# 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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### Rational points on fibrations

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### Embedding quadric bundles

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### Rationality of quadric bundles

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### Rational points on surfaces

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### Questions about Hironaka's example

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### Are Du Val singularities smoothable?

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### Number of rational points over finite fields mod $q$ is birational invariant

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### Existence of terminal $3$-fold flips

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### diagonal cubic hypersurfaces

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### Parameter spaces for conic bundles

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### What's the modification of a Calabi-Yau manifold?

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### How can I "see" that a map is birational?

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### Does a resolution of a rational singularity have rationally connected fibers?

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### Minimal $b_2$ in Sarkisov's construction

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### Is the automorphism group of a normal affine scheme a group scheme or an algebraic space?

**3**

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### Mori cone of Picard rank two varieties

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### Moduli spaces of horizontal curves

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### Dynamical degree and spectral radius

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### Cremona transformations and divisors

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### Divisors on projective bundles

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### Coefficients of elliptic curves over function fields

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### Does every Fano variety contain every abstract curve?

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### On the exceptional divisor of the resolution of indeterminacy locus of rational map

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### On tensor product of field extensions

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### Blow-ups of surfaces over a field

**8**

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### Divisors whose restriction is big

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### Construction of Jacobian Ideal

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### Fibers of period map

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### How to distinguish the singularities on moduli space?

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### Curves in conic bundles

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### Motivation for birational geometry

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### Simultaneous Log resolutions for both varieties and divisors

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### Global sections of multiples of a divisor

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### A question on effective divisors

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### Nef and pseudo-effective divisors over non algebraically closed fields

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### On the b-nefness of the moduli part of canonical bundle formula

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### Canonical covering stack of a flop

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### Pseudoeffective divisors on surfaces

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### Examples of complex manifolds with trivial Néron–Severi group?

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### Log canonical surface with an elliptic singularity

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### Restriction of small transformations

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### descent of nef divisors

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### Rationality in pencil of projective varieties

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### Rational and rationally chain connected surfaces

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### Demailly Campana Peternell Conjecture for isolated singularities

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### Subsets of $\mathbb{N}$ arising as genera of smooth projective curves in a variety

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### Cone and contraction theorems for certain sub-klt pairs

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### Identifying plane scrolls

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### An invariance property of rational singularities

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