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

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

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

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### Picard group of resolution

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### Lefschetz type theorems for linear sections

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

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

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### Does nefness carry over through flips?

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

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### Mori extremal contraction with small Betti number

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

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### Picard group modulo codimension 2

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### Is $\mathbb{Q}$-factoriality preserved under contraction?

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### Isomorphisms of weighted complete intersections

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### Jacobian fibration of an abelian fibration

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### Terminal $\mathbb{Q}$-factorial divisorial contractions

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### Concerning $\mathbb{C}(s_1,s_2,s_3,y)=\mathbb{C}(x,y)$, where $s_1,s_2,s_3$ are symmetric

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### On the birational equivalent class of algebraic surfaces with Picard number $1$

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### Characterizing subfields $\mathbb{C}(u,v) \subseteq \mathbb{C}(x,y)$ invariant under an involution

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

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### A formula on a generically finite morphism

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### Connected components of a codimension one fiber for a finite morphism

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### Kodaira dimensions of push-forward via finite map

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### What is known about lower etale cohomology of unirational varieties?

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### Log canonical centers of toric (and toroidal) varieties

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### Birational model of a log smooth pair

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

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### Are terminal singularities $ \mathbb{Q}$-factorial?

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### Finding divisors with canonical singularities in a moving linear system

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### Birational contraction of toric vector bundle

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### Explicit equations for rational elliptic surfaces (Halphen surfaces)

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### Strict transform by normalization

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

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### Some simple algebra of rational functions by André Weil

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### A question on Okounkov bodies

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### Has anyone researched this variant of separable rational connectedness?

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### Does anyone know a reference in the literature regrading a proof that every projective hypersurface with vanishing canonical divisor is uniruled

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### Does anyone know of a uniruled hypersurface over $ \mathbb{C} $, which is not rationally connected?

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### Finiteness of birational types for targets of algebraic fibrations

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### Are Gromov-Witten invariants birational invariants?

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### Is the boundary divisor of a smooth projective toric variety an snc divisor?

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### Push forward a Cartier is still Cartier

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

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### definition of discrepancy

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### Flawed argument and when is a sheaf that can be associated to any complete, normal variety a birational invariant?

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### $\operatorname{NEF}(X)\subset\operatorname{Big}(X)$?

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### Terminal and log canonical singularities

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### Motivic integration of an Abelian variety and its dual are same?

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### Integrality of divisors in the canonical bundle formula

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