**4**

votes

**1**answer

369 views

### Numerically equivalent effective divisors and semiampleness

Recall that a divisor $M$ on a variety $X$ is said to be semiample if $kM$ is base point free for a certain $k > 0$.
Being semiample is not a numerical property (take for example torsion and a ...

**2**

votes

**1**answer

281 views

### Do divisors of degree g with this property exist in general

I have the following question. It's a long shot, but worth the try.
Let X be a compact connected Riemann surface of genus $g\geq 2$. Does there exist an effective divisor $D$ on $X$ of degree $g$ ...

**3**

votes

**2**answers

271 views

### When is the Wendt binomial circulant determinant divisible by 3?

The Wendt binomial circulant determinant $W_n$ can be defined quite simply as a resultant:
$$ W_n = \operatorname{res}(x^n-1, (x+1)^n-1). $$
Truer to its name, one may also define it as the ...

**2**

votes

**1**answer

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 $E$ ...

**0**

votes

**0**answers

245 views

### Canonical divisor of a curve base point free (if g>0)

Is there a way to prove that the canonical divisor $W$ of an algebraic function field in one variable $F$ over a field $K$ (that is the function field of an algebraic curve) of genus $g>0$ is base ...

**3**

votes

**1**answer

390 views

### Is every Weil divisor on an arithmetic surface Q-Cartier

This question is about a technical issue I ran into.
Let $S$ be a connected 1-dimensional Dedekind scheme, and let $X\to S$ be a flat projective integral normal 2-dimensional scheme. (For simplicity, ...

**5**

votes

**3**answers

631 views

### Cone of movable curves

Let $X$ be a smooth complex projective variety of dimension $n$.
Under the duality between $N_1(X)$ and $N^1(X)$ we know that closure of cone of effective curves $\overline{NE}(X)$ is dual to closure ...

**1**

vote

**2**answers

293 views

### Top self-intersection of the tautological line bundle

Let $\mathcal E$ be a rank $n$ vector bundle over a curve $Y$ and let $X=\mathbb P(\mathcal E)$ and let $\pi: X \to Y$ be the projection. I would like to compute the value of the top self-intersection ...

**8**

votes

**0**answers

744 views

### Ample divisors on projective surfaces

Question: If $X$ is a projective surface and $U$ is an open affine subset of $X$, then is it true that $X \setminus U$ is the support of an (effective) ample divisor on $X$?
Background: I was reading ...

**3**

votes

**1**answer

330 views

### References about pseudoeffective cone

I'm looking for references of explicit computation of the pseudoeffective cone $\overline{\text{Eff}}(X)$ of a projective variety $X$.

**0**

votes

**0**answers

202 views

### Properties of morphisms induced by divisors on curves

There are a few properties from Hartshorne IV on curves that I am trying to verify. Let $D$ be an effective divisor on a curve (integral scheme of dimension 1, proper over $k$, regular) $X$, $\dim ...

**3**

votes

**2**answers

641 views

### Ample divisors on blown-up projective space

Let $\mathbb{P}=\mathrm{Proj}(\mathbb{C}[x_0,\ldots,x_n])$ be complex projective $n$-space. Assume I have linear subvarieties $L_1,\ldots,L_k\in\mathbb{P}$ of codimension $r_i\ge 2$, respectively. Let ...

**1**

vote

**1**answer

445 views

### Self intersection of blown up points and the lines which they lie on

I'm currently trying to understand the process of blowing-up, and a few things strike me as a little difficult to get an intuitive understanding of what's happening.
The current problem is on self ...

**3**

votes

**1**answer

156 views

### Boundedness of $C.K$ on a surface with $-K$ pseudoeffective

Let $S$ be a projective surface with pseudoeffective anticanonical divisor $-K_S$. Is it true that if $C$ is an integral curve with $C^2<0$ and $C \cdot K_S >0$, then $\max_C (C \cdot K_S)$ is ...

**3**

votes

**2**answers

654 views

### (Anti)Canonical divisor of a blow up

This question may be utterly trivial, or not, but as someone with hardly any knowledge of algebraic geometry I thought there could be a chance I get lucky.
Let X be a rational surface obtained by n ...

**2**

votes

**2**answers

893 views

### Generalisations of Riemann-Roch for surfaces

Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have
$$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$
This is the famous ...

**3**

votes

**2**answers

284 views

### Moving a canonical divisor on a normal surface away from the singular locus

In a previous question Moving a Weil divisor on a normal surface away from a finite set of closed points I probably asked for too much. As J.C. Ottem pointed out, it is not always possible to move a ...

**2**

votes

**3**answers

363 views

### Moving a Weil divisor on a normal surface away from a finite set of closed points

Let $Y$ be a normal surface and let $X$ be a closed subscheme of codimension 2, i.e., $X$ is a finite set of closed points.
Let $D$ be a Weil divisor on $Y$.
Question. Does there exist a Weil ...

**2**

votes

**1**answer

267 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 ...

**2**

votes

**2**answers

281 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 ...

**29**

votes

**10**answers

2k views

### Which 'well-known' algebraic geometric results do not hold in characteristic 2?

A smooth curve $X$ in $\mathbb{P}^n$ is strange if there is a point $p$ which lies on all the tangent lines of $X$.
Examples are $\mathbb{P}^1$ is strange and so is $y=x^2$ in characteristic $2$. ...

**1**

vote

**1**answer

958 views

### Cone of effective divisors!

Let $X$ be a smooth simply connected projective variety of dimension $n$ (over complex numbers of course). For such $X$ we have two famous cones which are cone of effective curves and ample cone and ...

**14**

votes

**2**answers

930 views

### Bertini theorems for base-point-free linear systems in positive characteristics

Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least ...

**3**

votes

**0**answers

250 views

### On generators of the Picard group of a projective smooth surface over a finite field

Let $X$ be a smooth projective surface over a finite field $k=\mathbb{F}_q$. Let us first review the proof of the finite generation of $Pic(X)$ (notice that the proof is valid for any smooth ...

**2**

votes

**2**answers

591 views

### Isolated conics on a del Pezzo surface

Is there anything known about isolated conics in a del Pezzo surface: their number, arrangement, and the corresponding elements of the class group of surface's minimal desingularization? (Isolated ...

**13**

votes

**2**answers

2k views

### Elementary short exact sequence of sheaves

This question arised when I was trying to use this answer to understand Reid's "Young Person's guide to Canonical Singularities". In particular page 352 when computing the blow-up $Y\rightarrow ...

**4**

votes

**2**answers

429 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 ... ...

**7**

votes

**1**answer

401 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

**1**answer

386 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

**0**answers

473 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

**1**answer

611 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

**1**answer

473 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

**0**answers

318 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

**3**answers

588 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 ...

**1**

vote

**1**answer

349 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

**0**answers

557 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

**2**answers

546 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 ...

**5**

votes

**2**answers

2k 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

**1**answer

572 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

353 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 ...

**11**

votes

**6**answers

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

**0**answers

232 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 ...

**7**

votes

**3**answers

1k 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

**3**answers

525 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

**2**answers

516 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

**7**answers

859 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.

**12**

votes

**5**answers

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

**3**answers

278 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

**4**answers

715 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 ...

**6**

votes

**2**answers

767 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?