# 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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### Action of birational map $f$ on the divisor class of line $[H]$

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### Bounding the denominator in the canonical bundle formula

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### Singularities of rational quartic surfaces

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### Is the generalized Kummer threefold rational in characteristics 3?

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### Behavior of geometric fibers under strict transforms

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### Breaking a morphism with generic fiber $\mathbb{F}_n$

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### Pull-back divisor being Cartier

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### $K_X+B \equiv 0$ implies $K_X + B \sim_\mathbb{Q} 0$?

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### Is an open subscheme of a rationally connected variety, rationally connected?

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### Reference request: birational automorphism group is finite

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### A paradox on the deformation of singularities

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### On a paper of Formanek about $PGL_4$

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### Blowing up the base of an elliptically fibered (non Weierstrass) threefold

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### Curves on rational surfaces and Lang's conjecture for M_g

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### Explicit descriptions of a flop

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### What English translations are there of work done by the Italian school of algebraic geometry?

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### Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic

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### Motivic homotopy theory and Noether problem

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### Conic bundles which are rational but with non-rational generic fibres

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### Is the $\hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?

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### Movable divisor with base locus on a hyperkahler variety

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### Deformation Vs. Blow up

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### Small contractions as blow ups

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### How to split a Multi-section into finitely many Sections via base-change?

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### Does a torus action with isolated fixed points imply rational?

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### Can one construct a ramified cover over prescribed divisors?

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### The specific elliptic fibration on the Kummer surface of the superspecial abelian surface

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### Liftability of varieties, after fpqc base change

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### “Reflexive” differentials on Gorenstein affine toric variety

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### Flatness of Fano Contractions

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

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### Spherical varieties as GIT quotients

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### $3$-fold which is a curve blow-up in two different ways

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### Linear sections of Segre varieties and rational normal scrolls

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

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### curve blow ups of toric Fano $3$-folds

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### Birational elliptically fibered variteties

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### Deformation invariance of rational connectedness in positive/mixed characteristic

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### Lifting a morphism

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### Unirationality over the base field

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### An explicit computation of the blow-up of curve over $\mathbb{F}_3$ at two points

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### surface with rational curve in the double locus

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### Compactifying morphisms and ample line bundles

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### Liftable rational varieties

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### Complete intersections in toric varieties

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### Rational map given by pfaffians

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### How can I describe in explicit geometric terms the (in general non-complete) linear system?

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### Automorphism groups of Hirzebruch surfaces

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### Blow-ups in étale cohomology

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