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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211 views

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 ...
12
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799 views

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 ...
10
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245 views

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 ...
9
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205 views

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 ...
7
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195 views

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 ...
5
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518 views

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 ...
4
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61 views

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 ...
4
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0answers
122 views

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 ...
4
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0answers
116 views

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 ...
3
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0answers
57 views

Psi-classes on moduli spaces of weighted curves

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the stack of weighted genus $g$ curves with weights $A[n]=(a_1,...,a_n)$, and let $\pi:\mathcal{C}\rightarrow \overline{\mathcal{M}}_{g,A[n]}$ be the universal ...
3
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185 views

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

Let $\ f:X\to Z$ be a surjective morphism between two smooth projective varieties with connected fibers $(f_*\mathcal{O}_X=\mathcal{O}_X)$. Let $F$ be a general fiber of $f$ and $\mbox{dim } F<(\...
3
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169 views

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 ...
3
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110 views

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 $X$ ...
3
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0answers
93 views

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 $\mathrm{char}(k) = p > 0$,...
3
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0answers
191 views

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 ...
3
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343 views

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, ...
2
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72 views

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 $X$. ...
2
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98 views

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 ...
2
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0answers
187 views

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?
2
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139 views

A question on resolution of singularities

I am wondering if it could be possible in particular cases to resolve a singularity of dimension $n$ by blowing-up a locus of dimension smaller than $n$. For instance consider a cubic surface $S\...
2
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0answers
87 views

singularities preserved by integral closure

Let $X$ be an affine variety. Let $A$ be the coordinate ring of $X$ and let $K$ be the fraction field of $A$. Given a Galois extension $K\subset L$, let $B$ be the integral closure of $A$ in $L$. Let ...
2
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0answers
162 views

Global sections of exceptional divisor in normalized blow-up

Let $(R, \mathfrak{m})$ be a Noetherian normal local domain and $I$ an $R$-ideal. Write $X$ for the normalization of $\mathrm{Proj}(R[It])$ and $E$ for the effective Cartier divisor defined by the ...
2
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0answers
359 views

Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces

Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
2
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0answers
148 views

Is there an example of index 1 picardnumber 1 Fano 4-fold with h^{21}\neq0

Are there any known examples of index 1 smooth Fano 4-folds X with $\mathsf{Pic}(X)\cong\mathbb{Z}$ with $h^2(\Omega^1_X)\neq0$?
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157 views

Non-vanishing of a higher direct image

Let $f:X \to Y$ be a small birational morphism between threefolds. Assume $X$ has terminal singularities and the relative Picard of $f$ is $1$. Suppose the exceptional locus of $f$ is a curve $C$, and ...
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0answers
72 views

Endomorphism of Chow goup induced by a birational map

Let $\phi:X\dashrightarrow Y$ be a birational map between smooth projective $k$-varieties ($k=\bar k$) and $\Gamma$ be the closure of the graph of $\phi$. In Fulton's intersection theory example 16.1....
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99 views

A birational compatification problem

Let $P_1$ be a projective variety over $\mathbb{C}$, and $Y_1 \subseteq P_1$ be a codimension $1$ (irreducible) subvariety. Suppose there is a blowdown morphism $Y_1 \to Y_2$. Then can I find a ...
1
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0answers
139 views

Resolution of singularities of projective varieties

Let $X\subset\mathbb{P}^n$ be an irreducible variety, and let $Sing(X)$ be its singular locus. Let $Y$ be the blow-up of $\mathbb{P}^n$ along $Sing(X)$. Assume that we know that the strict transform ...
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0answers
124 views

How to Calculate Minimal Log Discrepancy on a Toric Variety?

Let $\sigma$ be a $3$ dimensional strongly convex rational polyhedral cone on $\mathbb{R}^3$ and $X_\sigma$, the corresponding affine toric $3$-fol. Also, assume that $\Delta$ is an anti-boundary $\...
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0answers
88 views

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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0answers
143 views

stably birational abelian varieties are isomorphic

Can anybody help me to prove the following result: Proposition. Let $A$ and $B$ be abelian varieties over a field $k$ of characteristic zero. Assume that $A \times \mathbb{P}_k^n$ and $B \times \...
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0answers
85 views

K-equivalence does not depend on the choice of the third variety

By K-equivalent of two smooth varieties $X,Y$, we mean there exist a smooth variety $Z$, and birational morphism $q: Z \to X,\quad p: Z \to Y$ , such that $q^* \omega_X \cong p^* \omega_Y$. Suppose $...
1
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0answers
260 views

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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0answers
149 views

Is always a Crepant birational map between smooth varieties a small modification

Lemma: A crepant birational map between quasi-projective normal varieties with only terminal singularities is in fact small modifitacions, i.e. isomorphism in codimension 1. So, if $f:X\...
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0answers
159 views

Fields over which cubic hypersurfaces are rational

All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...
1
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0answers
330 views

birational equivalence and mirror CYs

If a CY X is a mirror to Y then any CY Z which is birational to X is also a Mirror of Y. This is the motivation for the Kawamata's "moveable Kahler cone" which includes the Kahler cones of all the CYs ...
0
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0answers
96 views

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!
0
votes
0answers
115 views

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 $K(X)$ ...
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0answers
87 views

Compactification of the affine space with a del Pezzo surface

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 $X$ ...
0
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0answers
132 views

birational invariants of projective surfaces

I am studying Castelnuovo's rationality criterion for surfaces. Let $S$ be a projective surface and $K$ a canonical divisor on $S$. Let's use the notation $h^i(S,\mathcal{O}_S)=dimH^i(S,\mathcal{O}_S)...
0
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0answers
85 views

invariance of the dimension of severi varieties of surfaces

Suppose I have a smooth projective surface $S\subset P^n$ embedded by a very ample linear system $|L|$. Consider now the generalized Severi varieties that parametrize curves on $S$ belonging to $|L|$, ...
0
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0answers
260 views

Numerical dimension of nef divisors

Let $D$ be a nef divisor (moreover suppose it is effective if you prefer) on a normal projective variety of dimension $n$. Let $k\in[1,n-1]$. If $D^k\cdot V=0$ for generic subvarieties $V\subseteq X$ ...