**1**

vote

**2**answers

282 views

### Numerically negative exceptional divisor on a surface.

Suppose $S$ is an algebraic surface (possibly projective) over an algebraically closed field $k$. Suppose $D_i$ are irreducible smooth curves (rational, if you want) with negative self-intersection ...

**5**

votes

**1**answer

794 views

### Minimal Model Program for surfaces over algebraically closed fields of characteristic p

Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...

**2**

votes

**1**answer

343 views

### Log canonical pairs and ample divisors

Suppose $(X,\Delta)$ is a log canonical pair and $A$ is an ample divisor on $X$. Could you give me an esay proof of the fact that there exists $A'$ that is $\mathbb{Q}$-linearly equivalent to $A$ such ...

**7**

votes

**1**answer

1k views

### A characterization of the blow-up

It is well-known that there is a universal characterization of blow-ups. However, this one is not so easy to use. According to Hartshorne-Theorem II.8.24. The blow-up of a nonsingular $X$ along a ...

**3**

votes

**2**answers

738 views

### Unirational implies rationally connected

It is evidently a well-known fact that a unirational variety $X$ over an algebraic closed field (i.e. there is a dominant rational map from $\mathbb P^n$ to $X$) is rationally connected (by which I ...

**3**

votes

**1**answer

290 views

### Applications of the boundedness of birational automorphisms

Recently the paper http://de.arxiv.org/PS_cache/arxiv/pdf/1011/1011.1464v1.pdf by Hacon, McKernan and Xu appeared on the arXiv. There the authors prove that the number of birational automorphisms of a ...

**2**

votes

**1**answer

207 views

### Divisor intersecting non-negatively the negative part of its Zariski decomposition

Hi all. I'm looking for an example of a smooth projective surface $X$ and a pseudo-effective divisor $D$ on $X$ such that when I consider the Zariski decomposition $D=P+N$ there is some component $E$ ...

**5**

votes

**2**answers

609 views

### Is it possible to check two curves on birational equivalence by some computer algebra system?

I have two curves, for example hyperelliptic:
\begin{align}
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 18, \\\\
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 5x + 1
\end{align}
Is it possible to check them ...

**3**

votes

**1**answer

300 views

### Sections of a fibration in intersections of quadrics

Suppose that we have a smooth variety $X$ of dimension $n$ that fibers (a flat morphism) over a curve $Y$, and s.t. the fibers of $X \to Y$ are all complete interesections of two quadric hypersurfaces ...

**4**

votes

**1**answer

536 views

### do geometric fibers determine scheme-theoretic image?

Let $X,Y,Z$ be reduced algebraic varieties, and let $Y$ and $Z$ be normal. Let $f:X \to Y$ and $g:X \to Z$ two surjective projective morphisms of algebraic varieties such that the geometric fibers of ...

**1**

vote

**1**answer

382 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 $X$ ...

**2**

votes

**1**answer

288 views

### resolution of singularities and a projection formula

Let $Y$ be a normal surface and let $p:X\longrightarrow Y$ be a resolution of singularities.
Let $f$ be a rational function on $Y$.
Do we have that $p_\ast$div $(d(f\circ p)) = $ div $df$ as ...

**3**

votes

**1**answer

404 views

### Exceptional loci are covered by rational curves: easy case

It's well-known that the exceptional locus of a birational morphism is covered by rational curves, in various degrees of generality. The best result I know in this direction is the following:
Theorem ...

**2**

votes

**2**answers

332 views

### Global sections of a linear system

Recently, following Beauville's book (exercises iv.(1),(2)) I have been working on Hirzebruch surfaces (from the algebraic geometry point of view) and I had to compute the space of global sections of ...

**5**

votes

**1**answer

654 views

### How to construct log-canonical (or Calabi-Yau), non-Cohen-Macaulay singularities of low codimensions?

(EDIT 07/06/11: although the question has not been settled definitely, Sándor's excellent answer and the comments by Angelo and ulrich have highlighted many potential obstructions to the constructions ...

**1**

vote

**1**answer

210 views

### Rationality of quadric fibrations

Let $Q\to S$ be a quadric fibration over a rational base $S$, over an algebraically closed field of non zero characteristic. Is it true the following?
$Q$ is rational if and only if $Q \to S$ has a ...

**2**

votes

**0**answers

147 views

### Is there an example of index 1 picardnumber 1 Fano 4-fold with h^{21}\neq0

Are there any known examples of index 1 smooth Fano 4-folds X with $\mathsf{Pic}(X)\cong\mathbb{Z}$ with $h^2(\Omega^1_X)\neq0$?

**4**

votes

**1**answer

401 views

### Is the strict transform of a smooth subvariety under the iterated blowup of a smooth varitiety still smooth?

Suppose $Y$ is a smooth variety, $X$ is a smooth hypersurface of $Y$, and $D_1, \ldots, D_n$ are smooth subvarieties of $Y$ that intersect transversely such that the sum of the codimensions of the ...

**8**

votes

**3**answers

471 views

### Birational automorphisms of canonical models

Let $X$ be a variety with canonical singularities such that $K_X$ is ample.
Do you have a reference of the fact that every birational map from $X$ to itself is biregular?
Thank you

**7**

votes

**1**answer

421 views

### Cones, monoids, and the space of (very) ample divisors

An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...

**2**

votes

**2**answers

755 views

### Rational map to a non-uniruled manifold

Let X, Y be smooth projective (over IC), let f:X..>Y be a rational map. Assume Y is not uniruled. Is it true that f will be regular over a non-empty open subset of Y? Funny..

**20**

votes

**1**answer

1k views

### Rationality of intersection of quadrics

Let $X \subset \mathbb{P}^n$ be a complete intersection of two quadrics. It is classical that, if $X$ contains a line, then it is rational. The proof is very simple and basically it is given by taking ...

**3**

votes

**1**answer

539 views

### Stable base loci

Let $\mathbb{B}(L):=\bigcap_{m \in \mathbb{N}} Bs(|mL|)$ be the stable locus of Cartier divisor. I have read the paper "Restricted volumes and base loci of linear sistems" in which it's proved that ...

**5**

votes

**0**answers

517 views

### Semistable sheaves on rationally connected manifolds

Let $(X,H)$ be a polarized projective manifold of dimension $n$ defined over $\mathbb C$, and let $\mathcal E$ be a reflexive sheaf on $X$.
If for every subsheaf $\mathcal F \subset \mathcal E$ the ...

**5**

votes

**3**answers

601 views

### When does f-nef imply nef (after twisting?)

For a surjective morphism $f \colon X \to Z$ of smooth projective varieties, a line bundle $L$ on $X$ is $f$-nef if is nef on the fibers of $f$. More precisely, for every curve $C$ that is mapped to a ...

**16**

votes

**3**answers

758 views

### What is known about the birational involutions of P^3?

Describing the group of birational automorphisms of $\mathbb{P}^n$, $\mbox{Bir}(\mathbb{P}^n)$, for $n\ge 3$ is a fundamental open problem in birational geometry. For $n=2$, the classical theorem of ...

**3**

votes

**1**answer

389 views

### Positivity of the anticanonical bundle of a rationally connected manifold

Let $X$ be a rationally connected smooth projective variety defined over $\mathbb C$.
(1) Can we find a surface $S \subset X$ such that $ (-K_X)^2 \cdot S > 0 ? $
If yes, can we assume that $S$ ...

**7**

votes

**1**answer

440 views

### Simplified treatment of resolutions of complex analytic varieties?

According to the article of Hauser:
The Hironaka theorem on resolution of singularities http://www.ams.org/journals/bull/2003-40-03/S0273-0979-03-00982-0/home.html
The existence of resolution of ...

**5**

votes

**2**answers

580 views

### 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 ...

**1**

vote

**0**answers

329 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 ...

**2**

votes

**1**answer

392 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 ...

**12**

votes

**0**answers

781 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 ...

**9**

votes

**2**answers

858 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 ...

**3**

votes

**2**answers

2k views

### 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 ...

**7**

votes

**1**answer

799 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 ...

**4**

votes

**2**answers

671 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 ...

**20**

votes

**3**answers

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 ...

**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, ...

**7**

votes

**2**answers

828 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 ...

**3**

votes

**1**answer

368 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 ...

**9**

votes

**1**answer

477 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 ...

**3**

votes

**1**answer

807 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 ...

**2**

votes

**1**answer

401 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 ...

**8**

votes

**4**answers

2k views

### 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 ...

**2**

votes

**3**answers

1k 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 ...

**10**

votes

**3**answers

2k views

### 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 ...

**7**

votes

**0**answers

192 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 ...

**7**

votes

**1**answer

549 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 ...

**9**

votes

**2**answers

502 views

### 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 ...

**1**

vote

**2**answers

460 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 ...