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6
votes
2answers
1k views

Training towards research on birational geometry/minimal model program

Being a not yet enrolled independently supervised graduate student in mathematics, with prospects of applying to American graduate schools hopefully in a 1-2 years' time, I have a background of having ...
1
vote
0answers
153 views

Fields over which cubic hypersurfaces are rational

All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...
5
votes
1answer
375 views

is Hasse principle a birational invariant?

...it is probably a very trivial question, but I am a beginner in arithmetics.
0
votes
0answers
201 views

Numerical dimension of nef divisors

Let $D$ be a nef divisor (moreover suppose it is effective if you prefer) on a normal projective variety of dimension $n$. Let $k\in[1,n-1]$. If $D^k\cdot V=0$ for generic subvarieties $V\subseteq X$ ...
5
votes
1answer
380 views

Can a birational morphism between smooth varieties be dominated by smooth blowup sequences?

Suppose $f:X\rightarrow Y$ is a birational morphism between smooth varieties and D is a snc divisor on $Y$. Can we find a smooth blowup sequence on $Y$ which dominates $f$ such that the preimage of D ...
2
votes
1answer
297 views

Birational maps and open subsets inducing isomorphisms

Let $\pi \colon X\to Y$ be a morphism, where $X$ is projective. Suppose the existence of an open subset $U\subset X$ such that $\pi^{-1}(\pi(U))=U$, that $\pi$ is injective on $U$, and that its ...
5
votes
2answers
361 views

Crepant resolutions of ODP's on a 3-fold

It seems to be a well-known fact that if we have a 3-fold $X$ with only ODP singularities (ordinary double point) and a smooth Weil divisor $E$ passing through them, then by blowing-up $X$ along $E$ ...
9
votes
3answers
915 views

Extending birational isomorphisms between planar curves to the P^2

Let $k$ be a field and let $C,D$ be two integral curves in $\mathbb{P}^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $\mathbb{P}^2_k$. To be precise, does there exist a ...
4
votes
1answer
381 views

Discussion of Luroth's problem in an article of Beauville

I am reading a wonderful article of Arnaud Beauville, called La théorie de Hodge et quelques applications http://math.unice.fr/~beauvill/conf/Bordeaux2.pdf There is one place on page 12 that I can ...
2
votes
1answer
207 views

extending biholomorphic maps to bimeromorphic maps

A meromorphic map of complex spaces (in the sense of Remmert) $f: X \to Y$ is a multivalued map such that its graph $\Gamma$ is an analytic subset of $X \times Y$ and off some analytic subset $Z ...
4
votes
2answers
651 views

Can one prove vanishing of higher direct images fiber-wise?

Let $\pi:X\to Y$ be a proper map of algebraic varieties (over $\mathbb C$) which is a bi-rational equivalence. are the following statements equivalent? The derived direct image of $O_X$ is $O_Y$. ...
7
votes
2answers
495 views

Why is $\mathbb{Q}$-factoriality not local in the étale topology?

I was reading Kollár and Mori's book today and stumbled on the following passage: "The $\mathbb{Q}$-factoriality assumption is a very natural one if we start with smooth varieties, and it makes many ...
2
votes
1answer
656 views

What can be said about a pullback of a very ample line bundle w.r.t birational maps?

Let $X$ be a smooth projective variety and $\phi: X \to \mathbb P^n$ be a map. If $\phi$ is an embedding then $E=\phi^*(O(1))$ is very ample. But can one say something if $\phi$ is birational (but not ...
1
vote
1answer
259 views

'Reference' request: Program to work with cyclic quotient singularities.

I'm looking for program code to deal with cyclic quotient singularities on normal surfaces. In particular, at the moment I need that given a singularity $p$ like $p=\frac{1}{n}(1,a)$ the code ...
2
votes
2answers
361 views

On the points of indeterminacy of a rational map

Let $X,Y$ be complex projective varieties with $X$ irreducible, and let $f:X\dashrightarrow Y$ be a rational map. If $U\subseteq X$ is the largest open set where $f$ can be defined, is it true that ...
4
votes
1answer
624 views

Example of cone of numerically effective curves which is not polyhedral

I think I have seen more than one reference in which the cone of numerically effective curves can be 'not polyhedral', i.e. with an infinite number of extremal rays I cannot remember where I read ...
1
vote
2answers
245 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
1answer
696 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
1answer
317 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 ...
6
votes
1answer
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
2answers
582 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
1answer
270 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
1answer
194 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
2answers
533 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
1answer
281 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
1answer
513 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 ...
2
votes
1answer
275 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
1answer
377 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
2answers
295 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
1answer
576 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
1answer
206 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
0answers
146 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
1answer
338 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
3answers
459 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
1answer
394 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
2answers
695 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..
18
votes
1answer
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
1answer
474 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
0answers
506 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
3answers
551 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 ...
15
votes
3answers
725 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
1answer
369 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
1answer
425 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
2answers
532 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
0answers
321 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
1answer
361 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 ...
10
votes
0answers
633 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
2answers
806 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
2answers
1k 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
1answer
753 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 ...