# 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.

119 questions
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $C\times \mathbb{P}^1$ 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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### 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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### 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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### 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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### 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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### 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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