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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251 views

### Breaking a morphism with generic fiber $\mathbb{F}_n$

Assume we are working over $\mathbb{C}$, and we have a projective morphism with connected fibers $f: X \rightarrow Z$ whose geometric generic fiber $X_\overline{\eta}$ is isomorphic to a Hirzebruch ...

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**1**answer

200 views

### Pull-back divisor being Cartier

Let $\pi \colon X \rightarrow Y$ be a projective morphism with connected fibers between normal quasi-projective varieties. Let $N$ be a $\mathbb{Q}$-Cartier divisor on $Y$ so that $\pi^*(N)$ is ...

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**1**answer

142 views

### $K_X+B \equiv 0$ implies $K_X + B \sim_\mathbb{Q} 0$?

Let $(X,B)$ be a projective log canonical pair (here I mean $B \geq 0$). Assume that the coefficients of $B$ are rational, and that $K_X+B \equiv 0$. Is it true that $K_X + B \sim_\mathbb{Q} 0$? I ...

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**1**answer

294 views

### Is an open subscheme of a rationally connected variety, rationally connected?

Let $X$ be a projective, irreducible variety over an algebraically closed field (of characteristic zero) which is rationally connected. Is it true that any open dense subvariety of $X$ is rationally ...

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362 views

### Reference request: birational automorphism group is finite

I am interested in having a look at the proof of the following fact: If $X$ is a smooth variety of general type, then $\mathrm{Aut(X)}$ is finite.
I know that this is proved in "On algebraic groups ...

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362 views

### A paradox on the deformation of singularities

Setup: $\pi: \mathcal X \to C$ is a flat morphism from a germ of a smooth curve $(C, o)$. Suppose the special fiber $\pi^{-1}(o) := \mathcal X_o$ has a certain class of singularities, one wants to ...

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183 views

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

In his paper "The center of the ring of 4 × 4 generic matrices" (Journal of Algebra, 1980) Formanek shows that, if $V$ is the representation of $PGL_4$ given by simultaneous conjugation of pairs of ...

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57 views

### Blowing up the base of an elliptically fibered (non Weierstrass) threefold

Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $\sigma_i$ ,$i=1 \dots n$. None of these "sections" are honestly a section, they ...

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133 views

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

There are a group of related conjectures associated to Lang's name - for this question I'll consider only the weakest one, namely that rational curves in a projective variety of general type are not ...

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152 views

### Explicit descriptions of a flop

I want to know how to describe explicitly the flop of the following flop contraction. Because the construction is so natural and simple, I was wondering such descriptions should already exist in the ...

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**1**answer

430 views

### What English translations are there of work done by the Italian school of algebraic geometry?

What English translations are there of work done by the Italian school of algebraic geometry?
Perhaps I'm being too spoiled here, given that mathematical French, German, Italian are much easier to ...

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132 views

### Tie-Breaking Trick for Log Canonical Pairs and F-pure pairs in Positive Characteristic

Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor ...

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117 views

### Motivic homotopy theory and Noether problem

Let $G$ be a finite group, and let $V$ be a faithful representation of $G$. The Noether problem asks whether $V/G$ is rational (stably rational, retract rational) or not.
To construct ...

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122 views

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

Let $n\ge 1$ be an integer and let us work over the field of complex numbers. Let $\mathcal{R}_n$ denote the set of rational conic bundles $\pi\colon X\to \mathbb{P}^n$ (morphisms such that the ...

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256 views

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

The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ ...

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**1**answer

138 views

### Movable divisor with base locus on a hyperkahler variety

I'm looking for an example of the following:
$X$ is a hyperkahler fourfold (deformation equivalent to $K3^{[2]}$);
$D$ is a movable divisor on $X$ with $D^4=0$; and the base locus of
$D$ is a ...

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**0**answers

168 views

### Deformation Vs. Blow up

Consider a surface $X$ (which is a projective variety over $\mathbb{C}$) with a double point singularity. To make $X$ smooth, we can either blow up the singularity with a $(-2)$ - curve ($\hat{X}$), ...

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**0**answers

125 views

### Small contractions as blow ups

To improve my chances of getting answers/ comments I post my mathstackexchange question https://math.stackexchange.com/q/2808852/42548 also here.
I am trying to learn a bit about birational morphisms:...

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**1**answer

169 views

### How to split a Multi-section into finitely many Sections via base-change?

Let $:f:X\to Y$ be a projective surjective morphism between two normal varieties over $\mathbb{C}$. Assume that $f$ has only $1$-dimensional fibers. Let $D$ be a multi-section of $f$, i.e., $D$ is a ...

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**1**answer

317 views

### Does a torus action with isolated fixed points imply rational?

Suppose that $X$ is a smooth projective variety $/\mathbb{C}$ with a $\mathbb{C}^{*}$-action with isolated fixed points. Must $X$ be rational?

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123 views

### Can one construct a ramified cover over prescribed divisors?

Suppose $X/\mathbb C$ is a smooth projective variety of dimension $d>1$, and $D \subset X$ is a simple normal crossing divisor. For any $k>0$, can I construct a ramified $K$-th cover $f: \tilde ...

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93 views

### The specific elliptic fibration on the Kummer surface of the superspecial abelian surface

Consider two copies $E_1$, $E_2$ of the supersingular elliptic curve
$$
y^2 = x^3 - 1\qquad (y^2 = x^4 - 1)
$$
over a finite field $\mathbb{F}_{p^2}$ of odd characteristics $p$ such that
$$p
\...

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**0**answers

155 views

### Liftability of varieties, after fpqc base change

Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$, with $S$ liftable.
Suppose there exists an fpqc cover $S'\to S$, such ...

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102 views

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

Let $P \subset \mathbb{R}^{n-1}$ be a lattice polytope of dimension $n-1$ and let $\sigma \subset \mathbb{R} \times \mathbb{R}^{n-1}$ be the cone over $1 \times P$.
To the cone $\sigma$, we may ...

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**1**answer

91 views

### Flatness of Fano Contractions

In his 1984 paper Cone of Curves, Kawamata asks on p 629 if a Fano Contraction is flat. ( an extremal ray contraction is called a Fano Contraction if the dimension of the target is less than the ...

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**1**answer

263 views

### Two bivariate polynomials (or rational functions) that generate $\mathbb{C}(x,y)$

Let $f=f(x,y),g=g(x,y) \in \mathbb{C}[x,y]$, each of degree $\geq 1$, and $f,g$ are algebraically independent over $\mathbb{C}$ (= their Jacobian $\in \mathbb{C}[x,y]-\{0\}$).
(1) Is there a ...

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125 views

### Spherical varieties as GIT quotients

Let $X$ be a normal projective variety with finitely generated Cox ring. Consider its characteristic space $p:\widehat{X}\rightarrow X$. This means that there is a torus $T$ acting on $\overline{X}=...

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117 views

### $3$-fold which is a curve blow-up in two different ways

Let $X$ be a smooth divisor of degree $(1,1,2)$ in $\mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{2}$.
According to the list of Fano $3$-folds of Ivskovskikh-Prokhorov, this is a curve blow-...

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189 views

### Linear sections of Segre varieties and rational normal scrolls

In a projective space $\mathbb{P}^{k+2}$ consider two complementary subspaces $\mathbb{P}^1,\mathbb{P}^k$, and let $C\subset\mathbb{P}^k$ be a degree $k$ rational normal curve. Fixed an isomorphism $\...

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**1**answer

290 views

### Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?

Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $...

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148 views

### curve blow ups of toric Fano $3$-folds

Suppose $X$ is a smooth toric Fano $3$-fold, and $D$ is a torus invariant divisor corresponding to a face of the polytope associated to $X$. I would like to search for (smooth) curves $C \subset D$, ...

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**1**answer

141 views

### Birational elliptically fibered variteties

Does anyone know any example of two elliptically fibered toric variety (3-fold) that are birational to each other?

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**0**answers

110 views

### Deformation invariance of rational connectedness in positive/mixed characteristic

Let $f:X \to S$ be a smooth morphism and $S$ the spectrum of a discrete valuation ring. If the generic fiber of $f$ is rationally (chain) connected then is the special fiber of $f$ also rationally (...

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**1**answer

124 views

### Lifting a morphism

Suppose we have a morphism $\phi : S_{1} \rightarrow S_{2}$, between quasi-projective varieties of dimension $2$ over $\mathbb{C}$ with at worst quotient singularities. Suppose furthermore that $\phi$ ...

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125 views

### Unirationality over the base field

Let us say that a variety $X$ over a field $k$ is unirational if there exists a dominant map $\mathbb{P}^n_k\to X$ and geometrically unirational if $X_{\bar{k}}$ is unirational.
Assume that $k$ is ...

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**1**answer

172 views

### An explicit computation of the blow-up of curve over $\mathbb{F}_3$ at two points

I would like to work through computing the blow-up of a particular curve along a subvariety consisting of just two points, both of which are ordinary double points.
Let $$F(X,Y,Z) = X^4 + Y^4 - X Y^2 ...

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**1**answer

318 views

### surface with rational curve in the double locus

I am interested in the existence of a surface $X$ over $\mathbb{C}$ with the following properties (or a reason for why one cannot exist):
$X$ is slc (and not-normal)
There is rational curve $C \...

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**0**answers

100 views

### Compactifying morphisms and ample line bundles

Let $f:X\to Y$ be a projective morphism between two normal quasi-projective varieties, and $L$ a $f$-ample line bundle on $Y$. Then the claim is: There is a compactication $\bar{f}:\overline{X}\to\...

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**1**answer

196 views

### Liftable rational varieties

Is there an example of a rational smooth projective variety over a perfect field of characteristic $p$, that is not liftable to characteristic zero?

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**1**answer

330 views

### Complete intersections in toric varieties

Let $X$ be a smooth projective variety over the complex numbers. Is $X$ a global complete intersection inside a smooth projective toric variety?

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**1**answer

130 views

### Rational map given by pfaffians

Consider a general skew-symmetric $(n+1)\times (n+1)$ matrix $Z$, and let su map $Z$ to the point of $\mathbb{P}^n$ determined by $[pf_0(Z):\dots:pf_n(Z)]$ where the $pf_i(Z)$ are the principal ...

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### How can I describe in explicit geometric terms the (in general non-complete) linear system?

Let $\varphi\!: S \to S^\prime$ be a birational morphism of projective non-singular irreducible surfaces over a (algebraically closed) field $k$ and let $D \in \mathrm{Div}(S)$. Also, let $\big(\...

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**1**answer

196 views

### Automorphism groups of Hirzebruch surfaces

The Hirzebruch surfaces are the $\mathbb{P}^1$ bundles $\mathbb{F_n}$ ($n\geqslant 0$) which can be obtained projectivizing the rank $2$ vector bundles $\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\...

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223 views

### Blow-ups in étale cohomology

If $f:X\to\text{Spec}(k)$ is a smooth projective variety over a separably closed field, with $i:Z\subset X$ a closed smooth sub variety, with complement $j: U\to X$, and $f':X'\to \text{Spec}(k)$ is ...

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**1**answer

191 views

### Closure of quasi projective scheme

If $S$ is a scheme, $X$ is a smooth quasi-projective $S$-scheme, is the $S$-projective closure of $X$ a smooth $S$-scheme with $X$ an open subscheme?

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171 views

### Intermediate moduli spaces of stable maps

In the following paper:
A. Mustata, M. A. Mustata, "Intermediate moduli spaces of stable maps", Invent. math. 167, 47–90 (2007)
the authors introduced a variation on moduli spaces of stable maps ...

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117 views

### Picard group finitely generated

Let $X$ be a smooth projective variety over a finite field $k$. Is the Picard group finitely generated? Equivalently, is $\text{Pic}^0(X)$ finitely generated?
(I am not assuming $k$ is separably ...

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151 views

### Complete intersections in projective spaces

Let $X$ be an arbitrary smooth projective variety over a field $k$.
Do there exist:
a smooth complete intersection $X'$ in a projective space.
a surjective morphism of $k$-varieties $X'\to X$ ?

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**1**answer

445 views

### Birational invariants in étale cohomology

If $k$ is a perfect field (not necessarily separably closed), $f : X\to Y$ a proper birational map of smooth projective $k$-varieties, $\ell$ a prime invertible in $k$, is the induced map, for some $j$...

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224 views

### Pullback of the canonical bundle

Let $f : X\to Y$ be a morphism of smooth projective varieties over a field $k$.
Assume $f^*\omega_{Y/k} \simeq \omega_{X/k}$.
I'd like to collect a bestiary of the properties $f$ has, or even ...