# 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.

**37**

**5**answers

### Surfaces in $\mathbb{P}^3$ with isolated singularities

**27**

**3**answers

### Birational invariants and fundamental groups

**20**

**1**answer

### Rationality of intersection of quadrics

**18**

**3**answers

### What is known about the birational involutions of P^3?

**17**

**3**answers

### Contracting divisors to a point

**16**

**1**answer

### Generalization of the rigidity lemma in birational geometry

**16**

**1**answer

### Applications of derived categories to “Traditional Algebraic Geometry”

**16**

**1**answer

### The Order of Approximation of a Rational Map

**15**

**1**answer

### Birational automorphisms of varieties of Picard number one

**14**

**1**answer

### If $X\times X$ is rational, must $X$ also be rational?

**14**

**1**answer

### Degree of secant varieties of Veronese varieties

**13**

**2**answers

### is the Hodge conjecture birationally invariant?

**13**

**2**answers

### Training towards research on birational geometry/minimal model program

**13**

**0**answers

### Relative canonical divisors

**12**

**7**answers

### When does $Aut(X)=Bir(X)$ hold?

**12**

**2**answers

### Blowups of Cohen-Macaulay varieties

**12**

**2**answers

### Rationality of GIT quotients

**12**

**0**answers

### birational geometry of moduli spaces: why work on the coarse space?

**12**

**0**answers

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

**11**

**4**answers

### Is the Euler characteristic a birational invariant

**11**

**2**answers

### Minimal model which is necessarily singular

**11**

**2**answers

### Blow-ups of $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$

**11**

**1**answer

### Why is the standard flop a flop?

**11**

**2**answers

### Geometrically unirational varieties that are not unirational

**10**

**1**answer

### Is the Hasse principle a birational invariant?

**10**

**3**answers

### Extending birational isomorphisms between planar curves to the P^2

**10**

**1**answer

### Are stably rational surfaces all rational?

**10**

**3**answers

### Birational automorphisms of canonical models

**10**

**1**answer

### Quadrics in the Grothendieck ring

**10**

**1**answer

### $K_0$-equivalence of varieties

**10**

**1**answer

### A property of varieties between unirational and retract rational

**10**

**1**answer

### Conditions for the contractibility of subvarieties

**10**

**0**answers

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

**10**

**0**answers

### Letters of a bi-rationalist

**9**

**2**answers

### Reference request: birational automorphism group is finite

**9**

**2**answers

### Reference request on birational invariance of Chow group of zero cycles of degree zero

**9**

**1**answer

### Liftable rational varieties

**9**

**1**answer

### Number of conditions imposed by fat points to a linear system

**9**

**2**answers

### Fibrations of projective varieties

**9**

**2**answers

### How much can small modifications change the nef cone?

**9**

**2**answers

### Intersection numbers in $\mathbb{P}^1$-bundles

**9**

**1**answer

### Degree of equations of secant varieties of Veronese varieties

**9**

**1**answer

### Concerning $k \subset L \subset k(x,y)$

**9**

**1**answer

### Rationality of moduli spaces of rational curves

**9**

**1**answer

### Schemes as a model category

**9**

**1**answer

### Is the number of minimal models finite

**9**

**1**answer

### Dimension-specific phenomena in algebraic geometry

**9**

**1**answer

### Why is proving fully-faithfulness of an integral functor locally analytically sufficient?

**9**

**0**answers

### Motivic homotopy theory and Noether problem

**9**

**0**answers