The birational-geometry tag has no wiki summary.

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### Topology of the preimage of a point for degree one holomorphic maps

Let $M^n$ and $N^n$ be two compact complex (or complex projective) manifolds. Let $f: M\to N$
be a holomorphic map of degree one. How to prove that for each $x\in N$ the set
$f^{-1}(x)$ is ...

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322 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 ...

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**1**answer

364 views

### Numerically rigid nef divisor

Is it possible to find an example of an $\mathbb{R}$-Cartier divisor $D$ on an irreducible variety $X$ that is non-trivial, nef, effective and numerically rigid?
By "numerically rigid" I mean that ...

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651 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 ...

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812 views

### Blowups of Cohen-Macaulay varieties

Suppose that $X$ is a Cohen-Macaulay normal scheme/variety and $\pi : Y \to X$ is a proper birational map with $Y$ normal.
Question: Is $Y$ also Cohen-Macaulay? Are there common conditions which ...

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### Technique to prove basepoint-freeness

Let $X$ be a smooth projective variety over $\mathbb{C}$.
And let $L$ be a big and nef line bundle on $X$.
I want to prove $L$ is semi-ample($L^m$ is basepoint-free for some $m > 0$).
The only way ...

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**1**answer

761 views

### Appropriate journal to publish a determinantal inequality

I have recently made the following observation:
Let $v_i := (v_{i1}, v_{i2})$, $1 \leq i \leq k$, be non-zero positive elements of $\mathbb{Q}^2$ such that no two of them are proportional. Let ...

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604 views

### Possible singularities of the base of a Mori fiber space

Suppose X is a normal projective complex variety, (X, $\Delta$) is a klt pair and f : X $\to$ Z is a Mori fiber space given by a contraction of an extremal ray for this pair. Here I mean that the ...

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2k views

### Birational invariants and fundamental groups

In pondering this MO question and people's efforts to answer it, and recalling also something that I learned in my youth about using Morse theory ideas to prove some results of Lefschetz in the ...

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### 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, ...

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751 views

### How much can small modifications change the nef cone?

First let me give a precise formulation of the question; I'll give some background/motivation at the end.
If X is a projective variety which is Q-factorial (meaning X is normal, and some sufficiently ...

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**1**answer

350 views

### A question about the construction of Francia flip

Here is the construction.
Start with $U$ a normal variety of dimension 3 with a unique singular point locally isomorphic to the quotient of $(x, y, z) \to (-x, -y, -z)$. Also assume we have a small ...

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votes

**1**answer

428 views

### Schemes as a model category

I'm just learning some basics of model categories, so please forgive me if my question turns out to be trivial. I hope it does at least make sense.
A natural temptation is to relate this machinery to ...

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**1**answer

713 views

### Birational pullbacks of divisors on singular varieties

Actually I have two related questions.
Here is the first...
Suppose $X$ is a, possibly singular, complex projective variety.
Let $D$ be an effective Cartier divisor on $X$ and $x\in X$ a closed ...

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votes

**1**answer

376 views

### About b-divisors

In the last period I have studied a number of papers of O. Fujino and F. Ambro using the language of b-divisors.
So far it seems to me that every proof I have studied can be translated in the ...

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### Is the Euler characteristic a birational invariant

Suppose that $X$ and $Y$ are smooth projective varieties which are birationally equivalent. I would like to have that $$\textrm{deg} \ \textrm{td}(X) = \textrm{deg} \ \textrm{td}(Y).$$ Invoking the ...

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990 views

### The Riemann correspondence for riemann surfaces made explicit and its generalizations

Riemann showed (not proved rigorously) that there is a correspondence between compact riemann surfaces and algebraic function fields in one variable (does anyone know the year?).
To construct the ...

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### Contracting divisors to a point

This is quite possibly a stupid question, but it is pretty far from what I normally do, so I wouldn't even know where to look it up.
If $X$ is a projective variety over an algebraically closed field ...

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185 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 ...

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**1**answer

518 views

### Are groups in (Var/k, rational maps) necessarily algebraic groups?

Let $RVar$ be the category which has complex algebraic verieties as objects and rational maps as morphisms [Edit: for $RVar$ to be a category, the rational maps have to be dominant].
Let's consider a ...

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### Minimal model which is necessarily singular

I was told during a summer school on the MMP a nice example (which I have also mentioned here on MO) that I'm not able to figure out anymore.
The example (due, I think, to Miles Reid) is a smooth ...

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409 views

### Relative minimality for conic bundles

Due to difficulties in finding references I am not sure of what relative minimality is about for conic bundles. Suppose we're working on con. bun. over surfaces.
The definition is ok: the fiber over ...

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**1**answer

419 views

### Does a birational involution of C^n always have a fixed point?

The title describes completely the question.
For n=1 it is an easy exercise. For n=2 the statement is still true, and depends on the classification of birational involutions of P^2 (see e.g. ...

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### Rationality of GIT quotients

I recently worked through most of the proof of the rationality of the moduli of genus 3 curves, which seemed to have the following structure:
Every nonhyperelliptic genus 3 curve is a smooth plane ...

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**3**answers

288 views

### Nefness of $h-e$ in the blowup of $\mathbb{P}^n$

Let $S$ be the blow up of $\mathbb{P}^n$ in a point $P$. Let $h$ be the class of the pullback of an hyperplane of $\mathbb{P}^n$ and $e$ the class of the exceptional divisor. Why is the divisor ...

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517 views

### Reference for explicit calculation of blowups (of ideals) and strict transforms

Can one suggest some references where explicit calculations for blow up technique(along an ideal) and strict transformation is done in different examples?

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### Why is the Hodge class of \bar{M_g} big and nef?

Let pi: \bar{Mg,1} \to \bar{M_g} be natural projection of compactified moduli stacks of curves and let omega be the relative dualizing sheaf. Then the Hodge class \lambda of \bar{M_g} is the first ...