# Tagged Questions

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### 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 ...
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### 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 $O_Y$. ...
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### 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 ...
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### 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 ...
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### '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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $E$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...