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.

Filter by
Sorted by
Tagged with
3 votes
0 answers
174 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 ...
Kheled-zâram's user avatar
3 votes
0 answers
92 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, ...
psubodiosa's user avatar
1 vote
0 answers
58 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 ...
Malkoun's user avatar
  • 4,981
12 votes
1 answer
1k views

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: $...
Rédoane Daoudi's user avatar
5 votes
2 answers
226 views

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 =...
swalker's user avatar
  • 713
3 votes
1 answer
201 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 ...
Yromed's user avatar
  • 63
2 votes
1 answer
239 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)...
Galois group's user avatar
1 vote
1 answer
208 views

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'...
Elías Guisado Villalgordo's user avatar
3 votes
1 answer
219 views

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 ...
Puzzled's user avatar
  • 8,832
2 votes
1 answer
254 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 ...
locallito's user avatar
1 vote
0 answers
183 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 ...
user124771's user avatar
3 votes
1 answer
177 views

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 ...
red_trumpet's user avatar
  • 1,061
3 votes
0 answers
127 views

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 ...
Boaz Moerman's user avatar
0 votes
1 answer
131 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 ...
user577413's user avatar
2 votes
2 answers
187 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 ...
Puzzled's user avatar
  • 8,832
3 votes
1 answer
273 views

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 ...
rfauffar's user avatar
  • 643
-1 votes
1 answer
155 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 ...
Puzzled's user avatar
  • 8,832
2 votes
1 answer
150 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{...
TCiur's user avatar
  • 469
1 vote
0 answers
174 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 ...
user333644's user avatar
0 votes
1 answer
287 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 ...
user267839's user avatar
  • 5,938
0 votes
0 answers
107 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 $\mathbb{Q}$-...
Boris's user avatar
  • 501
1 vote
1 answer
201 views

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 ...
Colin Tan's user avatar
  • 251
1 vote
1 answer
109 views

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 ...
Luca Francone's user avatar
1 vote
0 answers
198 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 $s_0\in S$. ...
Puzzled's user avatar
  • 8,832
1 vote
1 answer
151 views

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 ...
Bogdan Grechuk's user avatar
0 votes
0 answers
125 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 ...
OOOOOO's user avatar
  • 357
2 votes
1 answer
217 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 ...
Jackson Morrow's user avatar
2 votes
0 answers
65 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 ...
Jackson Morrow's user avatar
3 votes
0 answers
334 views

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 ...
Daebeom Choi's user avatar
1 vote
0 answers
76 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 $Y = \mathbb{P}(\mathcal{L}^{\oplus 3})$...
Doyoung Choi's user avatar
1 vote
0 answers
146 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 ...
Konstantce's user avatar
1 vote
0 answers
164 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 ...
map's user avatar
  • 11
0 votes
1 answer
146 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 ...
Zhaoting Wei's user avatar
  • 8,657
2 votes
0 answers
84 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 $...
Somatic Custard's user avatar
2 votes
1 answer
357 views

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 $X$?
user45397's user avatar
  • 2,195
7 votes
1 answer
524 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 $x$, $y$, $z$ ...
Dimitri Koshelev's user avatar
2 votes
1 answer
237 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 ...
HARRY's user avatar
  • 267
4 votes
1 answer
164 views

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 ...
Dimitri Koshelev's user avatar
3 votes
1 answer
190 views

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 $...
user45397's user avatar
  • 2,195
4 votes
0 answers
775 views

Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?

Given a morphism between normal varieties $f: X \to Y$, we can push forward a Cartier divisor $D$ to get a cycle $f_* D$. On the other hand, we can form the line bundle $\mathscr{L}(D)$ and push that ...
Kim's user avatar
  • 4,034
2 votes
0 answers
52 views

Conditions for long exact sequence for line bundles on curve to degenerate?

Let $\varphi:X\to Y$ be a morphism of schemes of relative dimension 1, and $\mathcal{L}' \xrightarrow{g} \mathcal{L}$ an injection of line bundles on $X$. The sequence $$0\to \mathcal{L}' \xrightarrow{...
PrimeRibeyeDeal's user avatar
4 votes
1 answer
217 views

Volume of conic bundles

Consider a smooth conic bundle $X\rightarrow \mathbb{P}^1$ with discriminant of degree $d$ (the locus of $\mathbb{P}^1$ over which the fibers are reducible conics). There is a formula for $(-K_X)^2$ ...
Puzzled's user avatar
  • 8,832
4 votes
1 answer
336 views

Zsigmondy's Theorem Generalization

Zsigmondy's Theorem states that if $a>b>0$ are coprime integers then for any integer $n\geq 1$ there is a prime $p$ that divides $a^n-b^n$ and does not divide $a^k-b^k$ for any positive integer $...
thebogatron's user avatar
2 votes
0 answers
119 views

Average length of consecutive integers which have an increasing number of divisors

Consider the nine consecutive natural numbers starting from $1584614377$. ...
Nilotpal Kanti Sinha's user avatar
3 votes
0 answers
101 views

Detecting non-principal Weil divisors on normal varieties using curves

Let $X$ be a normal projective variety over an algebraically closed field $k$. Given any morphism $f:Y\to X$, there is a pullback homomorphism $f^*:\text{Cl}(X)\to\text{Cl}(Y)$, where $\text{Cl}(X)$ ...
Jonathan Love's user avatar
6 votes
1 answer
576 views

Hartshorne's proof of Halphen's theorem

Apologies if this is not quite at the level of MathOverflow, but it has already been asked at MSE and gone unresolved for several years despite a bounty. Hartshorne states the theorem as follows: ...
Hank Scorpio's user avatar
1 vote
0 answers
47 views

Torsion order on Prym variety

Consider two hyperlliptic curves $C_1,C_2$ over $\mathbb{Q}$, and a morphism $\phi:C_1 \rightarrow C_2$. Lifting this morphism on the Jacobians of $C_1,C_2$ and taking its kernel defines a Prym ...
T. Combot's user avatar
  • 231
11 votes
1 answer
851 views

Is the divisibility graph of the proper divisors of n more often planar than not?

Define the divisibility graph of a set of positive integers as the graph whose vertices are the integers, two of which are joined by an edge if one divides the other. For all N, is it true that ...
Bernardo Recamán Santos's user avatar
5 votes
1 answer
401 views

Square root of a line bundle up to a finite surjective morphism

Given a projective variety $X$ over a field of any characteristic, consider a line bundle $\mathcal{L}$ over $X$. The existence of a line bundle $\mathcal{L}^\prime$ with an isomorphism ${\mathcal{L}^...
user158892's user avatar
2 votes
1 answer
187 views

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 ...
Puzzled's user avatar
  • 8,832

1
2 3 4 5
7