# 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.

**5**

**1**answer

### Isomorphisms of complete intersections

**5**

**1**answer

### Weak Fano varieties and small transformations

**4**

**1**answer

### Push-forward of divisors and intersections

**-1**

**0**answers

### Is there a rational projective hypersurface of degree three?

**2**

**1**answer

### Cohomology of normal bundle and tangent bundle on Gushel-Mukai threefold

**5**

**0**answers

### Divisorial contractions and singularities

**4**

**0**answers

### Dominant rational maps and compositions

**5**

**0**answers

### The structure of the Hilbert scheme of conics contained in hypersurfaces in $\mathbb P^3$

**2**

**1**answer

### Kahler cone of blow up of $\mathbb{C}P^1 \times \mathbb{C}P^n$

**4**

**0**answers

### When is the dualizing sheaf globally generated?

**3**

**1**answer

### Normal bundle and small contraction in threefolds

**1**

**0**answers

### Is there a direct way to show Fano surface of lines and conics on the pairs of Fano threefolds isomorphic?

**1**

**1**answer

### Log resolution of a variety of log general type

**1**

**0**answers

### Blow up along codimension 1 subscheme

**3**

**1**answer

### Existence of birational sections over a center of an exceptional divisor

**4**

**1**answer

### (stable)-base locus on fibres

**5**

**1**answer

### Variety of negative Kodaira dimension contains a projective line

**5**

**0**answers

### Steps of the MMP “in family”

**2**

**0**answers

### Non-uniruled connected smooth fibers implies flat

**2**

**1**answer

### How to find a rational $\mathbb{F}_{\!q}$-curve on a quite classical Calabi–Yau threefold?

**7**

**1**answer

### Multiplication maps for big line bundles

**12**

**0**answers

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

**2**

**1**answer

### Decomposition of a morphism with positive dimensional fibers

**1**

**0**answers

### Unirationality of $\mathrm{Sym}^2(C)$ of genus $3$ curve

**5**

**1**answer

### Extending rational maps of nodal curves

**8**

**0**answers

### Which field extensions do not affect Chow groups?

**3**

**0**answers

### Whether a particular fiber product of varieties is integral

**4**

**1**answer

### Certain endomorphisms of $\mathbb{C}(x,y)$

**1**

**2**answers

### Example of a projective variety over a field of characteristic zero which is uniruled but not ruled

**7**

**1**answer

### An explicit negative solution to the Lüroth problem for non-algebraically closed fields

**2**

**0**answers

### An analogue of Noether's Problem for non-rational varieties

**7**

**0**answers

### Log Calabi-Yau variety diffeomorphic to an algebraic torus

**2**

**0**answers

### Wonderful compactification of $\mathrm{SL}(2)/\mathrm{SO}(2)$

**14**

**1**answer

### Degree of secant varieties of Veronese varieties

**2**

**0**answers

### Clemens-Griffiths component birational invariant

**1**

**0**answers

### Realising flips as quotients of flops by an involution

**2**

**0**answers

### Determining if a morphism is a blowup along a given subvariety

**10**

**1**answer

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

**6**

**0**answers

### Lefschetz type theorems for linear sections

**1**

**1**answer

### Endomorphism of Chow group induced by a birational map

**4**

**0**answers

### Isomorphisms of weighted complete intersections

**1**

**0**answers

### Picard numbers of isogenous K3 surfaces over a non-closed field

**1**

**0**answers

### Does nefness carry over through flips?

**3**

**0**answers

### What is the exceptional divisor of a divisorial contraction?

**1**

**1**answer

### Mori extremal contraction with small Betti number

**2**

**1**answer

### Cremona transformation $\sigma: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2 $ and pushforward of divisor

**9**

**1**answer

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

**3**

**2**answers

### When is a monomial rational map on the projective space birational?

**2**

**1**answer

### Picard group modulo codimension 2

**1**

**1**answer