Questions tagged [divisors]
For questions related to divisors in the sense of algebraic geometry (Cartier divisors, Weil divisors and so on). For question on divisors in the number theoretic sense please use the tag divisors-multiples.
361 questions
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Congruence restrictions on prime divisors of polynomial values
Consider the polynomial $f(x)= x^2+1$. Can you prove that there are infinitely many integers $x$ such that $f(x)$ has no prime divisor congruent to $1 \bmod 3$? Obviously the prime divisors are ...
2
votes
1
answer
343
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Is it possible to define $d(n)$ function which defines the number of divisors for non-natural numbers?
I was exploring recently Lambert series $\Theta(x)=\sum_{n=1}^{\infty}\frac{x^n}{1-x^n}$ and its analytic continuation. Alternatively $\Theta(x)=\sum_{n=1}^{\infty}x^n d(n)$, where $d(n)$ is the ...
- 189
2
votes
1
answer
178
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Effective Weil divisors: existence of minimal models & projection formulae
Background: Let $D$ be a Weil divisor on a nice variety $X$ (normal, $\mathbb{Q}$-factorial, etc.). If one can run Mori's program on $D$ — by which I mean identify a birational variety $X_D$ (which ...
- 23
0
votes
0
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105
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Are there a finite numbers of zeros in this integer sequence?
Consider the triangular array $T(n,k)_{1 \le k \le n}$ defined by the recurrence \begin{align*} T(n,1) &= 1, \\ T(n,k) &= 1+\sum_{i=1}^{k-1} T(n - i, k - 1) -\sum_{i=1}^{n-1} T(n - i, k). \end{...
- 1,203
1
vote
1
answer
184
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Normality of the canonical dual of a very small line bundle
Let $C$ be a generic curve of genus $8$ and $D$ a $g^4_{11}$. Is $D$ $2$-normal? In other words, is the natural multiplication
$$H^0(D)\otimes H^0(D)\longrightarrow H^0(2D)$$
surjective?
More ...
- 443
1
vote
0
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148
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The existence of special line bundles on algebraic curves
This is the problem that I encounter with when reading the proof of Theorem 17 of this paper.
Let $C$ be an algebraic curve of genus $g\geq3$. Assume that $L$ is a line bundle with $h^0(C,L)=3$. There ...
- 443
1
vote
1
answer
198
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Curves contracted by a morphism are numerically proportional
I am trying to prove that
If $f:X\to Y$ is a birational morphism between smooth projective varieties with exceptional locus a prime divisor $E\subset X$, then $E\cdot C<0$ for any curve $C\subset ...
- 121
10
votes
2
answers
569
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When do horizontal integrals (of $\zeta(s)$ and the like) matter?
I was just browsing through Chapter XII (Divisor problems) of Titchmarsh's Theory of the Riemann Zeta function. In there, to estimate sums of the divisor function $d(n)$ and the $k$-divisor function $...
- 22k
7
votes
1
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421
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Can exceptional components be numerically equivalent?
Assume we have a birational morphism $\pi:X\to Y$ between smooth projective varieties with non-empty exceptional locus $E$ and irreducible components $E_1,\dots,E_r$.
My question:
Is it possible that ...
- 73
4
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0
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127
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Is there a fast way to find median divisors of a number?
Example: 72 has the following divisors:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
The median (middle) divisors are 8 and 9.
Provided we already have the prime factors of a number x, what would be an ...
- 41
0
votes
0
answers
157
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Excision exact sequence for divisor class group on locally Noetherian scheme with no extra assumptions
Let $X$ be an arbitrary locally Noetherian scheme, and let $Z \subset X$ be an arbitrary closed subset. Let $U = X - Z$. Denote by $i: Z \to X$ and $j: U \to X$ the inclusions. Since $X$ is locally ...
- 321
3
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0
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166
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Pushforward of Line Bundles under Birational Morphisms
I'm considering a birational morphism $f:X \to Y$ between smooth projective varieties, and a divisor $D$ on $X$.
I'm trying to study $f_*(\mathcal O_X(D))$. Its structure depends on $D$ and on how $f$ ...
- 31
1
vote
0
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102
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Relationship between the complete linear system of a line bundle and the Proj of its section ring
Let $X$ be a projective variety over $\mathbf C$ and assume it has all the good hypotheses one can wish for, and let $ \mathcal{L} $ be a line bundle on $ X $. One can consider the complete linear ...
- 121
7
votes
1
answer
373
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Locally principal Weil divisor that is not associated to a Cartier divisor?
If $X$ is an integral separated Noetherian scheme that is regular in codimension 1, then there is a natural map $\text{Cart}(X)\to \text{Weil}(X)$ that sends a Cartier divisor to its divisor of zeros ...
- 173
3
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1
answer
275
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Surjectivity on cohomology of normal crossings divisor
Let $X$ be a smooth, projective variety and $D \subset X$ be a reduced simple normal crossings divisor. In an article of Steenbrink, he says that the natural morphism from $H^i(D,\mathbb{C})$ to $H^i(\...
- 1,122
1
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0
answers
114
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It seems natural to try to construct a non-degenerate pairing generalizing the Weil pairing
It is well known that a principally-polarized abelian variety $A$ has a non-degenerate (e.g., Weil) pairing of the form $A[n]\times A[n] \to \mu_n$, where $n \in \mathbb{N}$. However, I have never ...
- 1,873
1
vote
0
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68
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Restriction of a torus invariant divisor to an affine toric subvariety
Using the notation from Cox, let $\Sigma$ be a fan and $X_\Sigma$ the correcponding toric variety. Let $D=\displaystyle\sum_{\rho\in\Sigma(1)}a_\rho D_\rho$ be a Weil divisor. For every open subset $U\...
- 465
2
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1
answer
407
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Divisor on compact Riemann surface
For a compact Riemann surface $\Sigma$, we denote $\text{Jac}(\Sigma)$ be its Jacobian, i.e.
$$
\text{Jac}(\Sigma) := H^{1,0}(\Sigma)^{*}/H_{1}(\Sigma,\mathbb{Z}).
$$
Denote $\text{Pic}_d(\Sigma)$ be ...
- 635
1
vote
0
answers
201
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Pushing/pulling divisors along small birational contractions
Let $f: X \to Y$ be a morphism of proper normal varieties (i.e. reduced and irreducible over algebraically closed $k$) which is a small birational contraction, meaning that $f_* \mathcal{O}_X = \...
- 4,242
2
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0
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336
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Intersection signature on divisors of Calabi-Yau threefolds
After over one week and quite a lot of views on this question, I would like to ask a refined version here.
Let X be a minimal Calabi-Yau threefold in the sense of [1] and let $D$ be a Weil divisor on $...
- 21
0
votes
1
answer
164
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on distribution of prime divisors of random integer
To slightly strengthen a result, we use the following lemma.
Lemma
For a fixed large $C>0,$ the density of (positive) integers $n$, for which its ordered prime factorization $p_1p_2 \ldots p_r$ ...
- 59
0
votes
0
answers
139
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 ...
- 229
1
vote
0
answers
166
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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
105
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The space of virtual Cartier divisors on a classical scheme over a closed immersion is discrete
I am currently reading the paper Virtual Cartier divisors and blow-ups where the virtual Cartier divisor on an $X$ scheme $S$ over a quasi-smooth closed immersion $Z\rightarrow X$ is defined to be the ...
- 982
2
votes
0
answers
178
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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
5
votes
1
answer
342
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Computing the divisor class group of toric varieties over an arbitrary field
Let $k$ be an arbitrary field and let $X$ be a toric variety over $k$, coming from a fan $\Sigma$. If $k$ is algebraically closed, then theorem 4.1.3 of Cox ,Little and Schenck’s Toric Varieties book ...
- 721
0
votes
1
answer
457
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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
2
votes
0
answers
118
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
1
vote
1
answer
223
views
Divisors on product abelian fourfolds
Given a principally polarized abelian surface $A$ with CM of signature $(1,1)$ by an imaginary quadratic number field $K$, I am interested in studying the Néron-Severi group $\text{NS}(A\times A)$. ...
- 91
1
vote
0
answers
66
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Positivity of self-intersection of dicisor associated to meromorphic function
In the book "Holomorphic Vector Bundles over Compact Complex Surfaces" by Vasile Brînzănescu, in the proof of theorem 2.13 there is the following claim
Let $X$ be a compact non-algebraic ...
- 111
0
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2
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558
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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
2
votes
0
answers
373
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
4
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0
answers
602
views
Is it always true that the complement of an ample divisor is affine?
Consider a proper and integral scheme $X\rightarrow\operatorname{Spec}(A)$ over a Noetherian ring $A$ and $D\in\operatorname{Div}(X)$ an effective ample Cartier divisor on $X$. Is it true that its ...
- 41
3
votes
0
answers
149
views
Error function of the second moment of the divisor function
It is easy to show that the second moment of the divisor function has asymptotics:
$$\sum_{n\leq x} d_0(n)^2 = xP(\log(x))+E_2(x)$$
Where $P$ is some polynomial and that:
$$E_2 = o(x)$$
Previously, ...
- 531
1
vote
1
answer
177
views
A criterion for divisors of degree $n$ on the projective line to belong to a linear system of codimension 1
My question is essentially about linear dependence/independence of polynomials, but I will formulate it in the language of algebraic geometry, hoping someone may suggest a result in algebraic geometry ...
- 5,377
5
votes
2
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311
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Characterize the space of all ramification divisors of degree $d$
Let $X$ be a compact Riemann surface of genus $g>0$, and let $f\colon X \to \mathbb{P}^1$ be a branched covering of degree $d$. Define the ramification divisor $R_f$ on $X$ by $f$, where $\deg R_f =...
- 713
6
votes
2
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503
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Toric varieties as hypersurfaces of degree (1, ..., 1) in a product of projective spaces
I wonder if some hypersurfaces of multi-degree $(1, ..., 1)$ in a product of projective spaces are toric, with polytope a union of polytopes of toric divisors of the ambient space. This question is ...
- 428
2
votes
1
answer
374
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Exact sequence for relative cohomology + normal crossing divisors
Let $X$ be smooth algebraic variety over $\mathbb C$ and $D_1, D_2$ are snc divisors such that $D_1\cup D_2$ is also snc.
Is it true that there is an exact sequence
$$H^*(X, D_1\cup D_2)\to H^*(X, D_1)...
- 229
1
vote
1
answer
376
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A short exact sequence regarding Kähler differentials and an invertible ideal on an algebraic curve
$\def\sO{\mathcal{O}}
\def\sK{\mathcal{K}}
\def\sC{\mathscr{C}}$I am trying to understand what the maps are on a certain s.e.s. of sheaves of modules on an algebraic curve. It is \eqref{ses} on Conrad'...
3
votes
1
answer
319
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A question on "Ample subvarieties of algebraic varieties"
Corollary 3.3 in Chapter IV of "Ample subvarieties of algebraic varieties" by R. Hartshorne asserts the following:
Let $X$ be a smooth projective variety and $Y\subset X$ a smooth subvariety ...
- 9,110
2
votes
1
answer
323
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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
1
vote
0
answers
417
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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 ...
- 543
3
votes
1
answer
320
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How to determine the type of a divisor on a product of elliptic curves?
I already asked this on Math.SE, but didn't receive an answer yet.
Say $E_1, \dotsc, E_n$ are elliptic curves (everything over $\mathbb C$), and $D \subset E_1 \times \dotsc \times E_n$ is an ...
- 1,356
3
votes
0
answers
167
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Inverse image Weil divisor on a toric variety as a Cartier divisor
Let $X$ be a normal toric variety over an algebraically closed field and let $D$ be a torus invariant (prime) divisor. Assume $\pi\colon \tilde{X}\rightarrow X$ is a toric resolution of singularities ...
- 339
0
votes
1
answer
207
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 ...
- 526
2
votes
2
answers
275
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 ...
- 9,110
3
votes
1
answer
343
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Differential of a specific morphism to a Grassmannian
This is a problem that's been bugging me for some time, and therefore I've decided to ask it here. Let $X$ be a smooth projective (irreducible) variety over an algebraically closed field of ...
- 673
0
votes
1
answer
385
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 ...
- 9,110
2
votes
1
answer
193
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{...
- 719
1
vote
0
answers
218
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