All Questions
Tagged with divisors ag.algebraic-geometry
313 questions
1
vote
0
answers
144
views
Existence and injectivity of a map from $\mathrm{Num}(X) \otimes_\mathbb{Z} \mathbb{C}$ to $H^1(X,\Omega^1_X)$
I am currently reading a paper by Mustata and Popa on the Van de Ven theorem, you can read the paper here.
On page 51 there is the following map
$$\alpha_\mathbb{C} : \mathrm{Num}(X) \otimes_\mathbb{Z}...
- 201
1
vote
0
answers
74
views
A criterion for divisors of degree $n$ on the projective line to belong to a linear system of codimension 1
My question is essentially about linear dependence/independence of polynomials, but I will formulate it in the language of algebraic geometry, hoping someone may suggest a result in algebraic geometry ...
- 5,215
1
vote
0
answers
277
views
How to define Cartier divisor and Weil divisor on algebraic stack?
How to define Cartier divisor and Weil divisor on algebraic stack? Do they correspond to line bundles on stack like the case of schemes? In case of a Deligne-Mumford stack, can we have a simpler ...
- 523
1
vote
0
answers
184
views
Divisor cohomology through spectral sequences
I don't know if it belongs here but anyway, I need to compute arithmetic genus of divisors pulled back from a Fano base space to a bundle (which may or mayn't be trivial) defined through the ...
- 11
1
vote
0
answers
81
views
How to calculate the divisor given by closure of subscheme
Let $X \subset \mathbb{P}^N$ be a nonsingular projective variety over algebraically closed field which is embedded by very ample line bundle $\mathcal{L}$. Let $Y = \mathbb{P}(\mathcal{L}^{\oplus 3})$...
- 121
1
vote
0
answers
175
views
Derivation for genus-degree formula from algebraic functions field theory
This is a copy of my question from math.stackexchange: https://math.stackexchange.com/questions/4517289/derivation-for-genus-degree-formula-from-algebraic-functions-field-theory. I didn't get any ...
- 81
1
vote
0
answers
190
views
How to define a principal divisor on general complex spaces?
[I am not a native English speaker, so my sentences may sound strange. ]
I'm studying about complex analytic spaces. For meromorphic functions, I don't know how to define their principal divisors ...
- 11
1
vote
0
answers
202
views
Mori cone of Picard rank two varieties
Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that
$$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$
is an isomorphism, where $i:S\...
- 8,998
1
vote
0
answers
168
views
Rational classes of $(-2)$-curves in a minimal surface of general type
Let $X$ be a minimal surface of general type over $\mathbb{C}$. One can show that if for any set of $(-2)$-curves $C_1,\cdots,C_l$ on $X$, there exists $k$, $1\le k\le l$ such that $$\sum_{i=1}^k\...
- 41
1
vote
0
answers
183
views
Problem regarding existence of a divisor representing line bundle
We consider a normal irreducible variety $X$ and a line bundle $L$. The question is when $L$ is induced by a Cartier divisor $D$. We know that if $s$ is a rational section of $O_X(D)$, where $D$ is a ...
1
vote
0
answers
157
views
The morphisms induced by two Cartier divisors
Let X be a projective variety. We consider two Cartier divisors $D,E$ such that $E\geq D$ and the relative morphisms
$\phi_D: X - - -> \mathbb{P}(H^0(X, O_X(D))^*)$ and $\phi_E: X- - -> \mathbb{...
1
vote
0
answers
212
views
Dimension of a linear system of divisors on singular curve
Consider an singular irreducible plane curve $C \subset \mathbb{P}^2_k$ of degree $d>1$ over algebraically closed field $k$ which is given as vanishing locus $C=V(f(x,y,z))$ of a $f \in k[x,y,z]$ ...
- 6,016
1
vote
0
answers
91
views
Picard numbers of isogenous K3 surfaces over a non-closed field
Let $S_1, S_2$ be K3 surfaces defined over a field $k$ and $\phi\!: S_1 \dashrightarrow S_2$ a dominant rational $k$-map (so-called isogeny). It is known that $\rho(S_1) = \rho(S_2)$ for the complex ...
- 1,833
1
vote
0
answers
194
views
Nef divisors on abelian varieties are pullbacks of ample ones
It is well known that for any polarization ( that is, ample line bundle) $L$ on an abelian variety $A$, there is an isogeny $\phi\colon A \to B$ to another abelian variety with a principal ...
1
vote
0
answers
116
views
On the fixed and negative part of a linear system
Let $X$ and $Z$ be smooth complex projective varieties and let $f:X\rightarrow Z$ be a contraction (i.e. $f_\ast\mathcal{O}_X=\mathcal{O}_Z$). Let $F$ be an effective $\mathbb{R}$-divisor on $X$ such ...
- 599
1
vote
0
answers
120
views
Question about Local Henselian Rings
I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces":
Here the relevant excerpt:
Remark: ...
- 6,016
1
vote
0
answers
114
views
Iitaka dimension of a $\mathbb{Q}$-Cartier Prime divisor
Let $X$ be a normal projective variety and $D$ a prime divisor such that $mD$ is Cartier for some integer $m>0$.
Suppose $H^1(X,\mathcal{O}_X)=0$ and $mD|_D\sim 0$.
My questions are the following:
...
- 85
1
vote
0
answers
106
views
singular $m$-canonical divisors
[remark for v2] I began by considering curves in v1. I am convinced that the answer is positive. Thanks to Jason Starr and abx.
Let $X$ be a complex projective variety.
Let $K_X$ be its canonical ...
- 101
1
vote
0
answers
214
views
Twisting a line bundle with the zero section
Let $X$ be a smooth projective curve and $L$ be an invertible sheaf on $X$. Denote by $\mathbb{L}$ the line bundle associated to $L$, $\pi:\mathbb{L} \to X$ the natural morphism and $0_\pi$ the zero ...
- 1,981
1
vote
0
answers
290
views
Intersection with very ample divisor and linear equivalence
Let $X$ be a smooth, projective variety and $D, E$ two effective divisors of $X$ which correspond to distinct elements on the cohomology group $H^2(X,\mathbb{Q})$. Denote by $H$ a very ample divisor ...
- 2,126
1
vote
0
answers
70
views
Moving curves and small transformations
Let $f:X\dashrightarrow Y$ be an isomorphism in codimension one between smooth projective varieties. Let $C\subset X$ a curve generating an extremal ray of the cone of moving curves $Mov_1(X)$, and ...
- 141
1
vote
0
answers
312
views
Cone of moving curves
Let $X$ be a projective variety and $C\subset X$ be a moving curve, that is the curves numerically equivalent to $C$ cover a dense open subset of $X$.
How can we detect when $C$ is an extremal ray ...
- 141
1
vote
1
answer
242
views
Number of conditions imposed by fat points to a linear system
Let $|D|$ be the linear system of degree $d$ hypersurfaces in $\mathbb{P}^n$ having multiplicity at least $m$ at $s$ general points.
Then $|kD|$ is the linear system of degree $kd$ hypersurfaces in $...
user97096
1
vote
0
answers
217
views
Family $(X_y,D_y)$ with trivial canonical bundles
Let $i:D\hookrightarrow X$ and $f : X \to Y$ be holomorphic mappings of complex manifolds
such that $i$ is a closed embedding and $f$ as well as$ f \circ i$ are proper and smooth and $D$ is a divisor. ...
user21574
1
vote
0
answers
210
views
On triviality and numerical triviality of (classes of) divisors
Let $X$ be a smooth irreducible threefold, and let $H$ be an ample divisor on $X$.
Assume that $D$ is a divisor on $X$ such that $D\cdot H^2=D^2\cdot H=D^3=0$.
Question 1: Is $D$ numerically trivial?...
- 1,159
1
vote
0
answers
351
views
A question on the secondary fan
I am studying the secondary fan decomposition of the effective cone of a projective variety $X$. Let as assume that $X$ is a Mori Dream Space. As far as I understand passing from a cone of maximal ...
user61586
1
vote
1
answer
687
views
A question on klt pairs
Let $D$ be a $\mathbb{Q}$-divisor in a smooth variety $X$. In Lazarsfeld book "Positivity in Algebraic Geometry 2" I found Proposition 9.5.13 saying that if for any $x\in D$ we have $mult_xD < 1$ ...
user58018
1
vote
0
answers
445
views
Pull-back of globally generated sheaves
Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...
- 1,981
1
vote
0
answers
83
views
lift sections on a thickened curve
Let $X$ a curve over an algebraically closed field $k$ and $D$ a divisor on X.
Fix an integer $N$ and a closed point $x$ on $X$, we assume that $\deg(D)$ is big enough such that we have a surjective ...
- 3,472
1
vote
0
answers
355
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,
$D_J=\...
- 11
1
vote
0
answers
1k
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 ...
- 11
0
votes
1
answer
384
views
Relation between canonical bundles under étale maps
Let $X$ and $Y$ be two integral separated Noetherian Gorenstein schemes over a base field $k$ of arbitrary characteristic whose local rings are unique factorization domains and $f: X\to Y$ an étale ...
- 6,016
0
votes
1
answer
249
views
Behavior of divisors under push forward and pull back
Consider a birational morphism between smooth projective varieties $f:X\to Y$. I would like to understand the behavior of push-pull/pull-push of effective divisors under $f$. I know that if $D$ is an ...
- 109
0
votes
1
answer
184
views
Curves in conic bundles
Consider a smooth minimal $3$-fold conic bundle $f:X\rightarrow\mathbb{P}^2$. Then $X$ has Picard rank two and consequently also the vector space of $1$-cycles is $2$-dimensional. Then the cone of ...
- 9
0
votes
1
answer
147
views
Are maps into a smooth curve equivalent to relative effective Cartier divisors?
Let $X$ be a smooth curve and $S$ an affine scheme, both over an algebraically closed field $k$.
Given a morphism $f : S \to X$, is it true that the graph morphism $\Gamma_f : S \to X \times S$ is a ...
- 426
0
votes
1
answer
257
views
Definition of canonical pair
Let $(X,D)$ be a pair and $f:Y\rightarrow X$ a log resolution. Write
$$
K_Y + \widetilde{D} = f^{*}(K_X) + \sum_{i}a_iE_i
$$
where $\widetilde{D}$ is the strict transform of $D$. I found the following ...
- 8,998
0
votes
2
answers
490
views
Small birational maps and singularities of the pair
Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair $(X,\...
- 8,998
0
votes
1
answer
259
views
Dimension of image of a hyperplane section
If we have a surjective morphism $f：X\to Y$, where $X$ is $n$ dimensional projective variety and $Y$ is $m$ dimensional projective variety.
If $m<n$, Can we choose a general hyperplane section $H$ ...
user39380
0
votes
2
answers
400
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 $k(x)=k$.
...
0
votes
1
answer
152
views
What is the cone of $\mathcal{F}\to i_*i^*\mathcal{F}$ for a divisor $i: D\hookrightarrow X$?
Let $X$ be a smooth projective variety of dimension $n$ and $i: D\hookrightarrow X$ be a smooth, anti-canonical divisor in $X$. In other words,$[D]+[\omega_X]=[0]$ where $\omega_X$ is the canonical ...
- 9,019
0
votes
1
answer
271
views
Pseudoeffective divisors on surfaces
Consider a minimal smooth conic bundle $S$ of dimension two. Assume that there are two curves $C,F$ on $S$ such that $C^2 < 0$ and $F^2 = 0$. Let $D$ be a pseudoeffective divisor on $S$ such that $...
- 8,998
0
votes
1
answer
131
views
On relating $l(A), l(B)$ and $l(A+B)$ for Weil divisors on a smooth projective curve where one of the divisors is effective
Let $X$ be a smooth projective curve over an Algebraically closed field $k$. Let $k(X)$ denote its function field.
If $A, B$ are Weil divisors on $X$ such that $A$ is effective (i.e. $A\ge 0$) , then ...
- 279
0
votes
2
answers
332
views
Vakil exercise on sheaf associated to the divisor of rational section
This is exercise 15.4.G. of Vakil's notes.
Let $\mathscr{L}$ be an invertible sheaf on an irreducible normal scheme $X$ with $s$ a rational section of $\mathscr{L}$. We want that $\mathscr{O}_X(\text{...
- 29
0
votes
1
answer
281
views
Question about Correspondences from Mumford’s Complex Projective Varieties
I study David Mumford's Algebraic Geometry I - Complex Projective Varieties
and have some problems to understand a step in the proof of Lemma 6.7 (b).
Firstly, the general setting & preparations ...
- 6,016
0
votes
1
answer
342
views
Intersections of divisors in blow-ups of $\mathbb{P}^n$
Let $p_1,p_2,p_3\in\mathbb{P}^n$ be three general points, $X$ the blow-up of $\mathbb{P}^n$ at $p_1,p_2,p_3$, then along the lines $\left\langle p_i,p_j\right\rangle$, and finally along the plane $\...
user91659
0
votes
1
answer
411
views
Intersection Matrix of a resolution
Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that
$$K_X = f^{*}K_S+\sum_ia_iE_i$$
with $a_i>0$. By Grauert-Mumford theorem the ...
user58018
0
votes
0
answers
80
views
Chow moving lemma with additional property
All varieties are over algebraically closed field of characteristic zero. Let $S$ be a smooth projective surface. Let $D$ be an irreducible divisor on $S$, $H$ be another divisor and $Z\subset S$ be a ...
- 219
0
votes
0
answers
137
views
Intersection product of $\mathbb{Q}$-Cartier divisors with irreducible complete curves is well-defined
I am learning the notion of intersection product of a $\mathbb{Q}$-Cartier divisor with an irreducible complete curve on a normal variety. The definition I learned is that if $D$ is a $\mathbb{Q}$-...
- 639
0
votes
0
answers
130
views
Is a closed subsecheme contained in a Cartier divisor?
Let $X$ be a variety over a field $k$. For a closed subscheme $Z\hookrightarrow X$ and a closed point $x\in X$ such that $\text{codim}_XZ \geq 1$ and $x\notin |Z|$, is there an effective Cartier ...
- 349
0
votes
0
answers
201
views
Intersection product when one factor is contained in the exceptional divisor
I am trying to calculate some intersection numbers and would appreciate help on the following problem:
Consider two divisors $D_1$ and $D_2$. Blowing up their intersection yields $\varphi^{*}(D_i) = \...
- 1