# 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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### Relative canonical divisors

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

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

**10**

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

**10**

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### Letters of a bi-rationalist

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### Motivic homotopy theory and Noether problem

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### Rationality of a certain real algebraic variety

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

**8**

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### Maximally nodal degree 6 Fano threefolds

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### How do I make the components of a Cartier divisor again Cartier divisors?

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### Status of the Sarkisov program a la Corti

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

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### Compactifications of reductive groups via representation theory

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

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### References about conic bundles

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### Projectivity of flops

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### Lefschetz type theorems for linear sections

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

**6**

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### Extremal rays in Picard rank two

**6**

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### Pseudo-effective divisor which is not nef in any birational model

**5**

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### Divisorial contractions and singularities

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

**5**

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### Steps of the MMP “in family”

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### Jacobian fibration of an abelian fibration

**5**

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### Unirationality of universal Jacobian over special strata of moduli space of pointed genus 3 curves

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

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### When does a Cartier divisor a pull-back of a Cartier divisor?

**5**

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### In search for examples concerning pushforward of nef divisors and lc-trivial fibrations

**5**

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### Description of flop as graded algebra

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### Restriction of the Canonical Divisor $K_X$ to a general fiber

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### Semistable sheaves on rationally connected manifolds

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### Dominant rational maps and compositions

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### When is the dualizing sheaf globally generated?

**4**

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### Isomorphisms of weighted complete intersections

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### Are there any explicit (prime-to-l) alterations for interesting varieties (or schemes)?

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### Behavior of geometric fibers under strict transforms

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### Complete intersections in projective spaces

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### Pullback of the canonical bundle

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### Singularities of Viehweg's fiber product

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### Can Kummer surfaces coming from the same abelian surface be Cremona equivalent / isomorphic?

**4**

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### A question on the Kodaira dimension of 3-folds

**4**

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### Singularities of fibrations 2

**4**

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### What is the fundamental group of Kontsevich's space of stable maps?

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### A theorem about log Calabi-Yau pairs

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### A question about potentially birational divisor

**4**

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### Rational connectedness of certain subvarieties of the linear series

**4**

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### Blow-up of $\mathbb{P}^4$ along a smooth surface

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### Is this $S$-birational map an open immersion on its domain of definition?

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### Adjunction Formula for Weil Divisors on a Normal Variety X

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