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Questions tagged [birational-geometry]

For questions on birational geometry, a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets.

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56 views

Blow Up of a Surface

I have a question a step in the example demonstating the blowing up in Liu's "Algebraic Geometry and Arithmetic Curves" in the excerpt below (or look up at page 320): We blow up the surface $X= ...
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27 views

Extend Morphism between Birationally Equivalent Curves

I have a question which arises from this one: Non Constant Morphism to Projective Line Assume we have two curves $C,D$ ($1$-dimensional, proper $k$-schemes) which are integral and birationally ...
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28 views

Blowup of non-singular varieties

How to construct a blowup $f$: X $\rightarrow$ Y between non-singular quasi-projective varieties? This is an exercise from Shafarevich "Basic Algebraic Geometry 1". I want to construct for any $n \...
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Extension of a rational map in codimension one - relative version

Suppose $X$ and $Y$ are smooth projective $T$-varieties, where $T$ is a smooth affine curve. Let $\phi:X\dashrightarrow Y/T$ be a rational map over $T$. My question is: is there a closed subset $Z\...
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Indeterminacy locus of an Iitaka fibration

This might be a trivial question, in this case I apologize. Let $X$ be a smooth projective complex algebraic variety of dimension $n$ and Kodaira dimension $n-1$. Let $\phi:X\dashrightarrow Z$ be the ...
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1answer
98 views

Non- simplicity of $\text{Bir}(\mathbb{P}_\mathbb{R}^2)$.

It was proven by Serge Cantat and Stéphane Lamy that the Cremona group $\text{Bir}(\mathbb{P}_\mathbb{C}^2)$ is not simple. It is later proven separately that the real Cremona group $\text{Bir}(\...
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1answer
40 views

A question on the proof of Rigidity Lemma in birational geometry

I am reading the book, Birational Geometry of Algebraic Varieties, by J$\acute{\mathrm a}$nos Koll$\acute{\mathrm a}$r et al.. (Rigidity Lemma) Let $Y$ be an irreducible variety and $f:Y\to Z$ a ...
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Lifting of automorphism of rational surface to that on abelian variety

The paper I am referencing is "Normal Subgroups of the Cremona Group." https://arxiv.org/abs/1007.0895. In theorem 5.14, at the bottom of page 52, the author stated for the abelian surface $Y= \mathbb{...
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1answer
44 views

Linearly Equivalence of pullback of divisors

Let $X_0=\mathbb{P}^2$ and $\eta: X_r \mapsto X_0$ be the blow-up of $p_1,\cdots, p_r$, where $p_1 \in X_0$. In a paper I am reading, the author states the following: If $C\subset X_r$ is an ...
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1answer
41 views

Multiplicity of Birational Map

Let $\varphi\in \text{Bir}(\mathbb{P}^2)$ be a birational map of degree $d$ sending the pencil of lines passing through a point $p\in \mathbb{P}^2$ onto a pencil of lines. How can we show that the ...
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Birational Morphism to a Normal Scheme Isomorphism

My question refers the answer of @user18119 in following thread: Are these two notions of Galois morphism the same We have $f:X\to Y$ a finite morphism of integral schemes and $G$ letautomorphism ...
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51 views

Tangent lines of elliptic curve with a fixed point and Weierstress form

I'm reading Ian Connell's Elliptic Curve Handbook for the details of Nagell's algorithm, which can construct the birational map from an elliptic curve to its Weierstress form. At the bottom of Page ...
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Definition of a rational map [duplicate]

Let $\phi:X \rightarrow Y$ and $\varphi: Y \rightarrow Z$ be rational maps between varieties and choose representatives $(U,\phi_U)$ and $(V,\varphi_V)$ respectively According to a set of notes I ...
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1answer
58 views

Reference request: Projection from a general point on a variety

Let $X$ be a smooth projective variety in $\mathbb{P}^n$, over the field of complex numbers or an algebraically closed field of characteristic $0$. EDIT (After Stefano's response): Assume $\dim X &...
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1answer
68 views

Is every complex algebraic variety birationally equivalent to a complex projective manifold?

Chow's Lemma states that every algebraic variety is birationally equivalent to a projective variety, and Hironaka's work on resolutions of singularities implies every projective variety is ...
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Maps between projective spaces induced by singular matrices

In this question, $V(a,k)$ denotes the $a$-dimensional vector space over the field $k$. Now consider $V(b,k)$ and $V(c,k)$, and let $M$ be a singular $(c \times b)$-matrix over $k$. Let $\ell_M: V(b,k)...
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A question on the base change of elliptic threefolds

Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $\sigma_i$ ,$i=1 \dots n$. None of these (in this specific example) "sections" are ...
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1answer
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The rationality theorem in birational geometry

I am now reading the proof of rationality theorem in birational geometry of algebraic varieties written by Kollár-Mori. (pp.86)The main confusing thing is the first step which reduced the big and nef ...
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1answer
97 views

Blow up of one point is isomorphic to $\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(1))$

My question comes from an example in Hartshorne (Example V.2.11.4) which I'm having trouble following. It is claimed that the Blow up of a point $p \in \mathbb{P}^n$ is isomorphic to $\mathbb{P}(\...
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1answer
98 views

Strict transforms after blowing up “complicated” ideals

I understand the concept of blowing up of an affine scheme along an arbitrary center but when it comes to compute strict transforms I have some difficulties in very simple examples. I would appreciate ...
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Computing $R^1f_*\mathcal{O}_\hat{X}(\pm E)$ for blow-up of $\mathbb{A}^2_k$ at the origin

Consider the blowup of the affine plane at the origin: $f:\hat{X}\to X=\mathbb{A}^2_k$. I want to show that $\mathcal{L}:=R^1f_*\mathcal{O}_\hat{X}(\pm E)=0$ in the most elementary way possible. It's ...
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1answer
143 views

Small contractions as blow ups

I am trying to learn a bit about birational morphisms: $f: X\rightarrow Y$, between (projective) normal varieties. In particular, it is well known that every such morphism is a blow-up (e.g ...
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1answer
92 views

Toric surfaces among rational surfaces

I'm learning about rational surfaces. Precisely, I've just read that all minimal rational projective, smooth surfaces are $\mathbb{P}^2$ and the Hirzebruch surfaces $\mathbb{F}_n$, for $n=0,2,3,\ldots$...
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Contraction of blow up of $\mathbf{P}^n$ at a linear subspace

Let $L=\mathbf{P}^m\subset\mathbf{P}^n$ be a linear subspace with $1\leq m\leq n-2$, say defined by $x_{m+1}=\ldots=x_n=0$. Let the blow up $X$ of $\mathbf{P}^n$ with center $L$. The exceptional ...
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Why is the torsion subgroup of the Neron Severi group a birational invariant?

Let $X$ be a smooth projective variety over $\mathbb{C}$, $NS(X)$ be its Neron-Severi group, i.e., the abelian group generated by divisors modulo algebraic equivalence, and $NS_{tor}(X)$ be the ...
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59 views

Compute multiplicity by intersections

Let $x\in X$ be a closed point in a normal variety $X/\mathbb C$, also assume $X$ is Cohen-Macaulay. Let $m_x$ be the maximal ideal of $x$. Suppose $f: Y \to X$ is a resolution of $m_x$ such that the ...
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1answer
163 views

Shafarevich's problem 1.7, showing the existence of a rational function.

Prove that for any two distinct points of an irreducible curve there exists a rational function that is regular at both, and takes the value 0 at one and 1 at the other. I think I can construct such ...
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1answer
52 views

Some problems related to unirational varieties

I am trying to understand the following contents from the notes and have several questions : If $X$ is unirational, that is, there is a dominant rational map $\mathbf{P}^N \dashrightarrow X$, then ...
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1answer
108 views

Birational transformation of conic to line (Cremona inversion)

I am following Discriminants, resultants, and multidimensional determinants and I am having trouble following the geometric argument in Lemma 2.11. I will outline the lemma and the proof here: ...
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1answer
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$C\times \mathbb{P}^1$ birational to $C'\times \mathbb{P}^1$ implies that $C$ and $C'$ are isomorphic.

Let $C$ and $C'$ be algebraic curves (complete and nonsingular over an algebraically closed field). Let us suppose that $C\times \mathbb{P}^1$ is birational to $C'\times \mathbb{P}^1$. I want to prove ...
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1answer
160 views

Complete Intersection of a Veronese embedding as Five Quadrics

Consider the Veronese embedding given by the linear system $|\mathcal{O}_{\mathbb{P}^2}(2)|$, $\phi:\mathbb{P}^2\to \mathbb{P}^5$. In exercise IV.5 of Beauville, Complex Algebraic Surfaces, we prove ...
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1answer
117 views

Example of a 3-dimensional rationally connected variety that is not rational.

We will call a proper variety $X$ rationally connected if any two general points of $X$ are in the image of some map $\boldsymbol{P} \to X.$ A variety is rational if it is birationally equivalent to $\...
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1answer
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Birational affine schemes

My question has to do mostly with the definition/terminology behind what a birational map is, in terms of affine schemes. For instance, if we have two affine schemes and a morphism between them say, $...
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156 views

Bijection morphism to be isomorphism?

Let $S$ be a smooth projective surface. $Bl_p(S)$ be the blow up of $S$ at a point $P$. Let $Y$ be a variety. If we have: $f:Bl_p(S)\to Y$, $g:Y\to S$ such taht $f\circ g$ is the natural projection, ...
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127 views

Characterization of surface blow up

Let $X$ be an connected surface which set-theoretically is the union of $\mathbb{P}^2-pt$ and $\mathbb{P}^1$. Suppose there is a birational morphism $\pi:X\to\mathbb{P}^2$ such that all of $\mathbb{P}^...
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1answer
144 views

Vanishing of derived pushforward

Let $f:Y\to X$ be a contraction of a rational curve between projective threefolds. Let $E$ be a sheaf on $Y$. Is it true that $Rf_*(E)=0$ iff $E$ is supported on the contracted curve? If so how to ...
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669 views

Understanding pullback of pushforward

Sorry if this is sort of a soft question. I added an example at the end to mitigate. So I hope you can bear with me here. Let $\phi : X \rightarrow Y$ be a morphism of noetherian schemes, $\mathscr F$...
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Atiyah flop, flip and related toric computation

I am trying to understand the definitions of flips and flops by studying examples in this article of Hacon and McKernan. I would like to ask why the toric varieties constructed in Ex. 1.13 are indeed ...
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1answer
103 views

Rigid and nef implies numerically trivial [closed]

If $X$ is a smooth projective variety and $L$ is an effective divisor that is both nef and rigid ($h^0(nD)=1$ for all $n\geq 0$), is $L$ numerically trivial?
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1answer
301 views

Normal bundle of a section of a projective bundle

Let $X=\mathbb{P}^1$, $E=\mathcal{O}\oplus\mathcal{O}(1)\oplus \mathcal{O}(2)$ and let $Y=\mathbb{P}(E)$. Denote $\pi:Y\to X$ the projection map. There is a section of $\pi$ corresponding to the ...
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Counting rational curves in a Calabi-Yau space of genus 0?

A Calabi-Yau manifold of genus 0 is topologically a sphere of some kind. Given that, how or why would we expect there to be so many rational curves in that space? Why is the total number of rational ...
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2answers
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Example of a non extendable rational map

A rational map is a class of equivalence on $$F=\{f:U \rightarrow Y \mid \mbox{morphisms with } U\subset X \mbox{ open subvariety}\}$$ given by $$f\sim g \iff f_{| (\operatorname{Dom}f \cap \...
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1answer
203 views

unique minimal models and birational automorphisms of algbraic surfaces

In his book on cubic forms Manin writes that if an algebraic surface has unique minimal model $X$ then "it follows easily from definitions" that the group of automorphisms of the field $k(X)$ is ...
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1answer
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Mori's “projective manifolds with ample tangent bundles”, Proposition 3

See http://www.jstor.org/stable/1971241?seq=6#page_scan_tab_contents, pages 597 and 598. At the bottom of page 597, we get the following diagram: $\begin{array}{ccccccccc} A/I & \xrightarrow{nat....
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Bend and break, the breaking part

See http://www.jstor.org/stable/1971241?seq=6#page_scan_tab_contents, pages 598 and 599. Suppose we have $\pi: \widetilde Y \rightarrow Y \rightarrow X\times \overline D \rightarrow \overline D$, ...
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1answer
89 views

Birational morphisms of surfaces

I read that in the category of projective smooth models of a function field $K = k(x,y)$ ($k$ algebraically closed), there is at most one morphism between two such models (the morphism being a ...
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1answer
196 views

Finite generated sheaf of graded $\mathcal{O}_X$ algebra

Let $X$ be a projective normal variety. I am having trouble finding out what it means for a graded sheaf of $\mathcal{O}_X$ algebra $R$ to be finitely generated over $\mathcal{O}_X$. Does it mean ...
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1answer
150 views

Criterion on nefness of a divisor on algebraic surfaces

Question 1: Let $X$ be a smooth rational surface with anti-canonical cycle, i.e, $-K_X$ is effective and its irreducible components form a polygon. Say, assume that $-K_X=\sum_\limits{i=1}^N D_i$ and $...
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1answer
101 views

Contraction morphism for surface

I am reading Mori and Kollár's 'Birational geometry of algebraic varieties'. I am confused when reading the proof of Thm 1.28, which is the classification of contraction morphism for surfaces. Here ...
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334 views

Pullback of pushforward of ample divisor under birational morphism

Suppose $f: Y\rightarrow X$ is a birational morphism and $H$ is an ample divisor on $Y$. Suppose $X$ is $\mathbb{Q}$-factorial (so $f_{*}H$ is $\mathbb{Q}$-cartier) and we take $f^{*}f_{*}H$. Why is ...