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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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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 ...
divergent's user avatar
2 votes
0 answers
212 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 ...
Ron Z.'s user avatar
  • 89
1 vote
1 answer
144 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 $\text{NS}(A\times A)$. ...
Fra's user avatar
  • 91
1 vote
0 answers
41 views

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 ...
JerryCastilla's user avatar
0 votes
2 answers
251 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{...
Teddy's user avatar
  • 29
2 votes
0 answers
172 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 $...
Don's user avatar
  • 271
3 votes
0 answers
196 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
101 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
62 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
  • 5,041
10 votes
1 answer
2k 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: $...
Craw Craw's user avatar
5 votes
2 answers
237 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
6 votes
2 answers
296 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
  • 173
2 votes
1 answer
260 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
222 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
233 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
  • 8,902
2 votes
1 answer
260 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
224 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
191 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
  • 1,071
3 votes
0 answers
129 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
135 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
194 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
  • 8,902
3 votes
1 answer
280 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
  • 653
-1 votes
1 answer
190 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
  • 8,902
2 votes
1 answer
152 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
  • 557
1 vote
0 answers
176 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
319 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
121 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
  • 569
1 vote
1 answer
210 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
113 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
206 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
  • 8,902
1 vote
1 answer
154 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
128 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
227 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
358 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
78 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
151 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
174 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
148 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
  • 8,737
2 votes
0 answers
87 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
383 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
  • 2,303
7 votes
1 answer
530 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
240 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
202 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
  • 2,303
4 votes
0 answers
889 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
  • 4,134
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
220 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
  • 8,902
4 votes
1 answer
395 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
125 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

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