# 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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### Why $\overline{\operatorname{Mov}(X)}\subset \overline{\operatorname{Eff}(X)}$, where $X$ is any normal variety?

I am trying to understand why $\overline{\operatorname{Mov}(X)}\subset \overline{\operatorname{Eff}(X)}$, where $X$ is any normal variety. Here $\operatorname{Mov}(X)$ is the convex cone generated by ...
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A variety $X$ is $F$-split if there exists an $\mathcal{O}_{X}$-linear map $\phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X}$ such that $\phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{... • 741 3 votes 1 answer 245 views ### Normal bundle of a linear subspace Let$X\subset\mathbb{P}^N$be a smooth scheme theoretical complete intersection, and$H\subset X$a linear subspace. Denote by$N_{H,X}$the normal bundle of$H$in$X$. If$\dim(H) = 1$, that is$H$... • 8,804 1 vote 0 answers 90 views ### Does nefness in analytic setting depend on Hermitian metric? I'm reading Demailly's book 'Analytic Methods in Algebraic Geometry'. Let$X$be a compact complex manifold with a Hermitian metric. A line bundle$L$is said to be nef if for every$\epsilon>0$, ... • 283 2 votes 0 answers 86 views ### When is a smooth point of a projective, simplicial, toric variety$ X_{\Sigma} $compatibly$ F $-split? A variety$ X $is$ F $-split if there exists an$ \mathcal{O}_{X} $-linear map$ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $such that$ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...
The following is a criterion by Fujita (On the structure of polarized varieties with $\Delta$-genera zero). Consider a complex smooth projective variety $X$ of dimension $n$ and an ample divisor $H$, ...