All Questions
Tagged with divisors ag.algebraic-geometry
313 questions
2
votes
1
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511
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Rigid effective divisors
Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$.
Now, let $f:X\...
user77919
2
votes
2
answers
208
views
Mori cones and projective morphisms
Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties, and $NE(X),NE(Y)$ the Mori cones of curves of $X$ and $Y$. Assume that $NE(X)$ is finitely generated. Then is $NE(Y)$ finitely ...
- 8,998
2
votes
1
answer
249
views
Two conditions on divisors on surfaces
Let $X$ be a smooth projective surface and $D$ be an effective Cartier divisor (not necessarily ample) on $X$. Is there a connection between these two conditions?
$(i)$ for a large enough $n$, the ...
- 267
2
votes
2
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674
views
Standard plane Cremona transformation
Let us consider nine general points $p_1,...,p_9\in\mathbb{P}^2$ and the line $L = \left\langle p_1,p_2\right\rangle$. Take the standard Cremona $f_1$ centred in $p_3,p_4,p_5$, then $C_1 = f_1(L)$ is ...
user58018
2
votes
2
answers
416
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?
- 427
2
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1
answer
218
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Existence of terminal $3$-fold flips
Does there exist a terminal $3$-fold $X$ with a curve $C\subset X$ such that $K_X\cdot C < 0$ admitting a Mori flip $X\dashrightarrow Y$, flipping $C$ to a curve $C'\subset Y$, where the singular ...
- 8,998
2
votes
2
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515
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Intuition behind Kawamata's definition of a relative movable Cartier divisor
I am trying to develop a good geometric intuition and to understand the motivation behind Kawamata's definition of a relative movable Cartier divisor in Section 2 of reference [1]:
[1] Y. Kawamata, ...
- 3,022
2
votes
1
answer
274
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 ...
- 14.3k
2
votes
1
answer
411
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 ...
- 2,232
2
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1
answer
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
2
votes
2
answers
1k
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 ...
- 2,215
2
votes
1
answer
276
views
Varieties with disjoint prime divisors
I've been following the works of Totaro, Pereira, and Bogomolov/Pirutka/Silberstein about algebraic varieties over complex numbers with families of disjoint divisors. The last one generalizes results ...
- 91
2
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1
answer
489
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Nef and pseudo-effective divisors over non algebraically closed fields
Let $X$ be a projective variety over a field $K$. As a consequence of Kleiman's criterion, when $K$ is algebraically closed, we have that if $D$ is a nef divisor on $X$ then $D$ is pseudo-effective.
...
- 8,998
2
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1
answer
398
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Picard group of $\mathrm{GL}(n)$-orbits
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Mat{Mat}$Consider the general linear group
$$
\GL(n) = \left\lbrace
\left(\begin{array}{cc}
A & C \\
M & B
\end{array}\right) \text{ with } A\...
user168611
2
votes
1
answer
257
views
Flipping and flipped loci
Let $f:X\dashrightarrow Y$ be the flip of a small contraction $\phi:X\rightarrow Z$, and let $\psi:Y\rightarrow Z$ be the small contraction such that $\psi\circ f = \phi$. Let $Exc(\phi), Exc(\psi)$ ...
user117617
2
votes
1
answer
974
views
Inverse image of a divisor
Let $f:X\rightarrow Y$ be a morphism with connected fibers between projective varieties (not necessarily flat). Let $D\subset Y$ be an irreducible divisor.
Let us look at the cycle $f^{-1}(D)\subset X$...
user82886
2
votes
1
answer
367
views
Do divisors of degree g with this property exist in general
I have the following question. It's a long shot, but worth the try.
Let X be a compact connected Riemann surface of genus $g\geq 2$. Does there exist an effective divisor $D$ on $X$ of degree $g$ ...
- 9,497
2
votes
1
answer
304
views
Exact sequence for relative cohomology + normal crossing divisors
Let $X$ be smooth algebraic variety over $\mathbb C$ and $D_1, D_2$ are snc divisors such that $D_1\cup D_2$ is also snc.
Is it true that there is an exact sequence
$$H^*(X, D_1\cup D_2)\to H^*(X, D_1)...
- 219
2
votes
1
answer
487
views
A question on effective divisors
Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. Then ...
- 8,998
2
votes
1
answer
679
views
A Decomposition for Iitaka fibration
Let $\pi: X\to Y$ be an Iitaka fibration of projective varieties
$X,Y$, then is there always the following decomposition
$$K_Y+\frac{1}{m!}\pi_*\mathcal O_X(m!K_{X/Y})=P+N$$
where $P$ is ...
user21574
2
votes
1
answer
546
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 ...
- 23
2
votes
1
answer
557
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 ...
- 205
2
votes
1
answer
158
views
Reference for torsion-freeness of the group of correspondences on a smooth projective variety
In Beauville's "Variétés de Prym et jacobiennes intermédiaires", Proposition 3.5, it is claimed that $\textrm{Corr}(T)$ is torsion-free for a smooth projective variety $T$. Here $$\textrm{...
- 679
2
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1
answer
483
views
Divisors on projective bundles
Let $\pi:X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^n$ be a projective bundle, where $\mathcal{E}$ is a rank two vector bundle over $\mathbb{P}^n$.
If $n = 0$ then $X = \mathbb{P}^1$, and for $n ...
- 8,998
2
votes
1
answer
543
views
Divisors on the symmetric product of an elliptic curve
Assume that $C$ is an elliptic curve and $C_p$ is the $p$-fold symmetric product. Let $\beta:C_p\to C$ be defined by the addition on the elliptic curve. Let $u\in C$ be the zero in the additive group ...
- 439
2
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1
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118
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Effective semi-group of a singular abelian surface
Let $A$ be a singular abelian surface over $\mathbb{C}$; that is, an abelian surface of maximal Picard rank $\rho(A)=4$. By Shioda-Mitani we know $A \cong E \times E'$ where $E,E'$ are isogenous ...
- 1,701
2
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1
answer
160
views
Sections of Cartier divisors on toric varieties
Let $X_{\Sigma}$ be a projective toric variety. Consider the total coordinate ring
$$S = \mathbb{C}[x_{\rho}\: | \: \rho\in\Sigma(1)]$$
Define $\deg(x_{\rho}) = D_{\rho}$.
Now, take a divisor $D = \...
user75933
2
votes
1
answer
209
views
Curves contracted by a rational map
Let $D$ be a big but not nef divisor on a normal $\mathbb{Q}$-factorial projective variety. Assume that the section ring
$$R(D) = \bigoplus_{n\in\mathbb{N}}H^0(X,nD)$$
is finitely generated and ...
user114666
2
votes
1
answer
181
views
Anti-canonical divisorial contractions of weak Fano $3$-folds
Let $X$ be a smooth weak Fano but not Fano $3$-fold ($-K_X$ is nef and big but not ample). Then the anti-canonical morphism $\phi:X\rightarrow W$ (the morphsim induced by the linear system $|-mK_X|$ ...
user114198
2
votes
1
answer
1k
views
Negative degree line bundles over a singular projective curve have no sections?
Let $C$ be a local complete intersection projective curve in $\mathbb{P}^3$. Assume that $C$ is integral. Let $\mathcal{L}$ be a line bundle on $C$ of negative degree. We know that if $C$ is smooth ...
- 503
2
votes
1
answer
717
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Singularities of secant varieties of rational normal curves
Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper:
http://ac.els-cdn.com/...
user56312
2
votes
2
answers
394
views
Divisors with positive Iitaka dimension
Let $X$ be a non-singular projective variety, and $D$ a divisor on $X$.
Saying that $D$ has positive (meaning non-zero) Iitaka dimension is equivalent to the function $n \mapsto h^0(\cal{O}(D))$ ...
2
votes
1
answer
300
views
a question on the space of divisors on a curve
Let $X$ a complex curve and $x\in X$ a point.
We consider the space of effective divisors $D$ with fixed degree $d$, whic we know is isomorphic to $X^{d}/S_{d}$ where $S_{d}$ is the symmetric group.
...
- 3,472
2
votes
1
answer
563
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 ...
2
votes
1
answer
1k
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 ...
- 23
2
votes
1
answer
321
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 ...
- 1,213
2
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1
answer
269
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$ ...
- 1,150
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
2
votes
1
answer
243
views
Intersection of translate of divisors on abelian variety
Setup. Let $K$ be an algebraically closed field of characteristic zero, and let $A/K$ be a simple abelian variety of dimension $n$. Let $\{ x_1,x_2,\dots,x_{m^{2n}}\}$ denote the $m$-torsion points of ...
- 998
2
votes
1
answer
171
views
Restriction of small transformations
Let $\phi:X\dashrightarrow Y$ be an elementary small transformation (isomorphism in codimension $1$) between normal and $\mathbb{Q}$-factorial projective varieties.
Then there are small contractions $...
user114666
2
votes
1
answer
465
views
Is the positive part of the Zariski decomposition of a big $\mathbb{R}$-divisor big?
I can't understand why the positive part of the Zariski decomposition of a big class is itself big. More concretely:
let $X$ be a smooth projective surface over $\mathbb{C}$. Let $N^1_{\mathbb{R}}(X)$...
- 115
2
votes
1
answer
230
views
Linear systems and 2-torsion shifts on hyperelliptic curves
Let $C$ be a hyperelliptic curve of genus $g$ and let $D$ be a divisor on $C$ of degree $g+1$. Assume that the linear system $|D|$ is base-point-free. Now add a $2$-torsion point $[E]$ to $D$. I would ...
- 174
2
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0
answers
149
views
Non-proper intersection between divisors on $\mathbb{P}^1$-bundle of Hirzebruch surfaces
We are working on algebraic closed field $k$. Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$, $C_0$ and $C_{\infty}$ are its zero and infinity sections ...
- 41
2
votes
0
answers
85
views
Branched covers of real algebraic varieties
Let $X$ be a smooth complex algebraic variety and $L$ be an $n$-torsion line bundle on $X$, i.e., a line bundle $L$ such that $L^n=\mathcal{O}_X(B)$, where $B$ is a divisor $B$ on $X$. Such a bundle ...
- 85
2
votes
0
answers
232
views
Chern classes and rational equivalence
Let $X$ be a complex variety and let $l_1$ and $l_2$ be line bundles on $X$. Let $f_1$ and $f_2$ be sections of $l_1$ and $l_2$ respectively, and let $Z_1$ and $Z_2$ be their zero-sets.
I would like ...
- 89
2
votes
0
answers
244
views
On the definition of the relative canonical divisor
Fix a field $k$. Let $X,Y$ be normal integral $k$-schemes of finite type and $h: Y \to X$ a proper birational $k$-morphism. Moreover, assume that $X$ is Gorenstein, i.e. it admits a canonical divisor $...
- 293
2
votes
0
answers
242
views
Semi-continuity of the Picard number
Let $f:X\rightarrow S$ be a family of smooth projective varieties. For $s\in S$ set $X_s := f^{-1}(s)$, and let $\rho(X_{s})$ be the Picard number of the fiber over $s\in S$. Fix a point $s_0\in S$.
...
- 8,998
2
votes
0
answers
67
views
Irreducible components over a singular divisor
Setup. Let $K$ be an algebraically closed field of characteistic zero, let $X/K$ be a smooth projective surface and let $Z \subset X$ be an integral curve which is nonsingular except for a finite set ...
- 998
2
votes
0
answers
92
views
Linear system of a relative effective divisor on an arithmetic surface contains vertical divisors
I am puzzled by the behavior of some divisors in my attempt to understand the relative Picard functor $\mathrm{Pic}_{X/S}$ of an arithmetic surface $\pi:X\to S$. This is defined by relative divisors $...
- 1,142
2
votes
0
answers
155
views
subspace of the global sections of $\mathcal O$$(D)$
Let $X$ be a smooth projective surface and $D$ an effective divisor whose complete linear system $|D$ is base point free and $D^2=1$. Suppose the dimension of $|D|$ is greater than or equal to 3. Is ...
user169803