The birational-geometry tag has no usage guidance.

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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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129 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 $X$ ...

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349 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 ...

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votes

**1**answer

203 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 ...

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114 views

### Canonical and Minimal model for Sasakian Varieties

Let $X$ and $Y$ be two Sasakian varieties, what is the definition of holomorphic fiber space between $ X$ and $Y$ and also holomorphic submersion.
If $X$ bea projective Variery then we have ...

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225 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 ...

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64 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 ...

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59 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 ...

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137 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 ...

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141 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 } ...

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192 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 $H^1(Y,\mathbb Z_l)\stackrel{f^*}\simeq H^1(X,\mathbb Z_l)$ ...

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212 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 ...

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**1**answer

229 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 ...

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179 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 ...

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137 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 ...

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295 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 ...

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200 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 ...

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128 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 ...

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182 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?

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220 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 ...

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306 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 ...

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213 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 ...

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233 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 ...

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98 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 ...

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370 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 ...

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178 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 ...

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112 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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158 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 ...

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227 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 $${\rm{H}}^1(X, K_X + D) = 0\quad?$$
...

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108 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 ...

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179 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.

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219 views

### Is an anticanonical Weil divisor in $\mathbb{Q}$-Gorenstein variety Calabi-Yau?

Let $P$ be a normal, $\mathbb{Q}$-Gorestein variety with terminal singularities. Let $X \subseteq P$ be a normal, irreducible Weil divisor such that $X \sim_{\mathbb{Q}} - K_P$, that is ...

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560 views

### Movable Divisors

Let $X$ be a projective variety. Does anyone know an example of a movable reducible divisor $D\in Mov(X)$ such that any element in the linear system $|D|$ of $D$ is reducible?

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213 views

### Relative form of Kodaira's lemma?

If $X$ is a smooth projective variety, Kodaira's lemma states that a big line bundle $D$ can be decomposed (as $\mathbb Q$-divisors) as $A+E$, with $A$ ample and $E$ effective.
I am wondering what ...

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383 views

### Blow-ups of $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$

Let us consider the points
$$p_1=[1:0:...:0],p_2 = [0:1:...:0],...,p_{n-2} =[0:...:0:1],\\
p_{n-1}=[1:1:...:1]\in\mathbb{P}^{n-3}$$
and the blow-up $X = Bl_{p_1,...,p_{n-1}}\mathbb{P}^{n-3}$.
...

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261 views

### Standard plane Cremona transformation

Let us consider nine general points $p_1,...,p_9\in\mathbb{P}^2$ and the line $L = \left\langle p_1,p_2\right\rangle$. Take the standard Cremona $f_1$ centred in $p_3,p_4,p_5$, then $C_1 = f_1(L)$ is ...

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189 views

### Log Canonical pairs

Let $X$ be a normal scheme ad $D = \sum_id_iD_i\subset X$ be a $\mathbb{Q}$-divisor such thay $K_X+D$ is $\mathbb{Q}$-Cartier. Let $f:Y\rightarrow X$ be a log resolution of the pair $(X,D)$ and let us ...

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281 views

### Rationality of moduli spaces of rational curves

Let $\overline{M}_{0,n}$ be the moduli space of Deligne-Mumford stable pointed rational curves, and let us consider the quotient $\widetilde{M}_{0,n} = \overline{M}_{0,n}/S_n$.
Clearly, there is a ...

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167 views

### Finite generation of certain $\mathcal{O}_X$-algebra

It is proved in this paper by Kawamata (Theorem 6.1) that for a 3-dimensional normal algebraic variety $X$ which has at most canonical singularities, and a Weil divisor $D$ on it, the ...

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114 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 ...

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142 views

### Run MMP between varieties of isomorphic in codimension 1

Let $X, Y$ be two birational projective varieties which are isomorphic in codimension 1. Suppose $H$ is an ample divisor on $Y$, and $H'$ be its strict transform on $X$, suppose we can run MMP with ...

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127 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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215 views

### A question about running MMP with scaling

Let $\pi:X \to U$ be a projective morphism, and $(X, \Delta = A + B)$ be a KLT pair, where $A$ is a general ample divisor and $B$ is effective.
Suppose $K_X + \Delta$ is not nef (over $U$) and there ...

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102 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 $X$ ...

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223 views

### Is pushforward of an ample divisor under small birational map nef?

Let $X, Y$ be $\mathbb{Q}$-factorial, projective, normal varieties. Let $f: X --> Y$ be a small birational map. I have two related questions about pushforward of an ample divisor:
(1) Let $H_X$ be ...

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201 views

### Decompose a big divisor as nef big divisor and effective divisor

Let $W_n$ be a set of a log pair having the following property:
For any $(X, D) \in W_n$
(1)$X$ has dimensional $n$ with tirvial canonical divisor (i.e.$K_X = 0$). Moreover, $X$ is a ...

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861 views

### Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map ...

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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 > ...

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146 views

### A question on klt pairs

Let $D$ be a $\mathbb{Q}$-divisor in a smooth variety $X$. In Lazarsfeld book "Positivity in Algebraic Geometry 2" I found Proposition 9.5.13 saying that if for any $x\in D$ we have $mult_xD < 1$ ...

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142 views

### Blowing up along birational equivalent subvarieties

Let $X$ be an algebraic variety (not necessarily projective) over $\mathbb{C}$, and $V_1,V_2\subset X$ two projective subvarieties of $X$, with $\textrm{codim}(V_1)=\textrm{codim}(V_2)=2$. Suppose ...