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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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
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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
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1 vote
0 answers
226 views

Does analytic isomorphism imply local isomorphism?

If $ \mathfrak{p} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(A) $, and $ \mathfrak{q} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(B) $ such ...
Schemer1's user avatar
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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 ...
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
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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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
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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\...
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9 votes
0 answers
369 views

How can I "see" that a map is birational?

This came up with the Euler brick. Let $T=(p,q,r)$ be a Randall triple, i.e. $$(p^2-1)(q^2-1)(r^2-1)=8pqr\ \qquad\text{[eq.1]}.$$ There are tons of maps that map a triple $T$ to another $T'=(p',q',r')$...
Hauke Reddmann's user avatar
8 votes
1 answer
1k views

Progress on Bondal–Orlov derived equivalence conjecture

In their 1995 paper, Bondal and Orlov posed the following conjecture: If two smooth $n$-dimensional varieties $X$ and $Y$ are related by a flop, then their bounded derived categories of coherent ...
mathphys's user avatar
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7 votes
1 answer
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How to construct log-canonical (or Calabi-Yau), non-Cohen-Macaulay singularities of low codimensions?

(EDIT 07/06/11: although the question has not been settled definitely, Sándor's excellent answer and the comments by Angelo and ulrich have highlighted many potential obstructions to the constructions ...
Hailong Dao's user avatar
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7 votes
1 answer
674 views

Proving a variety is not unirational

It is known that if a variety is unirational then it is rationally connected. However, there are no known examples of rationally connected varieties which are not unirational. In these notes, at the ...
Derek Allums's user avatar
7 votes
1 answer
488 views

Field extensions over which algebraic varieties cannot acquire points

The following fact (slightly reworded here) is proven in this answer: If $K$ is a purely transcendental extension of an infinite¹ field $k$, then whenever a separated scheme $X$ of finite type over $...
Gro-Tsen's user avatar
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7 votes
1 answer
411 views

Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?

Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $...
Dimitri Koshelev's user avatar
6 votes
3 answers
792 views

Pseudo-automorphisms on Fano varieties

Is every pseudo-automorphism (self-birational map which does not contract any hypersurface) of a smooth Fano variety of Picard rank $1$ equal to a biregular automorphism? Remark: For $\mathbb{P}^n$, ...
Jérémy Blanc's user avatar
6 votes
1 answer
2k views

Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $...
Mikhail Skopenkov's user avatar
6 votes
1 answer
248 views

Iitaka dimension is invariant under surjective morphism between smooth projective varieties

I would like to prove the following result (working on $\mathbb{C}$) but get trouble with the other direction. Let $f:Y'\rightarrow Y$ be a surjective morphism between smooth projective varieties, ...
user avatar
6 votes
1 answer
2k views

Top self-intersection of exceptional divisors

Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...
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5 votes
2 answers
298 views

3-folds with "simple" Betti numbers and positive Kodaira dimension

I am interested to know an example of a simply connected smooth projective 3-fold $X$ (over $\mathbb{C}$) satisfying the following two constraints: $X$ has the same Betti numbers as $\mathbb{C}\...
Nick L's user avatar
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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 ...
Nati's user avatar
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4 votes
1 answer
994 views

Bertini's type theorems over imperfect fields

Let $X$ be a projective variety over an imperfect (hence infinite and char(k)=p>0) field $k$. If the local rings of $X$ are all regular, then can we say that a general hyperplane section $H$ is also ...
Omprokash Das's user avatar
4 votes
0 answers
185 views

Can Kummer surfaces coming from the same abelian surface be Cremona equivalent / isomorphic?

Assume we are given a simple abelian surface $A$ which has 2 non-equivalent principal polarizations $D_1$ and $D_2$ in $NS(A)$ (up to isomorphism), thus giving rise to two non-isomorphic smooth ...
Bernie's user avatar
  • 1,015
4 votes
1 answer
304 views

Singularities of fibrations

Let $f:X\rightarrow \mathbb{P}^2$ be a fibration, here $X$ is a projective variety of dimension three. Assume that there exixts a smooth curve $C\subset\mathbb{P}^2$ such that for any $p\in\mathbb{P}...
Puzzled's user avatar
  • 8,832
4 votes
1 answer
532 views

Discussion of Luroth's problem in an article of Beauville

I am reading a wonderful article of Arnaud Beauville, called La théorie de Hodge et quelques applications http://math.unice.fr/~beauvill/conf/Bordeaux2.pdf There is one place on page 12 that I can ...
aglearner's user avatar
  • 14k
4 votes
1 answer
344 views

Two bivariate polynomials (or rational functions) that generate $\mathbb{C}(x,y)$

Let $f=f(x,y),g=g(x,y) \in \mathbb{C}[x,y]$, each of degree $\geq 1$, and $f,g$ are algebraically independent over $\mathbb{C}$ (= their Jacobian $\in \mathbb{C}[x,y]-\{0\}$). (1) Is there a ...
user237522's user avatar
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4 votes
1 answer
477 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, ...
Li Yutong's user avatar
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3 votes
2 answers
768 views

Irreducible divisor in a basepoint free linear system

Let $X$ be a projective, normal variety over complex field with canonical singularities. Suppose $|D|$ is a basepoint free linear system, then is it true that the generic elements in $|D|$ are ...
Li Yutong's user avatar
  • 3,362
3 votes
1 answer
450 views

Automorphisms of Cartesian products

Let us consider the Cartesian product $X^r$, where $X$ is a smooth projective variety. There is a subgroup $Aut_{\Delta}(X^r)\subset Aut(X^r)$ of automorphisms of $X^r$ mapping a $k$-dimensional ...
user avatar
2 votes
2 answers
837 views

Rational maps and Kodaira dimension

Let $\phi:X\dashrightarrow Y$ be a generically finite, dominant rational map between smooth projective varieties over $\mathbb{C}$. Assume that $Y$ is of general type. May we conclude then that $X$ ...
Puzzled's user avatar
  • 8,832
2 votes
2 answers
598 views

Numerically negative exceptional divisor on a surface.

Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection ...
Jesus Martinez Garcia's user avatar
2 votes
1 answer
283 views

F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$

It is conjectured that a divisor on $\overline{\mathcal{M}}_{g,n}$ would be ample if $D\cdot C >0$ for all F-curves $C\subset \overline{\mathcal{M}}_{g,n}$. Does the intersection degree with all F-...
IMeasy's user avatar
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1 vote
0 answers
97 views

Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM

Does anyone know an example of a $ \mathbb{Q} $-factorial, normal, Cohen Macaulay, projective, Mori dream space $ Z $ over a field $ k $ of arbitrary characteristic such that the Cox ring of $ Z $ is ...
Schemer1's user avatar
  • 789