**1**

vote

**1**answer

137 views

### Methods of showing a variety is stably rational

As anyone who follows the algebraic geometry tag on arXiv will probably know, there has been a lot of papers recently showing various varieties are non-stably rational. What I am interested in however ...

**7**

votes

**2**answers

183 views

### Obstruction to rationality of del Pezzo surfaces of degree 4

Let $X$ be a del Pezzo surface over a number field $k$. (A del Pezzo surface over $k$ is a smooth, projective, geometrically connected surface whose anti-canonical class $K_X$ is ample.) Let $d := ...

**4**

votes

**1**answer

296 views

### Are stably birational varieties birational?

We say that two (irreducible) algebraic varieties $X$ and $Y$ are stably birational if $X \times \mathbb{P}^n$ is birational to $Y\times \mathbb{P}^n$ for some $n\ge 0$.
The natural question is then ...

**9**

votes

**0**answers

195 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 ...

**4**

votes

**1**answer

188 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 ...

**3**

votes

**1**answer

87 views

### Sections of a linear system splitting as a product of degree one polynomials

Let $X\subset\mathbb{P}^n$ be a hypersurface of degree $d$ and with multiplicities $m_1,...,m_k$ at $p_1,...,p_k\in\mathbb{P}^n$ general points.
Let $S\subseteq |\mathcal{O}_{\mathbb{P}^n}(d)|$ be ...

**14**

votes

**1**answer

328 views

### Generalization of the rigidity lemma in birational geometry

Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected.
If there exists ...

**9**

votes

**0**answers

236 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 ...

**0**

votes

**0**answers

95 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!

**5**

votes

**1**answer

142 views

### Automorphisms of Cartesian products

Let us consider the Cartesian product $X^r$, where $X$ is a smooth projective variety. There is a subgroup $Aut_{\Delta}(X^r)\subset Aut(X^r)$ of automorphisms of $X^r$ mapping a $k$-dimensional ...

**1**

vote

**2**answers

187 views

### Automorphisms of locally trivial fibrations

Let $f:X\rightarrow Y$ be a locally trivial fibration with a variety $F$ as the fiber. Here $X, Y, F$ are smooth, projective varieties.
Does any automorphism of $F$ induce an automorphism of $X$?
In ...

**2**

votes

**1**answer

157 views

### Inverse image of a divisor

Let $f:X\rightarrow Y$ be a morphism with connected fibers between projective varieties (not necessarily flat). Let $D\subset Y$ be an irreducible divisor.
Let us look at the cycle $f^{-1}(D)\subset ...

**3**

votes

**1**answer

208 views

### Generic Smoothness Type of Results in Positive Characteristic

Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth.
We know that ...

**1**

vote

**1**answer

137 views

### Rigid effective divisors

Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$.
Now, let ...

**4**

votes

**1**answer

98 views

### $Ex(f)$ has codimension at least 2

The following is a part of proof of lemma 6.2 in the book.
$f:X \to Y$ a projective birational morphism of normal varieties
$D$: Weil divisor on $Y$, $E$: exceptional divisor of $f$
...

**3**

votes

**1**answer

193 views

### A Bertini-type result for hypersurfaces containing a subvariety

Let $P$ be a smooth projective variety of dimension $4$ and let $Z$ be an irreducible subvariety of dimension $2$ ($Z$ is not necessarily smooth, but you can assume it).
Is there a smooth, ...

**1**

vote

**0**answers

154 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 ...

**35**

votes

**5**answers

2k views

### Surfaces in $\mathbb{P}^3$ with isolated singularities

It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, ...

**2**

votes

**0**answers

95 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 ...

**4**

votes

**1**answer

143 views

### A moduli problem inspired by Stein factorization

Let $f:X \to Y$ be a proper, birational morphism with connected fibers, $X$ is non-singular and $Y$ is normal. Does there exist a moduli space parametrizing all invertible sheaves $\mathcal{L}$ on $X$ ...

**4**

votes

**1**answer

388 views

### Is every complex rational algebraic variety simply connected for the Euclidean topology?

Is it true that every quasi-projective rational irreducible algebraic complex variety is simply connected for the Euclidean topology?
Of course, this is false if we replace "complex" with "real" or ...

**3**

votes

**1**answer

236 views

### Picard groups, ample cones, and proper birational maps

Let $f:Y\to X$ be a proper birational map of normal varieties over an algebraically closed field which is an isomorphism over the regular locus.
Q1: Is it the case that the pullback ...

**5**

votes

**1**answer

261 views

### A question on young persons guide to canonical singularities

In the Corollary at pag 407 of Young persons guide to canonical singularities there is a formula to compute the contributions $c_q(D)$ to Riemann-Roch of a divisor $D$ passing through a point $q\in ...

**2**

votes

**1**answer

622 views

### Resolution of the indeterminacy locus of a rational map away from an irreducible component

Suppose I have a rational map between two smooth, complex, projective varieties (or a meromorphic map between two compact, complex, manifolds)
$X$ and $Y$. Ordinarily one eliminates the indeterminacy ...

**1**

vote

**0**answers

70 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 ...

**4**

votes

**0**answers

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 ...

**3**

votes

**1**answer

154 views

### Standard techniques on rationally connected varieties

Is there some standard technique or approach to determine when a (irreducible) subvariety of a rationally connected variety is again rationally connected? Any reference/text dealing with this kind of ...

**3**

votes

**0**answers

177 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 } ...

**2**

votes

**2**answers

205 views

### Birational morphism and first cohomology group

Let $f:X\rightarrow Y$ be a proper birational morphism of smooth projective varieties over $k=\bar k$ ($char(k)>0$). Is it true that $H^1(Y,\mathbb Z_l)\stackrel{f^*}\simeq H^1(X,\mathbb Z_l)$ ...

**3**

votes

**2**answers

223 views

### Fibrations on blow-ups of $\mathbb{P}^2$

Let $X_n = Bl_{p_1,...,p_n}\mathbb{P}^2$ be the blow-up of $\mathbb{P}^2$ in $n$ general points $p_1,...,p_n\in\mathbb{P}^2$.
Let $f_i:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^1$ be the linear ...

**2**

votes

**1**answer

276 views

### pull back of an ample line bundle under a blow up

Suppose $\mu:X\rightarrow Y$ is a blow up of a smooth irreducible subvariety $Z$ of $Y$. Let $L$ be an ample line bundle on $Y$. Let $E$ be the exceptional divisor of $f$. Is it true that there ...

**2**

votes

**1**answer

217 views

### On the number of irreducible components of an exceptional divisor

Let $X$ be a complex, affine variety and $Z\subseteq X$ a closed subset of $X$ (i.e. a closed, reduced subscheme). Let $E$ be the exceptional divisor of the blow-up $\pi:\tilde X\to X$ of $X$ with ...

**3**

votes

**1**answer

374 views

### What is $h^0(\mathcal O_F)$ where $F$ is a fiber of a normal surface over a smooth curve?

Lately I am studying the bend-and-break, and I follow the proof in the following note written by Olivier Debarre:
http://www.math.ens.fr/~debarre/M2.pdf
There is a detail that I just cannot go ...

**1**

vote

**1**answer

384 views

### Why are the different definitions of minimal model equivalent?

I'm starting to learn the minimal model program. It seems there are two definitions for a variety $X$ with only terminal singularities to be minimal:
$K_X$ is nef.
Every birational morphism from $X$ ...

**3**

votes

**1**answer

147 views

### Graded ring of a genus 2 curve

Let $X$ be a smooth projective complex curve of genus 2 with canonical divisor $K$. $X$ of course is hyperelliptic and has an involution that I denote by $j$.
There exists 3 possibilities for ...

**9**

votes

**2**answers

314 views

### Fibrations of projective varieties

Let $f:X\rightarrow Y$ be a flat morphism of normal projective varieties with fibers of positive dimension (in particular all the fibers are connected and of the same dimension).
Let $g:X\rightarrow ...

**1**

vote

**2**answers

204 views

### Cohen-Macaulayness of the direct image of the canonical sheaf

Let $Y$ be a normal projective variety and let $f:X\to Y$ be a desingularization. Define $\mathcal K_X=f_*\omega_X$, the Grauert--Riemenschneider canonical sheaf of $X$. It is independent of the ...

**0**

votes

**1**answer

134 views

### Finiteness of geometric valuations

I feel the following fact has been used in many argument in algebraic geometry, but I was not be able to prove it or find the precise reference:
Let $X$ be a $\mathbb{Q}$-factorial variety with log ...

**2**

votes

**0**answers

186 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?

**4**

votes

**1**answer

313 views

### pull-back of canonical divisor under blow-up of a singular point

I was checking an example of canonical singularities from surface.
We consider the surface $X:(xz=y^2)\subset \mathbb A^3$. The only singular point is the origin. We write down one affine piece of ...

**2**

votes

**1**answer

362 views

### Is every surjective, birational transformation of projective varieties automatically proper?

Let $X$ and $Y$ be two complex, irreducible, normal, projective varieties (read: integral, projective, normal $\mathbb C$-schemes of finite type), projective in the sense of Hartshorne.
Let ...

**2**

votes

**2**answers

235 views

### Bertini theorem for big divisors and klt pairs

Let $X$ be a smooth projective variety and let $D$ be a big $\mathbb Q$-divisor on $X$. Assume that for $m$ large $|mD|$ has no fixed components. Is there a $\mathbb Q$-divisor $D'\equiv D$ so that ...

**4**

votes

**2**answers

248 views

### A question on the effective cone

Let $X$ be a projective variety and $G$ a finite group acting on $X$. We consider the quotient $\pi:X\rightarrow Y :=X/G$.
I'm interested in the relation between $Eff(X)$ and $Eff(Y)$. In ...

**1**

vote

**0**answers

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 ...

**3**

votes

**1**answer

246 views

### Definition and sigularity of Ramified covers

Let $X$ be a normal variety over $\mathbb{C}$.
In their book Birational geometry of algebraic varieties, Kollár and Mori define [Definition 2.50 and 2.51] a ramified m-th cyclic cover associate to a ...

**0**

votes

**0**answers

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 $K(X)$ ...

**3**

votes

**0**answers

167 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 ...

**6**

votes

**1**answer

252 views

### Vanishing theorem for big divisors

Let $X$ be a projective, smooth variety over $\mathbb{C}$, and let $D$ be a irreducible, big Cartier divisor(notice, I do not assume nefness). Then is it true that $${\rm{H}}^1(X, K_X + D) = 0\quad?$$
...

**4**

votes

**0**answers

118 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 ...

**2**

votes

**1**answer

186 views

### Reference request: log Fano varieties

I need a reference for a proof of the following fact: let $X$ be a toric variety then $X$ is log Fano.
Thanks a lot.