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2
votes
1answer
173 views

F-curves and divisors on $\overline{\mathcal{M}}_{g,n}$

It is conjectured that a divisor on $\overline{\mathcal{M}}_{g,n}$ would be ample if $D\cdot C >0$ for all F-curves $C\subset \overline{\mathcal{M}}_{g,n}$. Does the intersection degree with all ...
1
vote
1answer
133 views

On Non F-pure ideal and Sharp F-Purity for a pair $(X, \Delta)$ where $K_X+\Delta$ is NOT $\mathbb{Q}$-Cartier

Suppose $(X,\Delta\ge 0)$ is a pair such that $(p^g-1)(K_X+\Delta)$ is an Integral Weil Divisor for some $g>0$ and $X$ is a normal variety. Define $\mathcal{L}_{e,\Delta} = \mathcal{O}_X( ...
5
votes
2answers
313 views

Hn(X, OX) = 0 for X birational to a regular affine variety?

It is a basic fact that $H^n(X, F) = 0$ if $X$ is noetherian affine, $n > 0$, and $F$ a quasi-coherent sheaf. If $Y \to X$ is a blow-up of a smooth variety in a smooth center, then then ...
9
votes
1answer
591 views

Why is proving fully-faithfulness of an integral functor locally analytically sufficient?

More than once I've come across a statement in a paper about derived categories in which it says something to the effect of "in order to prove that $\Phi:D^b(X)\rightarrow D^b(Y)$ is fully-faithful we ...
8
votes
1answer
222 views

Conditions for the contractibility of subvarieties

One often finds statements of the sort "and one can contract this subvariety $E\subset X$ to a point in the projective variety $W$," without any explanation of the reasons such a contraction exists, ...
5
votes
1answer
108 views

Are there only finitely varieties of general type dominated by a given variety in the following sense

Let $X$ be a smooth projective complex algebraic variety. I believe the answer to the following question should be well-known. It is an instance of the Iitaka conjecture. There are only finitely many ...
11
votes
2answers
460 views

Geometrically unirational varieties that are not unirational

By a variety over a field $k$, I mean a scheme that is separated and of finite type over $k$. I indicate changes of the base ring by subscripts. Does there exist a smooth and projective variety $V$ ...
6
votes
1answer
327 views

Proving a variety is not unirational

It is known that if a variety is unirational then it is rationally connected. However, there are no known examples of rationally connected varieties which are not unirational. In these notes, at the ...
2
votes
0answers
232 views

Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces

Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
3
votes
0answers
241 views

Adjunction Formula for Weil Divisors on a Normal Variety X

Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
1
vote
0answers
65 views

K-equivalence does not depend on the choice of the third variety

By K-equivalent of two smooth varieties $X,Y$, we mean there exist a smooth variety $Z$, and birational morphism $q: Z \to X,\quad p: Z \to Y$ , such that $q^* \omega_X \cong p^* \omega_Y$. Suppose ...
0
votes
0answers
175 views

How to Construct a ''Nice'' Birational Model in Characteristic p>0

Let $X=Spec\ A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ be a prime Weil-divisor on $X$. Now, I ...
3
votes
1answer
173 views

About del Pezzo surfaces

First of all, I would like to apologize if my question is stupid or a well known fact. Let $F$ be a rational surface with $K_F^2=5$ and $f: F\rightarrow \mathbb{P_k^2}$ be a birational morphism ...
1
vote
0answers
129 views

Is always a Crepant birational map between smooth varieties a small modification

Lemma: A crepant birational map between quasi-projective normal varieties with only terminal singularities is in fact small modifitacions, i.e. isomorphism in codimension 1. So, if ...
2
votes
1answer
162 views

Lifting vector fields to its resolution in char $p$

In this paper 4.7, the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log ...
2
votes
1answer
173 views

Crepant Birational Map on the Blow-up

Let $ f: \mathbb{P}^n \dashrightarrow (\mathbb{P}^1)^n , [x_0:\dots :x_n] \mapsto ([x_0,x_1],[x_0,x_2], \dots ,[x_0,x_n])$ a birational map. In particular, if $X$ is the blow-up of $\mathbb{P}^n$ at ...
1
vote
1answer
140 views

Crepant Birational Map

Let $ f: \mathbb{P}^n \dashrightarrow (\mathbb{P}^1)^n , [x_0:\dots :x_n] \mapsto ([x_0,x_1],[x_0,x_2], \dots ,[x_0,x_n])$ a birational map. Is $f$ a Crepant birational map?
2
votes
1answer
162 views

On a Strongly F-regular Pair (X, \Delta)

Let $X$ be a normal projective variety over a field of characteristic $p>0$ and $(X, \Delta\geq 0)$ be a pair such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier whose index is not divisible by $p$. ...
4
votes
1answer
104 views

Strictness of the inequality relating the Iitaka dimension and algebraic dimension

For any (compact and connected) complex manifold $X$ and any line bundle $L$ on $X$ we have the well known inequalities $\kappa(X,L)\leq\alpha(X)\leq\dim(X)$ relating the Iitaka-dimension ...
10
votes
6answers
585 views

When does $Aut(X)=Bir(X)$ hold?

Let $X$ be a projective complex manifold. Under what condition do we have the equality $Aut(X)=Bir(X)$? Here $Aut(X)$ denotes the group of holomorphic automorphisms of $X$ and $Bir(X)$ the group of ...
6
votes
2answers
338 views

Quotients of rational surfaces

Let $X$ be a projective surface defined over a field $k$ of characteristic $0$, and let $G$ be a finite group acting biregularly on $X$. Assuming that $X$ is rational over $k$, is the quotient $X/G$ ...
5
votes
2answers
914 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
149 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
353 views

is Hasse principle a birational invariant?

...it is probably a very trivial question, but I am a beginner in arithmetics.
0
votes
0answers
182 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 $V\subseteq X$ ...
2
votes
1answer
277 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
294 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
728 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
351 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
191 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
565 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
463 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
528 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
241 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
307 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
555 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
240 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
638 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
307 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
967 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 ...
2
votes
2answers
542 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
256 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
190 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
508 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
271 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
499 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
268 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
364 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
285 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
554 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 ...