All Questions
Tagged with divisors ag.algebraic-geometry
313 questions
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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,
$D_J=\...
- 11
2
votes
1
answer
1k
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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 ...
- 23
5
votes
2
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3k
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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" ...
- 53
9
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2
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754
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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 ...
- 1,150
2
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1
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546
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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 ...
- 23
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194
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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 ...
- 16.9k
3
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1
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695
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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
...
- 3,555
4
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2
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2k
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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 $ ...
- 2,215
5
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1
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870
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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 interesting)....
- 7,722
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2
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416
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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?
- 427
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411
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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 ...
- 2,232
2
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1
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321
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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 ...
- 1,213
10
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1
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911
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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 non-...
- 205
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1
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367
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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 $g$ ...
- 9,497
2
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1
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269
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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 $E$ ...
- 1,150
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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 ...
- 11
3
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1
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736
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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, ...
- 9,497
9
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3
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3k
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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
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2
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985
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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 ...
- 427
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3k
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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 ...
- 5,359
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1
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1k
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References about pseudoeffective cone
I'm looking for references of explicit computation of the pseudoeffective cone $\overline{\text{Eff}}(X)$ of a projective variety $X$.
- 427
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331
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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 |D|...
- 419
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2k
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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 ...
- 4,902
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1
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2k
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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 ...
- 153
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228
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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 ...
- 427
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2
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4k
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(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 ...
- 153
6
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2
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2k
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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 ...
- 2,215
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2
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859
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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 ...
- 9,497
2
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3
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753
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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 ...
- 9,497
2
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1
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389
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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 cycles?...
- 21
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2
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1k
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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 ...
- 2,215
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10
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3k
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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 $2$. ...
- 2,215
13
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1
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4k
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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 ...
19
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2
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3k
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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 ...
- 20.5k
4
votes
2
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1k
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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 ...
- 1,488
21
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2
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11k
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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 A^2/\...
- 2,215
4
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2
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482
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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
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1
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697
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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 ...
- 9,497
7
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650
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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 ...
- 1,871
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0
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764
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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'. (...
- 468
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1
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2k
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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 ...
- 7,338
6
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1
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868
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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 ...
- 1,150
2
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0
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515
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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 ...
- 16.9k
6
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3
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1k
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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 ...
- 427
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1
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557
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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 ...
- 205
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0
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3k
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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 ...
- 20.5k
8
votes
2
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783
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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 ...
- 1,150
19
votes
2
answers
8k
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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 ...
- 2,215
3
votes
1
answer
1k
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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 ...
- 1,150
2
votes
1
answer
743
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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 ...
- 1,150