# 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.

322
questions

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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 ...

0
votes

1
answer

114
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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 ...

4
votes

2
answers

141
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 ...

3
votes

1
answer

237
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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 ...

0
votes

1
answer

97
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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 ...

2
votes

1
answer

138
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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{...

1
vote

0
answers

152
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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 ...

0
votes

1
answer

191
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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 ...

0
votes

0
answers

80
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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 $\mathbb{Q}$-...

1
vote

1
answer

182
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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 ...

1
vote

1
answer

90
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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 ...

2
votes

0
answers

159
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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 $s_0\in S$.
...

1
vote

1
answer

140
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 ...

0
votes

0
answers

116
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 ...

2
votes

1
answer

176
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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 ...

2
votes

0
answers

54
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 ...

3
votes

0
answers

228
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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 ...

1
vote

0
answers

74
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 $Y = \mathbb{P}(\mathcal{L}^{\oplus 3})$...

1
vote

0
answers

117
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 ...

1
vote

0
answers

109
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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 ...

0
votes

1
answer

133
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 ...

2
votes

0
answers

77
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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 $...

2
votes

1
answer

293
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 $X$?

7
votes

1
answer

471
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 $x$, $y$, $z$ ...

2
votes

1
answer

222
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 ...

4
votes

1
answer

146
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 ...

3
votes

1
answer

172
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 $...

4
votes

0
answers

489
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 ...

2
votes

0
answers

49
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{...

5
votes

1
answer

199
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 $(-K_X)^2$ ...

3
votes

1
answer

182
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 $...

2
votes

0
answers

110
views

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

Consider the nine consecutive natural numbers starting from $1584614377$.
...

3
votes

0
answers

83
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 $\text{Cl}(X)$ ...

6
votes

1
answer

485
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:
...

1
vote

0
answers

43
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 ...

11
votes

1
answer

826
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 ...

5
votes

1
answer

350
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}^...

3
votes

1
answer

155
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 ...

4
votes

1
answer

417
views

### Cohomology of divisors on Hirzebruch surfaces

Consider the Hirzebruch surface $\mathbb{F}_n = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(n))\rightarrow\mathbb{P}^1$. The Picard group of $\mathbb{F}_n$ is generated by ...

3
votes

2
answers

294
views

### Abelian varieties corresponding to Hodge substructures

In an exercise of Voisin book, says:
Let $j:C\rightarrow S$ the inclusion of a smooth curve on a smooth connected projective surface. Set
$H=ker(j_*:H^1(C,\mathbb{Z})\rightarrow H^3(S,\mathbb{Z}))$.
...

3
votes

0
answers

187
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\...

2
votes

1
answer

350
views

### Divisors on projective bundles

Let $\pi:X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^n$ be a projective bundle, where $\mathcal{E}$ is a rank two vector bundle over $\mathbb{P}^n$.
If $n = 0$ then $X = \mathbb{P}^1$, and for $n ...

5
votes

2
answers

466
views

### Divisors whose restriction is big

Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$.
...

7
votes

2
answers

328
views

### Nef divisors on surfaces

Let $X$ be a smooth projective rational surface over an algebraically closed field of characteristic zero, and $D$ a divisor on $X$ such that $D$ is nef and $D^2 = 0$ with the following properties:
$...

2
votes

1
answer

162
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 ...

7
votes

2
answers

437
views

### Global sections of multiples of a divisor

Let $D$ be an integral divisor on a smooth projective variety $X$. Consider the multiples $mD$ of $D$ for $m\geq 0$. Clearly, $h^0(X,mD) = 1$ for $m = 0$.
Is there any example where $h^0(X,mD) = 0$ ...

1
vote

0
answers

137
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\...

3
votes

1
answer

340
views

### A question on effective divisors

Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. Then ...

3
votes

1
answer

256
views

### Nef and pseudo-effective divisors over non algebraically closed fields

Let $X$ be a projective variety over a field $K$. As a consequence of Kleiman's criterion, when $K$ is algebraically closed, we have that if $D$ is a nef divisor on $X$ then $D$ is pseudo-effective.
...

1
vote

1
answer

234
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 $...