The birational-geometry tag has no usage guidance.

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### 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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votes

**1**answer

193 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

168 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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**1**answer

131 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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278 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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195 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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votes

**1**answer

127 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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181 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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205 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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votes

**1**answer

281 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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201 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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226 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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95 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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369 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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votes

**1**answer

156 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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111 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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153 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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votes

**1**answer

205 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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103 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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174 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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211 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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447 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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200 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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378 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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233 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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173 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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267 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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**1**answer

162 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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112 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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134 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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121 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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190 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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100 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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186 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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195 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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840 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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86 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 > ...

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140 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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**1**answer

140 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 ...

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258 views

### Is each rationally chain connected surface rational?

Consider a 2-dimensional smooth projective algebraic surface S over complex numbers. Could you recommend any exact references to the proofs of the following assertions (of course, if they are true):
...

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130 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 ...

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203 views

### Pseudo-automorphisms on Fano varieties

Is every pseudo-automorphism (self-birational map which does not contract any hypersurface) of a smooth Fano variety of Picard rank $1$ equal to a biregular automorphism?
Remark: For $\mathbb{P}^n$, ...

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175 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 ...

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183 views

### Which singularities of log pairs do not depend on the resolution?

Let $(X,\Delta)$ be a log pair (we assume the coefficients of $\Delta\leq 1$, but could be negative rationals), and I use the definitions in the book of "Birational Geometry of Algbebraic varieties" ...

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209 views

### Hypersurfaces with rational self-maps

I'm looking for interesting examples of hypersurfaces $X\subset \mathbb P^n$ with a rational self-map $X\dashrightarrow X$?
Are there such examples for cubic hypersurfaces?

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131 views

### Pushforward of a log canonical pair

This question comes from learning the paper "Existence of minimal models for varieties of log general type", where they define the log terminal model (See Definition 3.6.7). However, the question ...

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215 views

### Small birational maps and singularities of the pair

Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair ...

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### 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 ...

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204 views

### Cremona transformations

Let $f:\mathbb{P}^n_1\dashrightarrow\mathbb{P}^n_2$ be the standard Cremona transformation based on $p_1,...,p_{n+1}\in\mathbb{P}^n_1$ and $q_1,...,q_{n+1}\in\mathbb{P}^n_2$. That is, $f$ is the ...

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225 views

### Vanishing theorems for pluri-canonical bundle

I would like to know if Grauer-Riemenschneider vanishing theorem is still true in the setting of pluri-canonical bundle, i.e. the power of canonical bundle.
Let me recall
Grauer-Riemenschneider ...