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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34 views

Spectrum of closure of function field?

Take $V$ and $W$ to be projective varieties over $\mathbb{C}$ of dimension $n$ and $m$, respectively. Let $f : V \to W$ be a fibre space, i.e., $f$ is a surjective map whose generic fiber is connected....
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30 views

Dimension of the exceptional divisor of a weighted blow-up

Consider a vector $a=(a_0,\ldots,a_n)\in\mathbb{N}^{n+1}$, and suppose for simplicity the $gcd$ among $a_0,\ldots,a_n$ is $1$. Consider moreover the $n$-dimensional projective space $\mathbb{P}^n$. If ...
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1answer
54 views

Multiple blow-ups

Suppose I consider, inside $\mathbb{P}^6$, the subvarieties $Y=V(x_2,\ldots,x_6)\simeq \mathbb{P}^1$ and $Z=V(x_0,\ldots,x_3)\simeq \mathbb{P}^2 $. I want to understand what is the blow up of $\mathbb{...
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60 views

Blow-up along a linear subspace: explanation and computations

Consider the quadric $Q$ in $\mathbb{P}^6$ defined as $x_0x_4+x_1x_5+x_2x_6+x_3^2=0$, and consider the subspace $Y\simeq \mathbb{P}^1$ contained in the quadric with homogeneous coordinates $x_0,x_1$. ...
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62 views

Integral of a blow up of $\mathbb P^2$

In Demailly's icm2006 p21, there is a statement: If $X$ is the surface obtained by blowing-up $\mathbb P^2$ in one point, then the exceptional divisor $E ≃ \mathbb P^1$ has a cohomology class {$\...
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56 views

How to show the blowing down a ruled surface is projective

Let $X$ be an irreducible projective threefold, let $S$ be a ruled surface over a curve $C$, where $C$ has finitely many singular points. Let $\pi:X\rightarrow Y$ be a map such that $\pi(S)=C\subset Y$...
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31 views

Proving that a map is birational of $\mathbb{P}^2$ to itself

Prove that the map $y_0 = x_1x_2$, $y_1= x_0x_2$, $y_2 = x_0x_1$ defines a birational map of $\mathbb{P}^2$ to itself. At which points are $f$ and $f^{-1}$ not regular? What are the open sets mapped ...
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35 views

Proving that a map is birational

Prove that, given an algebraic plane curve $X$, the map $f: \mathbb{A}^1 \to X$ defined as $f(t) = (t^2, t^3)$ and the map $f: \mathbb{A}^1 \to X$ defined as $g(t) = (t^2 - 1, t(t^2 -1))$ are ...
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1answer
69 views

$f:S\to X$ is generically finite of degree $d$, then $f_*f^*D=dD$

In Beuville's Complex Algebraic Surfaces, chapter 1, he consideres a map $f:S\to X$ between smooth projective varieties over $\Bbb{C}$ ($S$ is a surface, but I don't think this will be important) ...
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1answer
65 views

When is the projection from a point on the variety smooth?

Let $X\subset\mathbb{P}^n$ be a smooth irreducible variety (over $\mathbb{C}$) and $p\in\mathbb{P}^n$ a point. Let $\pi:\mathbb{P}^n\setminus\{p\}\to\mathbb{P}^{n-1}$ be the linear projection with ...
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57 views

Rational singularity of Spec, Proj and Spec of localization of a standard graded $2$-dimensional ring

If $X$ is a two dimensional Noetherian reduced excellent scheme, then we know by a Theorem of Lipman that $X$ has a desingularization, i.e., there exists a regular scheme $Y$ and a proper birational ...
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41 views

Is this particular fiber product of varieties reduced?

Let $X, Y, Z$ be irreducible projective varieties over $\mathbb{C}$ (you can assume all of them are normal). Let $f:X\rightarrow Z, g: Y\rightarrow Z$ be two birational projective morphisms, such that ...
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42 views

Birational invariants on cotangent bundles

I think this question should have a well known answer, and I thanks in advance any help with it! Assume the base field is the complex numbers. Question: Let $X$ and $Y$ be affine smooth irreducible ...
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1answer
39 views

Reference request: unique minimal model in birational class for a complex surface of general type (which may not be smooth)?

Frankly I know nothing about this type of algebraic geometry (or any kind of graduate level AG), but I need at one point in an article to say "there exists a unique minimal model, since the ...
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58 views

general questions about algebraic surfaces and Castelnuovo's contraction theorem

I am not really sure where should I ask this question so feel free to move it to other more fit community or add more tags. My master thesis is about Algebraic surfaces and Castelnuovo's contraction ...
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139 views

Why the exceptional divisor of blowup of $\{x^2+yt=0\}$ has multiplicity one (but not two)?

Let $X$ be the affine surface $\{x^2+yt=0\}\subseteq \mathbb C^3$, then $X$ has an $A_1$ singularity at $0$. Consider $X$ as a family of curves via the projection to the last coordiate $$\pi:X\to \...
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46 views

How to write $x= \sqrt{n^2+2}$ as a continued fraction and prove that x is irrational?

I have to write $x= \sqrt{n^2+2}$ as a continued fraction, where $n \in N^*$. I tried something like this: $$n< \sqrt{n^2+2}<n+1 \text{ so } [a_{0}]=n\\ x_1= \frac{1}{x-a_0}=\frac{1}{2}(\sqrt{...
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19 views

Intuition for strong $n$-complements in birational algebraic geometry.

Let $(X,B)$ be a log canonical pair equipped with a projective morphism $X \to Z$. Following Birkar [B], define a strong $n$-complement of $K_X+B$ over a point $z\in Z$ to be of the form $K_X + B^+$ ...
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36 views

Iitaka dimension, direct sums and bigness

I've started to study bigness of vector bundles and I'm wondering how the Iitaka dimension behaves under direct sum. Specifically, I wonder if we can show that if $E$ is big and $L$ is a vector bundle ...
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1answer
49 views

Hartshorne's proof of Castelnuovo's theorem

For those of us who have forgotten, Castelnuovo's theorem is the following: Theorem: If $Y$ is a curve on a surface $X$ with $Y \simeq \mathbb{P}^1$ and $Y^2 = -1$, then there is a morphism $f: X \to ...
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29 views

Plane minus a projective line vs. plane minus a conic

Let $k$ be a field. In $\mathbb{P} = \mathbb{P}^2(k)$, an irreducible conic $C$ is isomorphic to a projective line $U = \mathbb{P}^1(k)$ (which we take to be a line in $\mathbb{P}$). If I am not ...
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1answer
71 views

Interpretation and Implications of the Abundance Conjecture

So I'm interested in working on the Abundance conjecture, which states that for every projective variety $X$ with Kawamata log terminal singularities over a field $k$, if the canonical bundle $K_{X}$ ...
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44 views

$P^2$ blow up nine points

I quote the following paragraph form Kollar-Mori on page 22: Let $X$ be obtained from $P^2$ by blowing up at the nine base points of a pencil of cubic curves, all of whose members are irreducible. ...
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36 views

Intersection of pullback divisor and exceptional divisor

Let $X,Y$ are nonsingular complex projective surfaces and $f:Y\to X$ be a birational morphism. Let $E_1,\ldots,E_k$ are irreducible exceptional divisors of $f$. In this situation, we have $K_Y=f^{\...
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1answer
77 views

Very simple characterisation of $(-1)$-curves

Let $X$ be a complex surface (i.e., complex dimension 2) and $C$ a $(-1)$-curve in $X$, i.e., a reduced, compact, connected curve $C$ with self-intersection $-1$. I'm trying to understand the proof of ...
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76 views

Why is the blow-up of 9 points an elliptic surface?

One example of elliptic fibration is obtained as follows: Let $Z(F),Z(G)\subset\Bbb{P}^2$ be two non-singular cubics intersecting in distinct points $P_1,...,P_9$ and take the rational map \...
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89 views

The image of the point at infinity under birational equivalence from an elliptic curve

To find rational points on the curve $C:y^2=P(x)$ where $P$ is a cubic or quartic with rational coefficients and no repeated roots, I can derive a birational equivalence between it and an elliptic ...
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92 views

Sections of the exceptional divisor of a blowup

Let $C$ be a smooth curve in a smooth threefold $X$. Denote by $Y$ the blowup of $X$ along $C$ with exceptional divisor $E$. Then $E \rightarrow C$ is a $\mathbb{P}^1$-bundle over $C$. Is it true ...
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35 views

$\mathbb{C}(s_1,s_2,k)=\mathbb{C}(x,y)$, where $s_1,s_2$ are symmetric and $k$ is skew-symmetric

Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution (= $\mathbb{C}$-algebra automorphism of $\mathbb{C}[x,y]$ of degree two) defined by $(x,y) \mapsto (x,-y)$. It is not difficult to ...
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91 views

Blow up the elliptic singularity $x^2+y^3+z^6=0$

It is mentioned in this question https://mathoverflow.net/questions/148826/do-there-exist-double-points-on-an-algebraic-surface-in-mathbbp-mathbbc that $X=\{x^2+y^3+z^6=0\}\subset \mathbb C^3$ defines ...
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53 views

Concerning $\mathbb{C}(s_1,s_2,s_3,y)=\mathbb{C}(x,y)$, where $s_1,s_2,s_3$ are symmetric

Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution on $\mathbb{C}[x,y]$ defined by $(x,y) \mapsto (x,-y)$. Let $s_1,s_2,s_3 \in \mathbb{C}[x,y]$ be three symmetric elements with ...
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48 views

The form of $w \in \mathbb{C}(x,y)$ satisfying $\beta(w)=w$, $\beta$ an involution on $\mathbb{C}[x,y]$

The symmetric case: Let $w \in \mathbb{C}(x,y)$, and write $w=\frac{u}{v}$, where $u,v \in \mathbb{C}[x,y]$. Let $\beta: (x,y) \mapsto (x,-y)$; $\beta$ is an involution (= an automorphism of order ...
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16 views

Form of $u,v \in \mathbb{C}[x,y]$ satisfying $\mathbb{C}(s_1,s_2,k_1,k_2)=\mathbb{C}(u,v)$

Let $\beta$ be an involution on $\mathbb{C}[x,y]$, namely, $\beta$ is a $\mathbb{C}$-algebra automorphism of $\mathbb{C}[x,y]$ of order two. Denote the set of symmetric elements with respect to $\beta$...
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56 views

A subfield of $\mathbb{C}(x,y)$ invariant under an involution

Let $u,v \in \mathbb{C}[x,y]$. Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution (= $\mathbb{C}$-algebra automorphism of order two) defined by $(x,y) \mapsto (x,-y)$. Denote the ...
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49 views

Smoothness of general fiber of flat morphism

Suppose $f: X \to Y$ is a surjective, flat morphism of varieties over an algebraically closed field $k$ of characteristic zero. Suppose $X$ is smooth over $k$ and $Y$ is integral with fraction field $...
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32 views

Proof of the projection formula $(\pi^*D)\cdot D'=D\cdot (\pi_*D')$

I'm trying to prove the following (simplified) version of the projection formula Let $S',S$ be non-singular algebraic surfaces and $\pi:S'\to S$ a birrational map between them. If $D\in \text{Div}(S)$...
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79 views

A doubt in Birational Geometry OF Algebraic Varieties

In the book Birational Geometry OF Algebraic Varieties (János Kollár) there is a corollary as stated Corollary 4.14. Let $f : Y \longrightarrow (P \in X)$ be a resolution of the germ a ...
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59 views

How do I invert the rational map $(x,y)\mapsto \frac xy$ over the variety $C=(y^3=x^4+x^3)$?

The following question comes from Miles Reid's Algebraic Geometry, Chapter $4$ problem $8$. I'm asked to come up with a polynomial map $\varphi$ which inverts $\phi:C \dashrightarrow\mathbb A^1$, $\...
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1answer
54 views

Resolution of the projection map $\text{proj}_P:\mathbb{P}^2\dashrightarrow\mathbb{P}^1$

If $P=(0:0:1)\in\mathbb{P}_{\mathbb{C}}^2$, we define the projection from $P$ as the rational map: \begin{align*} \text{proj}_P:\mathbb{P}^2&\dashrightarrow\mathbb{P}^1\\ (x:y:z)&\mapsto(x:y) \...
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1answer
50 views

Nonreduced quadric hypersurfaces

I am reading a text in which one speaks of quadric hypersurfaces $\mathcal{Q}$ (in $\mathbb{P}^n(k)$ with $k$ a finite field) which are "everywhere nonreduced." What does this mean ? (And is there a ...
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2answers
100 views

Blowup along the fundamental locus of a rational map

Assume $f:X\dashrightarrow Y$ is a rational map between varieties, where $X$ is normal and $Y$ is complete. Then, the fundamental locus the $f$ (which means cannot extend the definition of $f$ on it), ...
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1answer
88 views

Small resolution of threefold with a node

I heard that a one-parameter family of surfaces acquiring a node (explicitly below) can be made into a smooth family through a small resolution of the ambient threefold. I want to know why. Explicity,...
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56 views

Algebraic characterization (or sufficient condition) when a (graded) local hypersurface has rational singularity

Let $(S, \mathfrak n)$ be a regular local ring of dimension $d\ge 4$ and let $R=S/(f)$ , where $0\ne f \in \mathfrak n^2$. Then $\dim R=d-1\ge 3$. If $\mathfrak m$ is the maximal ideal of $R$ then $\...
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27 views

nontrivial compactifiable variety

In Kollar and Mori, after the first bend and break, saying that if a surjective map of curve $C$ to a proper variety $X$, moves with a fixed point, then it degenerates into a sum of rational curves ...
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1answer
73 views

Degree $1$ birational map of $\mathbb{P}_k^n$

I wonder why the degree $1$ birational maps of $\mathbb{P}_k^n$ are the automorphisms of $\mathbb{P}_k^n$? In particular, why are they defined everywhere on $\mathbb{P}_k^n$? I know each degree $1$ ...
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1answer
55 views

Why is this an automorphism of $\mathbb{P}^2$?

I know the group of automorphisms of $\mathbb{P}^n$ is equal to the degree $1$ birational transformations of $\mathbb{P}^n$ and every birational transformation of $\mathbb{P}^n$ can be written as $[...
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35 views

Birational complex smooth varieties and homotopy equivalence

Let $X$, $Y$ be two birational complex smooth projective varieties. Note they are also compact complex manifolds. I wonder if they are homotopy equivalent? If not, what topological properties do they ...
2
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1answer
98 views

Birational transformation of $\mathbb{P}^n_k$

I wonder why every birational transformation of $\mathbb{P}^n_k$ can be written as $[f_1:...:f_{n+1}]$, where the $f_i$ are homogeneous polynomials? I wonder how to deduce this from the definition of ...
3
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2answers
224 views

Birationally transforming general curve of genus 1 to Weierstrass form

What are general rules to birationally transform general curve of any degree of genus 1 to Weierstrass form, provided we have one rational point? Example of curve of degree 12: $$x^9 y^3+9 x^9 y^...
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30 views

Some special class of rational curves.

Let $G(z)$ and $S(z)=\sum_{k\geq 1}k s_k z^k$ as polynomial in $z$. Let $$X(z):=\frac{z}{ G(S(z))},\quad Y(z):=\frac{S(z)}{z} G(S(z)) \tag{$***$} $$ be a rational parametrization of a plane ...