Mathematics

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.

362 questions
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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, $\mathscr F$ a coherent sheaf on $X$. I am looking to better…
rollover
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How do people define flop?

Let $f:Y \to X$ be a small contraction morphism of projective normal varieties. Then what is the usual way (if any) to define the flop $f^+: Y^+ \to X$ ? I could find a clear and uniform way to define flips in the literature, however, for flops…
Li Yutong
  • 4,354
11
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2 answers

Understanding rational maps in Algebraic Geometry-Examples of birational equivalence between varieties

I am struggling to understand the concept of rational maps. In Hartshorne, the definition of rational maps is as follows Let $X$ and $Y$ be varieties. A rational map $\phi:X\rightarrow Y$ is an equivalence class of pairs $\langle U,\phi_{U} \rangle$…
Damo
  • 1,108
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Birational Equivalence of Diophantine Equations and Elliptic Curves

A while ago I saw this question Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$ which was very relevant to a undergraduate research paper I am currently working on. The answer given for this problem describes a method for determining…
AnotherPerson
  • 6,008
8
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2 answers

What is the example of pseudo-effective divisor which is not an effective divisor

By definition, a pseudo-effective $\mathbb{R}$-divisor is the limit of effective $\mathbb{R}$-divisors in $N^1(X)$, I was wondering what is the example of pseudo-effective divisor which is not an effective divisor? One thing I don't understand is…
Li Yutong
  • 4,354
8
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"Flipping fractions" and a strange involution

Apologies for the vague title. Part of my question is what the technically accurate language is for this function $f$ I'm asking about. The Function I found this map on $\mathbb C^3$ I'd like to learn more about. It takes a triplet of numbers…
Chris Wolird
  • 339
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Algebraic surface with infinitely many exceptional curves

I am learning about the classification of Projective Algebraic Surfaces (in fact, Compact Complex Surfaces) and I am troubled with the following point. If I understood correctly every surface $X$ admits a (not necessarily unique) minimal model…
jameson
  • 115
8
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1 answer

Blow-up of $\mathbb P^2$ in 2 points is isomorphic to blowup of quadric in 1 point

How to show that blow-up of $\mathbb P^2$ in 2 points is isomorphic to blowup of quadric in 1 point? I think it is a standard fact, but Google can not help me with it. Update : I found an answer here: J. Harris " Algebraic Geometry: A First Course…
evgeny
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7
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1 answer

Factoring a birational morphism through blowup

Let $X,Y$ be smooth, proper varieties, and $f: X \to Y$ be a proper birational morphism. Suppose $E$ is a smooth, irreducible exceptional divisor, with the image $f(E)$ also smooth. Let $I$ be the sheaf of ideal associated to $f(E)$. Now, let $\pi:…
Li Yutong
  • 4,354
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1 answer

How close are birational surfaces to being isomorphic?

Birational geometry is extremely difficult for me to comprehend, because this type of transformation is so general and seems to preserve so little of the geometry. For example cubic surfaces are birational to hyperboloids, but the former class has…
roymend
  • 553
7
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2 answers

About a theorem in Beauville's book.

Look at the following theorem (due to Castelnuovo) taken from Beauville's book on complex surfaces: I have a question about the behavior of $f$ on the exceptional curves generated by $\eta$. Suppose that $E$ is an exceptional curve coming from a…
Dubious
  • 14,268
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Is the preimage of the non-normal locus a divisor?

Let $X$ be a complex, affine variety. Let $\nu:\tilde X\to X$ be the normalization of $X$ and denote by $D\subseteq X$ the closed set of points where $\nu$ fails to be an isomorphism, i.e. $D$ is the non-normal locus of $X$. Question 1. Under which…
Jesko Hüttenhain
  • 15,286
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1 answer

The map $f\colon\mathbb{A}^2_k\to\mathbb{A}^2_k$ given by $f(x,y)=(x,xy)$ is birational?

I'm reading a bit about rational maps, and I'm still trying to get get my head around birational maps. Consider the map $f\colon\mathbb{A}^2_k\to\mathbb{A}^2_k$ on the affine $2$-space over $k$ algebraically closed, defined by $f(x,y)=(x,xy)$. I…
Ankita Desari
  • 239
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For this morphism of integral schemes $X\to Y$, if $X$ is geometrically reduced (irreducible) then is $Y$ also geometrically reduced (irreducible)?

Let $k$ be an arbitrary field. Suppose that $C/k$ is an integral curve which is birationally equivalent to a projective line. Is it true that $C$ is geometrically reduced and irreducible? This is of course true if $C$ is a normal, since then $C$ is…
Tomasz Lenarcik
  • 481
6
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1 answer

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 flips and flops. It is possible that my question…
L_b
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