All Questions
Tagged with divisors ag.algebraic-geometry
313 questions
35
votes
1
answer
2k
views
Are there topological versions of the idea of divisor?
I am trying to extract a particular, more lightweight and more focussed at the same time, case of my recent question Which of the physics dualities are closest in essence to the Spanier-Whitehead ...
- 18.4k
32
votes
10
answers
3k
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 $2$. ...
- 2,215
29
votes
7
answers
7k
views
Elementary proof of Riemann-Roch for compact Riemann surfaces
I am supposed to give a talk about the Riemann-Roch theorem to a seminar of first and second year graduate students. I want to do Riemann-Roch for compact Riemann surfaces, but I am open to perhaps ...
user100272
27
votes
5
answers
7k
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 ...
- 271
21
votes
2
answers
11k
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 A^2/\...
- 2,215
20
votes
2
answers
10k
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 ...
- 3,968
19
votes
2
answers
8k
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 ...
- 2,215
19
votes
2
answers
3k
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 ...
- 20.5k
17
votes
2
answers
1k
views
Line bundles vs. Cartier divisors on a non-integral scheme
It is well-known that if $X$ is an integral scheme, then there is an isomorphism $CaCl(X)\to Pic(X)$ taking $[D]$ to $[\mathcal{O}_X(D)]$. Does anyone know any simple examples where the above map ...
- 11.6k
16
votes
3
answers
4k
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 ...
- 4,450
15
votes
0
answers
3k
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 ...
- 20.5k
14
votes
1
answer
529
views
Birational automorphisms of varieties of Picard number one
Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism.
Must $f$ necessarily contract a divisor?
user91659
14
votes
0
answers
3k
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 ...
- 5,359
13
votes
1
answer
4k
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 ...
13
votes
1
answer
863
views
Generalization of the rigidity lemma in birational geometry
Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected.
If there exists ...
user75933
12
votes
1
answer
8k
views
Simple normal crossing divisors
I found the following definition.
A Weil divisor $D = \sum_{i}D_i \subset X$ on a smooth variety $X$ is simple
normal crossing if for every point $p \in X$ a local equation of $D$
is $x_1\cdot...\...
user47036
10
votes
2
answers
2k
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?
- 427
10
votes
2
answers
1k
views
Picard group of a cubic hypersurface
Consider the following cubic hypersurface in $\mathbb{P}^5$:
$$
X = \{z_0z_3z_5-z_1^2z_5-z_0z_4^2+2z_1z_2z_4-z_2^2z_3 = 0\}\subset\mathbb{P}^5
$$
The singular locus of $X$ is the Veronese surface $V\...
user168611
10
votes
1
answer
314
views
Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?
Is there a closed subscheme $D$ in $\mathbb P^2_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map
$$ \phi: X(\mathbb C) \to \mathbb P^...
- 103
10
votes
1
answer
911
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 non-...
- 205
10
votes
0
answers
217
views
Subvarieties with isomorphic complements
Let $X$ be a smooth irreducible projective variety over $\mathbb C$, $Y_1, Y_2$ are two closed smooth subvarieties. Assume $X-Y_1 \cong X-Y_2$, what do $Y_1$ and $Y_2$ have in common (at least ...
- 6,622
9
votes
2
answers
754
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 ...
- 1,150
9
votes
3
answers
3k
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 ...
9
votes
2
answers
329
views
Effectiveness of the distinguished theta characteristic in characteristic 2
Let $k$ be an algebraically closed field of characteristic 2. Let $C$ be a (smooth projective connected) curve over $k$. Can there exist a rational function on $C$ whose differential is holomorphic ...
- 231
9
votes
1
answer
443
views
Set theoretic equation for Veronese varieties
Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety ...
user125056
9
votes
1
answer
2k
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 ...
- 7,338
8
votes
1
answer
697
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 ...
- 9,497
8
votes
2
answers
783
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 ...
- 1,150
8
votes
0
answers
343
views
How do I make the components of a Cartier divisor again Cartier divisors?
Let $D$ be an effective Cartier divisor on a normal noetherian scheme $X$. Its irreducible components are codimension $1$ subschemes, i.e. Weil divisors, of $X$ but not necessarily Cartier divisors. I ...
- 227
8
votes
0
answers
167
views
On a smooth curve $C$, when is $K_C \sim_\mathbb{Q} (2g-2)P$?
Let $C$ be a smooth curve of genus $g$ over $\mathbb{C}$. I am interested in the following property:
There exists a point $P \in C$ such that $K_C \sim_\mathbb{Q} (2g-2)P$. Equivalently, $K_C - (2g-2)...
- 625
7
votes
7
answers
2k
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.
user2529
7
votes
3
answers
585
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 ...
- 3,968
7
votes
1
answer
880
views
Bertini's theorem over non-algebraically closed field
Let $K$ be a non-algebraically closed (infinite) field of characteristic $0$ and $X$ a smooth, projective $K$-variety. Does there exist an ample invertible sheaf $\mathcal{L}$ on $X$ such that a ...
- 1,981
7
votes
1
answer
760
views
Bertini's Theorem
Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...
user47036
7
votes
1
answer
543
views
A constructive proof of the theorem of the cube
Do you know a constructive proof of the theorem of the cube ? More precisely, let $X$, $Y$, $Z$ be projective varieties (e.g., over an algebraically closed field $k$) with points $x$, $y$, $z$ ...
- 1,833
7
votes
1
answer
945
views
Push-forward of nef divisors via finite morphisms
Let $f:X\rightarrow Y$ be a finite morphism between smooth projective varieties, and let $D$ be an effective nef but not ample divisor on $X$.
Consider the divisor $f_{*}D$ on $Y$. Is $f_{*}D$ nef ...
user58018
7
votes
1
answer
650
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 ...
- 1,871
7
votes
1
answer
1k
views
Pull-back divisor being Cartier
Let $\pi \colon X \rightarrow Y$ be a projective morphism with connected fibers between normal quasi-projective varieties. Let $N$ be a $\mathbb{Q}$-Cartier divisor on $Y$ so that $\pi^*(N)$ is ...
- 1,595
7
votes
1
answer
426
views
Degree of equations of secant varieties of Veronese varieties
Let $Sec_r(V)$ be the $r$-secant variety of a Veronse variety $V\subset\mathbb{P}^N$, that is
$$Sec_r(V) = \bigcup_{p_1,...,p_r\in V}\left\langle p_1,...,p_r\right\rangle\subset\mathbb{P}^N$$
where $...
user101890
7
votes
1
answer
333
views
Pencils on del Pezzo surfaces
Let $X$ be the blow-up of $\mathbb{P}^2$ at three general points $p_1,p_2,p_3$, that is a del Pezzo surface of degree six, and let $\pi_i:X\rightarrow\mathbb{P}^1$ be the morphism induced by the ...
user114198
6
votes
3
answers
1k
views
Are there (-2)-curves on an Enriques surface?
Let $X$ be an Enriques surface. A $(-2)$-curve is an irriducible rational curve on X such that $C^2 = -2$. By Proposition [VIII,16.1] from Barth-Peters-Van de Ven, we have that if $D^2 = -2$, then it ...
- 427
6
votes
2
answers
2k
views
Generalisations of Riemann-Roch for surfaces
Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have
$$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$
This is the famous ...
- 2,215
6
votes
1
answer
354
views
Fundamental groups of complements of divisors in $\mathbb P^2$
Let $D$ be a divisor in $\mathbb P^2_{\mathbb C}$ and let $X= \mathbb P^2_{\mathbb C} - D$.
Under what condition on $D$ is the fundamental group of $X$ infinite?
- 71
6
votes
2
answers
524
views
Global sections of multiples of a divisor
Let $D$ be an integral divisor on a smooth projective variety $X$. Consider the multiples $mD$ of $D$ for $m\geq 0$. Clearly, $h^0(X,mD) = 1$ for $m = 0$.
Is there any example where $h^0(X,mD) = 0$ ...
- 8,998
6
votes
2
answers
360
views
The kernel of a nef line bundle
Let $V$ be a complex projective variety and $L$ a nef line bundle on $V$ (i.e., $L$ is non-negative on every curve in $V$). Denote, as usual, $\deg_LX = c_1(L)^{\dim{X}}.[X]$ for $X$ a subvariety of $...
- 13.8k
6
votes
1
answer
3k
views
Top self-intersection of exceptional divisors
Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...
user49214
6
votes
1
answer
868
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 ...
- 1,150
6
votes
2
answers
345
views
Toric varieties as hypersurfaces of degree (1, ..., 1) in a product of projective spaces
I wonder if some hypersurfaces of multi-degree $(1, ..., 1)$ in a product of projective spaces are toric, with polytope a union of polytopes of toric divisors of the ambient space. This question is ...
- 183
6
votes
1
answer
316
views
First Chern class and field extensions
Let $X$ be a smooth, complex projective algebraic variety defined over a number field $K$.
Let $D$ be a divisor of $X$ defined over $K$ with the following property:
For any curve $C$ defined over $K$,...
- 321
6
votes
2
answers
483
views
Blow-ups of $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$
Let us consider the points
$$p_1=[1:0:...:0],p_2 = [0:1:...:0],...,p_{n-2} =[0:...:0:1],\\
p_{n-1}=[1:1:...:1]\in\mathbb{P}^{n-3}$$
and the blow-up $X = Bl_{p_1,...,p_{n-1}}\mathbb{P}^{n-3}$.
...
user58018