# Tagged Questions

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 ...

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### On factorization theorem of toric birational morphisms

Let $X_{Σ′}\to X_{Σ}$ be a toric birational morphism between smooth and complete toric varieties induced by a regular subdivision $Σ′\leq Σ$, i.e. every cone in $Σ′$ is contained in a cone in $Σ$ and ...
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### Relative canonical divisors

Suppose that $X$ is a Gorenstein variety and that $\pi : Y \to X$ is a birational map of varieties with normal $Y$. In this case the relative canonical divisor is defined to be $K_Y - \pi^*K_X$ (if ...
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### Letters of a bi-rationalist

V.V. Shokurov has written several papers over the course of about 10 years which are called "Letters of a bi-rationalist". Here are the ones that I could find: Letters of a bi-rationalist. I. A ...
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### Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even. Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...
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### Projectivity of flops

Say I have a projective (smooth, compact) irreducible symplectic variety $X$ over $\mathbb{C}$ and I perform a Mukai flop. It is well known that if the resulting variety $\widetilde{X}$ is Kahler, it ...
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### Semistable sheaves on rationally connected manifolds

Let $(X,H)$ be a polarized projective manifold of dimension $n$ defined over $\mathbb C$, and let $\mathcal E$ be a reflexive sheaf on $X$. If for every subsheaf $\mathcal F \subset \mathcal E$ the ...
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### Rational connectedness of certain subvarieties of the linear series

Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$, $|\mathcal{O}_X(a)|$ be the complete linear system for some integer $a>0$. Ofcourse, a general element of the linear system is a ...
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### Blow-up of $\mathbb{P}^4$ along a smooth surface

Let $\pi \colon X\to \mathbb{P}^4$ be the blow-up of a smooth surface $S\subset \mathbb{P}^4$. Is there a formula to compute $(K_X)^4$ ? (which should be dependent on invariants of $S$). In dimension ...
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### Is this $S$-birational map an open immersion on its domain of definition?

My question is about a claim on the bottom of p. 121 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud, so I will freely use the general terminology recalled in this book, but will ...
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### When can a blow-down be pushed out?

I am interested in constructing a morphism out of a blown down variety. Let $V$ be a scheme, $U\hookrightarrow V$ an open immersion. Let $\widetilde V$ be a blow-up of $V$, $\widetilde U$ its ...
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### motivic integration and jacobian ideal

When we consider the change of variables in motivic integration, we have a birational map $f:Y\rightarrow X$ with Y smooth and we have to consider two invariants the order of the Jacobian ideal of $X$ ...
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### birational classification of rationally connected 3-folds

What is the birational classification of (smooth projective) rationally connected 3-folds (over algebraically closed fields of characteristic $0$ or even $\mathbf{C}$, if $\mathrm{char}(k) = p > 0$,...
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### Uniruled degenerations of abelian varieties

Suppose I have a smooth projective variety $X$ over $\mathbb{C}$ with $K_X$ semiample, and consider the fiber space $f:X\to Y$ given by $|\ell K_X|$, for some $\ell>0$ large, where $Y$ is a normal ...
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### Adjunction Formula for Weil Divisors on a Normal Variety X

Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
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### GKZ decomposition for spherical varieties

If $X$ is a complete toric variety the GKZ decomposition of the effective cone $Eff(X)$ of $X$ corresponds to its Mori Chamber Decomposition, and therefore it encodes the birational geometry of $X$. ...
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### Can one control the ramification of a Brauer class under birational morphisms?

Assume we are given a Brauer class $\xi\in Br(k(\mathbb{P}^n))$ ramified at some divisor $D\subset \mathbb{P}^n$, here $k=\mathbb{C}$. If $f: \mathbb{P}^n\mathrel{-\,}\rightarrow \mathbb{P}^n$ is a ...
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### How to prove two manifolds are not birational?

Given a family of compact complex manifold $\mathcal{X} \rightarrow B$, what are the standard techniques to prove two distinct fibers $\mathcal{X}_a$ and $\mathcal{X}_b$ are not birational?
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### Can discrepancy change after pseudo isomorphisms?

Let $X$ and $Y$ be two projective irreductive algebraic varieties of dimension $3$ and let $f:X\dashrightarrow Y$ be a pseudo isomorphism, i.e. a birational map which restricts to an isomorphism ...
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### How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?

Let $X={\rm Spec} A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$. Now,...
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### rationality of Fano 3fold $X_{18}$

I need a reference for an explicit proof of the rationality of the Fano 3-fold $X_{18}$. By explicit I mean by a sequence of explicit birational transformations. Thank you!
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### How can every divisor be reached by a sequence of blow-ups?

The following is a result of Zariski [cf. Lemma 2.45 of Birational Geometry of Algebraic Varieties]. $X$ : an algebraic variety over a field $k$. $(R,m)$ : a DVR of the quotient field $K(X)$ ...
Is there some nice compactification of the affine space with a del Pezzo surface of degree $\le 4$ (for example with a cubic surface) ? More precisely, I would like a projective algebraic variety $X$ ...