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.
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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 $...
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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 ...
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2
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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 ...
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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 ...
- 11.9k
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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, ...
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answer
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Power of primes
$n$ is a natural number $>1$, $\varphi(n)$ denotes the Euler's totient function, $P_n$ is the $n^\text{th}$ prime number and $\sigma(n)$ is the sum of the divisors of $n$. Consider the expression:
$...
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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 ...
- 5,011
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2
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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
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1
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Strict transform under resolution of singularity along a singular $\mathbb{Q}$-Cartier divisor
$\DeclareMathOperator\Bl{Bl}$Let $f: Y=\Bl_0^\omega(\mathbb{C}^3)\to \mathbb{C}^3$ be a weighted blow up of $\mathbb{C}^3$ with weights $w(x,y,z)=(1,1,2)$. Then $Y$ and the exceptional divisor $E\cong ...
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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 ...
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2
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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?
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1
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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)...
- 137k
1
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1
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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'...
- 11.4k
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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 ...
- 1,266
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1
answer
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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 ...
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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 ...
- 5,632
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1
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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 ...
- 1,071
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0
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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 ...
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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 ...
- 337
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1
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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 ...
- 37.2k
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2
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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 ...
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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$ ...
- 27.3k
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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 ...
- 1,446
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1
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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 ...
- 5,217
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1
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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$.
- 1,446
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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
$$
where $\widetilde{D}$ is the strict transform of $D$. I found the following ...
- 1,266
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Reference for torsion-freeness of the group of correspondences on a smooth projective variety
In Beauville's "Variétés de Prym et jacobiennes intermédiaires", Proposition 3.5, it is claimed that $\textrm{Corr}(T)$ is torsion-free for a smooth projective variety $T$. Here $$\textrm{...
- 27.3k
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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 ...
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1
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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 ...
- 27.3k
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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 $\mathbb{Q}$-...
- 531
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1
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Explicit description of the space of global sections of a torus invariant Weil divisor over a real toric variety
Let $D$ be a Weil divisor on a normal toric variety $X$ with fan $\Sigma$ that is invariant under the action by the torus $T$. Then Proposition 4.3.2 of the textbook Toric Varieties by Cox, Little and ...
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1
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Can $P(z)$ have a divisor in a given congruence class?
In the answer to this previous question , Noam D. Elkies proved that for any integer $x$, $x^3-x^2-2x+1$ can only have a divisors equal to $-1$, $0$, or $1$ modulo $7$. I would like to know what is ...
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Homogeneous components of Cox RIngs
Let $X$ be an irreducible smooth projective variety over a field $k$ (algebraically closed and of characteristic zero if needed). Let $U \subseteq X$ an affine open such that $O_X(U)$ is factorial and ...
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0
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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 $s_0\in S$.
...
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1
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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 ...
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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 ...
- 357
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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 ...
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Cartier divisor that is not a difference of two effective Cartier divisors
Note: There are already several related questions, without any definite answer.
I want to find an example of a Noetherian integral scheme $X$ which contains a Cartier divisor that is not linearly ...
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0
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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 $Y = \mathbb{P}(\mathcal{L}^{\oplus 3})$...
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0
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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 ...
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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 ...
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1
answer
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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-...
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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 ...
- 37.2k
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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 $...
- 1,102
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Tri-homogenous polynomials of tridegree $(3,3,3)$ to add three points on an elliptic curve
Consider an elliptic curve $E \subset \mathbb{P}^2$ with the zero point $\mathcal{O}$. There are classical articles about complete systems of addition laws on $E$ (see
Lange and Ruppert - Complete ...
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1
answer
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Is the reduced scheme associated to a Cartier divisor always Cartier?
Let $X$ be a normal integral variety over $\mathbb{C}$ and $D \subset X$ be a Cartier divisor in $X$. Is the associated reduced scheme $D_{\mathrm{red}}$ also necessarily a Cartier divisor in $X$?
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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 ...
- 37.3k
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1
answer
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Extending effective Cartier divisors
Let $X$ be a non-singular, quasi-projective variety (over $\mathbb{C}$) of dimension at least $3$, $D_1, D_2$ are integral effective divisors in $X$ with $D_1 \cap D_2$ of codimension $2$ in $X$. Let $...
- 1,266
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1
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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...\...
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answer
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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 $...
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