delta)$ be a log pair (we assume the coefficients of $\Delta\leq 1$, but could be negative rationals), and I use the definitions in the book of "Birational Geometry of Algbebraic varieties" (...
- 3,362
5
votes
2
answers
354
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 $Y$ ...
- 9,095
5
votes
2
answers
571
views
Uniruled degenerations of abelian varieties
Suppose I have a smooth projective variety $X$ over $\mathbb{C}$ with $K_X$ semiample, and consider the fiber space $f:X\to Y$ given by $|\ell K_X|$, for some $\ell>0$ large, where $Y$ is a normal ...
- 73
5
votes
1
answer
1k
views
What goes wrong to use "bend-and-break" trick for singular varieties?
When $X$ is a smooth projective variety, one can use Mori's bend-and-break trick to establish the cone theorem. However, when $X$ has singularity (say klt. singularity), the cone theorem is obtained ...
- 3,362
5
votes
1
answer
1k
views
Example of cone of numerically effective curves which is not polyhedral
I think I have seen more than one reference in which the cone of numerically effective curves can be 'not polyhedral', i.e. with an infinite number of extremal rays
I cannot remember where I read ...
- 2,205
5
votes
1
answer
637
views
What English translations are there of work done by the Italian school of algebraic geometry?
What English translations are there of work done by the Italian school of algebraic geometry?
Perhaps I'm being too spoiled here, given that mathematical French, German, Italian are much easier to ...
5
votes
1
answer
1k
views
Definition and sigularity of Ramified covers
Let $X$ be a normal variety over $\mathbb{C}$.
In their book Birational geometry of algebraic varieties, Kollár and Mori define [Definition 2.50 and 2.51] a ramified m-th cyclic cover associate to a ...
- 3,362
5
votes
2
answers
662
views
Crepant resolutions of ODP's on a 3-fold
It seems to be a well-known fact that if we have a 3-fold $X$ with only ODP singularities (ordinary double point) and a smooth Weil divisor $E$ passing through them, then by blowing-up $X$ along $E$ ...
- 2,098
5
votes
1
answer
427
views
Rational contraction and Proj of section ring
I am reading the paper "Mori Dream Spaces and GIT" by Hu and Keel.
https://arxiv.org/abs/math/0004017
I cannot understand the proof of Lemma 1.6 in it.
Let $X$ be a normal projective variety....
5
votes
1
answer
404
views
Termination of a minimal model program
I am reading "The dual complex of
singularities" by de Fernex, Kollár
and Xu and in the proof of Corollary 24 I have encountered a bit of
reasoning that confuses me.
Let $(X, \Delta)$ be a $\...
- 4,060
5
votes
1
answer
157
views
Rational map given by pfaffians
Consider a general skew-symmetric $(n+1)\times (n+1)$ matrix $Z$, and let su map $Z$ to the point of $\mathbb{P}^n$ determined by $[pf_0(Z):\dots:pf_n(Z)]$ where the $pf_i(Z)$ are the principal ...
user117617
5
votes
1
answer
1k
views
Recognizing a Mukai flop
Let us first review the usual construction of a Mukai flop. Suppose $M$ is a smooth $2m$-dimensional projective variety over $\mathbb C$ containing a closed $m$-dimensional subvariety $W$, and suppose ...
- 497
5
votes
1
answer
691
views
A question about the construction of Francia flip
Here is the construction.
Start with $U$ a normal variety of dimension 3 with a unique singular point locally isomorphic to the quotient of $(x, y, z) \to (-x, -y, -z)$. Also assume we have a small ...
- 51
5
votes
1
answer
366
views
Étale covers and birationality of varieties
All varieties are assumed to be projective over $\mathbb{C}$. Let $f_1: Y \to X$ and $f_2: Y' \to X$ be étale morphisms with same finite Galois groups (to be honest, I don't know what does Galois ...
- 3,362
5
votes
1
answer
190
views
Strictness of the inequality relating the Iitaka dimension and algebraic dimension
For any (compact and connected) complex manifold $X$ and any line bundle $L$ on $X$ we have the well known inequalities $\kappa(X,L)\leq\alpha(X)\leq\dim(X)$ relating the Iitaka-dimension $\kappa(X,L)$...
- 1,114
5
votes
2
answers
1k
views
Unirational implies rationally connected
It is evidently a well-known fact that a unirational variety $X$ over an algebraic closed field (i.e. there is a dominant rational map from $\mathbb P^n$ to $X$) is rationally connected (by which I ...
- 2,098
5
votes
1
answer
450
views
Variety of negative Kodaira dimension contains a projective line
Does a smooth projective variety over an algebraically closed field of negative Kodaira dimension contain a projective line?
I do not want to assume any conjectures.
- 117
5
votes
1
answer
385
views
surface with rational curve in the double locus
I am interested in the existence of a surface $X$ over $\mathbb{C}$ with the following properties (or a reason for why one cannot exist):
$X$ is slc (and not-normal)
There is rational curve $C \...
- 379
5
votes
1
answer
233
views
Blowing-up an ideal generated by squares
Let $f_1,\dots,f_r$ be regular functions on a smooth projective variety $X$, and consider the ideals $I = (f_1^2,\dots,f_r^2)$ and $J = (f_1,\dots,f_r)$. Let $Y = Z(I)$ and $W = Z(J)$ be the ...
user114666
5
votes
1
answer
705
views
Are stably birational varieties birational?
We say that two (irreducible) algebraic varieties $X$ and $Y$ are stably birational if $X \times \mathbb{P}^n$ is birational to $Y\times \mathbb{P}^n$ for some $n\ge 0$.
The natural question is then ...
- 7,652
5
votes
1
answer
3k
views
Resolution of the indeterminacy locus of a rational map away from an irreducible component
Suppose I have a rational map between two smooth, complex, projective varieties (or a meromorphic map between two compact, complex, manifolds)
$X$ and $Y$. Ordinarily one eliminates the indeterminacy ...
- 51
5
votes
1
answer
233
views
Extending rational maps of nodal curves
Let $R$ be a discrete valuation ring with fraction field $K$ and $C, D$ two nodal (=prestable) curves over $\operatorname{Spec} R$. If I have a map $C_K \to D_K$ between the restriction of the curves ...
- 1,084
5
votes
1
answer
611
views
Conormal bundle of irreducible components of an exceptional divisor
Let $X$ be a smooth complex variety of dimension $3$ and $Y$ a (perhaps singular) normal complex variety also of dimension $3$ which is smooth outside a point $y \in Y$. If needed one may assume ...
- 8,738
5
votes
0
answers
171
views
Equations for conic del Pezzo surfaces of degree one
Let $X$ be a del Pezzo surface of degree one over a field $k$ of characteristic not $2$ equipped with a conic bundle $\pi: X \rightarrow \mathbb{P}^1$. By Theorem 5.6 of this paper, $X$ admits a ...
- 241
5
votes
0
answers
163
views
Strong factorisation conjecture for toric varieties
In this survey is remarked (see page 6 after Example 1.12) that to prove the
Conjecture 1.10 (Strong factorisation). Let $\phi: X \dashrightarrow Y $ be a birational map
between two quasi-projective ...
- 6,000
5
votes
0
answers
155
views
The structure of the Hilbert scheme of conics contained in hypersurfaces in $\mathbb P^3$
We work over a field of characteristic $0$. Let $X\hookrightarrow\mathbb P^3$ be a geometrically integral hypersurface of degree $\delta$. It is well known that the Hilbert scheme of conics in $\...
- 403
5
votes
0
answers
157
views
Steps of the MMP "in family"
Let $\pi\colon X\to Y$ be a morphism between irreducible varieties with terminal singularities (let us say smooth if you want). I suppose that I have an open subset $U$ of $Y$ over which the fibres ...
- 7,652
5
votes
0
answers
335
views
Jacobian fibration of an abelian fibration
Let $f \colon S \rightarrow C$ be a minimal elliptic surface and let $g \colon J \rightarrow C$ be its jacobian fibration. In this case, we know that the fibers of $g$ are better behaved that the ones ...
- 625
5
votes
0
answers
167
views
Unirationality of universal Jacobian over special strata of moduli space of pointed genus 3 curves
Let $M_{3,1}$ be the (coarse, non-compactified) moduli space of genus $3$ curves with a marked point over a field $k$ of characteristic zero. Throwing away the hyperelliptic curves, take the open ...
- 949
5
votes
1
answer
787
views
Definition of discrepancy
In Kollar and Mori "Birational geometry of algebraic varieties" discrepancy is defined as following way.
Let X be a normal variety and $D = \sum_i a_i D_i$ be a $\mathbb{Q}$ divisor. Assume ...
- 81
5
votes
0
answers
498
views
When does a Cartier divisor a pull-back of a Cartier divisor?
Suppose $f: Y \to X$ is a projective birational morphism between two varieties with mild singularities. For example, we can assume $X$ is normal and kawamata log terminal, $Y$ is $\mathbb Q$-factorial....
- 3,362
5
votes
0
answers
223
views
In search for examples concerning pushforward of nef divisors and lc-trivial fibrations
My question is motivated by ideas around the moduli b-divisor of an lc-trivial fibration (see for instance the following paper by Ambro https://arxiv.org/pdf/math/0308143.pdf).
In such a setup, one ...
- 625
5
votes
0
answers
209
views
Description of flop as graded algebra
I am looking for an example of a flop $Y \to X \leftarrow W$, possibly with exceptional locus at least a $\mathbb{P}^2$, where $X = \text{Spec } A$ is affine and $Y,W$ can be described as explicit ...
- 1,869
5
votes
0
answers
414
views
What is the fundamental group of Kontsevich's space of stable maps?
... at least in the case where the target is a rationally connected variety.
This question is a follow-up to question
Constructing embedded families of curves with general moduli
and Jason Starr's ...
- 1,971
5
votes
0
answers
542
views
Semistable sheaves on rationally connected manifolds
Let $(X,H)$ be a polarized projective manifold of dimension $n$ defined over $\mathbb C$, and let $\mathcal E$ be a reflexive sheaf on $X$.
If for every subsheaf $\mathcal F \subset \mathcal E$ the ...
- 8,738
4
votes
2
answers
744
views
Are Du Val singularities smoothable?
I am interested in when a Du Val surface singularity is smoothable. By Du Val singularity, I mean (the germ of a) isolated double point surface singularity admitting a resolution by blowups of ...
- 205
4
votes
1
answer
818
views
Complete intersections in toric varieties
Let $X$ be a smooth projective variety over the complex numbers. Is $X$ a global complete intersection inside a smooth projective toric variety?
user120812
4
votes
1
answer
615
views
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 $...
- 3,717
4
votes
2
answers
481
views
Smoothness of fibers over finite fields
Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties over a finite field of characteristic different from $2$. Is there any result on the existence of a point $y\in Y$ such that $X_y = ...
user58018
4
votes
1
answer
553
views
Chern classes of a vector bundle
Let $\mathcal{E}$ be a rank two vector bundle on $\mathbb{P}^2$ fitting in the following exact sequence
$$0\rightarrow \mathcal{O}_{\mathbb{P}^2}\rightarrow \mathcal{E}\rightarrow \mathcal{I}_p(-1)\...
user101890
4
votes
1
answer
480
views
A Bertini-type result for hypersurfaces containing a subvariety
Let $P$ be a smooth projective variety of dimension $4$ and let $Z$ be an irreducible subvariety of dimension $2$ ($Z$ is not necessarily smooth, but you can assume it).
Is there a smooth, ...
- 3,362
4
votes
1
answer
2k
views
pull back of an ample line bundle under a blow up
Suppose $\mu:X\rightarrow Y$ is a blow up of a smooth irreducible subvariety $Z$ of a smooth projective variety $Y$. Let $L$ be an ample Cartier divisor on $Y$. Let $E$ be the exceptional divisor of $\...
- 103
4
votes
3
answers
2k
views
The Riemann correspondence for riemann surfaces made explicit and its generalizations
Riemann showed (not proved rigorously) that there is a correspondence between compact riemann surfaces and algebraic function fields in one variable (does anyone know the year?).
To construct the ...
- 41
4
votes
2
answers
227
views
Non- simplicity of $\text{Bir}(\mathbb{P}_\mathbb{R}^2)$
In this paper: https://perso.univ-rennes1.fr/serge.cantat/Articles/nsgc-acta-c.pdf the authors said that their article 'directly implied' that $\text{Bir}(\mathbb{P}_\mathbb{R}^2)$ is not simple as ...
- 157
4
votes
1
answer
159
views
Rationality of $V_1$ fano threefold
In the book of Iskovskikh and Prokhorov it seems not known wether the $V_1$, an hypersurface of degree $6$ in the weighted projective space $\mathbb{P}(3,2,1,1,1)$, is rational or not. Is there any ...
- 372
4
votes
1
answer
1k
views
General fiber of a rational map
Let $f:X\dashrightarrow Y$ be a rational map, where $X,Y$ are reduced and irreducible varieties over a field of characteristic zero.
Is the general fiber of $f$ always reduced? Is this true if we ...
user75933
4
votes
1
answer
362
views
Applications of the boundedness of birational automorphisms
Recently the paper http://de.arxiv.org/PS_cache/arxiv/pdf/1011/1011.1464v1.pdf by Hacon, McKernan and Xu appeared on the arXiv. There the authors prove that the number of birational automorphisms of a ...
- 1,150
4
votes
1
answer
232
views
Cycles contained in ample enough hypersurfaces
Let $X$ be an irreducible smooth projective variety of pure dimension $d$ over the complex numbers and $Z\subset X\times X$ a codimension $d$ irreducible smooth closed subvariety.
Is there a smooth ...
user497064
4
votes
1
answer
292
views
Del Pezzo surfaces of degree four and complete intersections of two quadrics
Let $X = Q_1\cap Q_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y_0,y_1,y_2,y_3,y_4$.
Set $Q_1 = \{F_1 = 0\}$ and $Q_2 = \{...
- 8,852