**0**

votes

**2**answers

255 views

### “rationality” of divisors

Let $X$ be a smooth projective variety over some field $k$. Then each closed point $x$ has an associated residue field $k(x)$ which is a finite extension of $k$ and a point is rational when $k(x)=k$.
...

**2**

votes

**1**answer

294 views

### what are the singularities of a normal crossings divisor?

This is probably a very stupid question. I'm sorry.
Let $D$ be a simple normal crossings divisor on some smooth projective variety $D$. By this I mean that the irreducible components $D_i$ are smooth ...

**1**

vote

**1**answer

239 views

### a question on Euler characteristic of normal crossing divisors

Let $X$ be a smooth, projective complex algebraic variety. Let $D$ be a simple normal crossings divisor on $X$, with irreducible components $D_i$, for $i \in I$. For each non-empty subset $J \subset ...

**2**

votes

**1**answer

178 views

### Conical divisor over a $\mathbb Q$-Cartier divisor.

I would like to know if the following statement is correct.
Statement. Let $X$ be a normal projective variety with $Pic(X)=\mathbb Z+torsion$. Let $L$ be an ample line bundle on $X$ and let $D$ be an ...

**0**

votes

**0**answers

71 views

### sections of vector bundles transversal to a divisor

Let $X$ a smooth projective curve over $\mathbb{C}$, $S$ a finite subscheme of $X$.
$E$ a vector bundle over $X$ with a divisor $D$.
We look at the sections $A:=H^{0}(X,E)$ with $\deg E$ big enough.
...

**1**

vote

**1**answer

178 views

### On divisorial correspondences between curves

Assume we are given two smooth curves $C_1$ and $C_2$ over an algebraically closed field $k$. It is known that divisorial correspondences between them correspond to homomorphisms between their ...

**0**

votes

**0**answers

80 views

### invertible sheaf of a hypersurface

Let $X \hookrightarrow \mathbb{P}^n$ be a hypersurface of degree $d$. I am trying to prove that $\mathcal{O}_{\mathbb{P}^n}(X)=\mathcal{O}(d)$. My idea is the following: if one considers the $d$-uple ...

**1**

vote

**0**answers

129 views

### cohomology of a normal crossing divisor

Let $D$ be a simple normal crossing divisor on a smooth projective variety over a field $k \subset \mathcal{C}.$ Write $D_i$ with $i \in I$ for its irreducible components. Denote, as usual,
...

**2**

votes

**1**answer

393 views

### On morphisms to projective space arising from a linear system

Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this question should ...

**1**

vote

**2**answers

537 views

### Embedded resolution of singularities

I'd like to check with my colleagues whether I have correctly understood "embedded resolution of singularities".
Let $X$ be a nonsingular projective variety over $\mathbf C$ and let $D$ be a "nice" ...

**7**

votes

**2**answers

461 views

### Irreducible divisors containing an arbitrary closed set

Let $X$ be a normal complex projective variety, let $V$ be a closed subset of $X$ (possibly reducible), and let $I_V$ be its ideal sheaf (consider the reduced scheme structure for example).
If $A$ is ...

**2**

votes

**1**answer

185 views

### divisors and powers of line bundles

Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Let $D$ be an effective divisor on $X$ and $m \geq 2$ an ...

**2**

votes

**0**answers

143 views

### A nice way to verify whether the Neron-Severi group of a smooth affine variety is zero

Let $S$ be a smooth affine variety over an algebraically closed field (this could be the field of complex numbers). Is there an 'easy' way to verify whether $NS(S)=0$? Unfortunately, I don't know how ...

**3**

votes

**1**answer

311 views

### Roots of line bundles

Let $k$ be a fixed algebraically closed field and $X/k$ an irreducible scheme smooth and proper over $k$. Can there exist a line bundles $\mathcal{L}, \mathcal{M}$ and an integer $m > 0$ so that
...

**3**

votes

**2**answers

802 views

### Bertini's Theorem small print

Suppose $S\subset \mathbb{P}^n$ is a smooth del Pezzo surface and $C$ is an irreducible smooth curve (you can make it rational if it simplifies the setting) such that $\mathcal{L}=\vert -K_S-C\vert $ ...

**4**

votes

**1**answer

300 views

### Divisorial contraction: when is the image an algebraic space or a stack?

Let $X$ be a smooth projective surface (in the category of varieties, or schemes), and let $C\subset X$ be a curve (a priori not irreducible, but the irreducible case in itself is already ...

**1**

vote

**1**answer

179 views

### The pseudoeffective cone does not contain lines

It seems to be well-known that the pseudoeffective cone $\overline{\text{Eff}}(X)$ of a normal variety $X$ does not contain lines through the origin. How can it be proved? Is there a reference?

**2**

votes

**1**answer

282 views

### Picard/cohomology lattice of surfaces of low degree in $\mathbb P^3$

Let $S_{d>3}\subset\mathbb{P}^3_{\mathbb{C}}$ be a smooth surface of degree $d$. What is known (where to read?) about the Picard/cohomology lattice for small d?
e.g. for $d=4$ the cohomology ...

**2**

votes

**1**answer

188 views

### Constructing rational functions with ramification locus the divisor of some $n$-form

I'm still busy learning the theory of linear systems for compact Riemann surfaces. If the answer to the following question is negative, then there might not be any point in continuing.
Let $X$ be a ...

**0**

votes

**0**answers

100 views

### How to compute the Betti numbers of S-D for a surface S and a divisor D?

Let S be a projective non-singular surface and D a Cartier divisor which has a smooth representative. Can the Betti numbers of S-D be represented by the Betti numbers of S and D? In a paper ...

**5**

votes

**1**answer

412 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

285 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

288 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

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 $E$ ...

**1**

vote

**0**answers

272 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

415 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

751 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

313 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

871 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

368 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

211 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

743 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 ...

**2**

votes

**1**answer

514 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

160 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

886 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

1k 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

297 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

398 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

277 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

298 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 $2$. ...

**2**

votes

**1**answer

1k 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

1k 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

267 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

652 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

430 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

**1**answer

417 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

401 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

494 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'. ...