For questions related to divisors in the sense of algebraic geometry (Cartier divisors, Weil divisors and so on). For question on divisors in the number theoretic sense please use the tag divisors-multiples.

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4
votes
2answers
432 views

Vague question on $Pic^0$

For a smooth variety $X$ when $Pic^0(X)$ is trivial, we get an isomorphism between $N^1(X)$ and Picard group and life become easier. My question is whether in general there are theorems, criteria ... ...
8
votes
1answer
428 views

Does combining Abhyankar's Lemma and embedded resolution give horizontal normal crossings

Let $\pi:Y\longrightarrow \mathbf{P}^1_{\mathbf{Z}}$ be a finite surjective flat morphism of schemes, where $Y$ is a normal integral flat projective 2-dimensional $\mathbf{Z}$-scheme, with branch ...
7
votes
1answer
407 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
0answers
511 views

Riemann-Roch for ARBITRARY Function Fields

I know that on an algebraic function field in one variable over any base field, there is a good divisor theory for it and a Riemann-Roch Theorem; in particular, there is a 'good' notion of 'genus'. ...
7
votes
1answer
719 views

Why is (line bundle, appropriate rational section) not a standard kind of divisor?

In algebraic geometry, there are two standard "kinds" of divisors: Weil divisors and Cartier divisors. Weil divisors provide better geometric intuition, while Cartier divisors are more general (if ...
5
votes
1answer
502 views

Stable base loci cannot contain isolated points

Let $X$ be a normal projective complex variety. A theorem of Fujita-Zariski says that if $L$ is a Cartier divisor on $X$ such that the base locus $Bs(|L|)$ is a finite set then $L$ is semiample. It ...
2
votes
0answers
333 views

A motivic complex

By definition, Voevodsky's motivic complex (an object of his $DM^{eff}_-$) is a complex of sheaves with transfers whose cohomology sheaves are homotopy invariant. Now, I consider the complex (of ...
5
votes
3answers
646 views

Are there (-2)-curves on an Enriques surface?

Let $X$ be an Enriques surface. A $(-2)$-curve is an irriducible rational curve on X such that $C^2 = -2$. By Proposition [VIII,16.1] from Barth-Peters-Van de Ven, we have that if $D^2 = -2$, then it ...
2
votes
1answer
370 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
0answers
692 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 ...
8
votes
2answers
596 views

Base locus of divisors on blowings up of the projective space

Let $X$ be the blowing-up of $\mathbb{P}^n$ in $r$ points in general position. Let $\{H,E_1,...,E_r\}$ be the standard basis of $Pic(X)$. Further let $D=dH-\sum m_j E_j$, be an effective divisor, with ...
6
votes
2answers
3k views

The canonical line bundle of a normal variety

I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...
3
votes
1answer
739 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
1answer
388 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 ...
12
votes
6answers
1k views

Seeking Noetherian normal domain with vanishing Picard group but not a UFD

Once again, the question says it all. My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...
2
votes
0answers
243 views

Proof of Saito criterion

Does it exist another proof of saito's criterion for free divisors, other than the one in "K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo ...
9
votes
3answers
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 ...
1
vote
3answers
538 views

Terminology issue: meaning of 'ample class' ?

What is meant by an "ample class" in general? Motivation: In the document I am reading, the phrase in question is "fix an ample class $\alpha\in H^1(X,\Omega^1_X)$." I know what ampleness of a line ...
4
votes
2answers
653 views

Special divisors on hyperelliptic curves

I was reading a proof that used the following result Let $C$ be a hyperelliptic of genus $\ge 3$ and $\tau \colon C \to C$ the hyperelliptic involution. If $D$ is an effective divisor of degree ...
4
votes
7answers
963 views

Nef divisors with few global sections

Are there nef divisors D on a complex projective manifold X such that $h^0(X,D)$ is less than or equal to $\dim X$? Edit: In fact I'm interested in nef line bundles D, not just divisors.
13
votes
5answers
3k views

blowing up, -1 curves, effective and ample divisors

Lets say we're on a smooth surface, and we blow up at a point. Is there a simple explicit computation that shows to me the fact that the exceptional divisor E has self intersection -1 ? I don't ...
3
votes
3answers
290 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 ...
2
votes
4answers
753 views

Examples of divisors on an analytical manifold

I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...
7
votes
2answers
835 views

Anticanonical divisor of the blow up of P^2 in 9 points

Let $S$ the blow up of $P^2$ in nine points. Why is the anticanonical divisor $-K_S$ not semiample?
4
votes
1answer
558 views

A working generalization of Weil divisors

Hartshorne defines Weil divisors under the hypotheses "Noetherian integral separated scheme regular in codimension 1", which, for example, ensures that the divisor of a rational function is a finite ...
5
votes
1answer
284 views

Is there a canonical notion of principal divisor on a discrete dynamical system?

I hope this question is well-posed. Let (X, f) be a discrete dynamical system such that every x in X has finite period, i.e. there is some n such that f^n(x) = x. Let Div(X) be the free abelian ...
9
votes
2answers
3k views

does a line bundle always have a degree

For curves there is a very simple notion of degree of a line bundle or equivalently of a Weil or Cartier divisor. Even in any projective space $\mathbb P(V)$ divisors are cut out by hypersurfaces ...
5
votes
3answers
407 views

Weil divisors on non Noetherian schemes

Let X be an integral scheme that is separated (say over an affine scheme). Define a Weil divisor as a finite integral combination of height 1 points of X, where the height of a point of X is the ...