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Questions tagged [divisors-algebraic-geometry]

For questions involving Cartier and Weil divisors, the Riemann-Roch theorem and related topics (e.g. Chern classes and line bundles) on algebraic varieties.

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Example II.3.5 in Arithmetic of Elliptic curves

The example is Let $C$ be a smooth curve, let $f \in \overline{K}(C)$ be a nonconstant function, and let $f:C\rightarrow \mathbb{P}^1$ be the corresponding map (II.2.2). Then directly from the ...
Choon Lee Yan's user avatar
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Genus of function fields, Riemann-Roch and base field extension

Let $K$ be a one variable function field over any base field $k_0$. By that, I mean a field extension of transcendantal degree 1. Let $k$ denote the constant field of $K$. By that I mean the field ...
Flöven's user avatar
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Is the difference of two points a principal divisor?

The problem comes from my thought about Hartshorne's Algebraic Geometry, Example 6.10.1. Let $X$ be a nonsingular complete curve and $K$ is its function field. I think any two distinct points $P,Q \in ...
Functor's user avatar
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Multiplicity of Cartier divisor

If $A$ is a ring, the length $length(M)$ of a module $M$ over $A$ is defined to be the sup of the $n$ such that their exists a chain $M_{0} := 0 \subset M_{1} ... \subset M_{n} =: M$ with $M_{i+1}/M_{...
Analyse300's user avatar
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Explicit linear system corresponding to rational map to $\mathbb{P}^1$

Let $L$ be a line in $\mathbb{P}^3$. Then we can define a map $$ \pi\colon \mathbb{P}^3 \dashrightarrow \mathbb{P}^1, \qquad x \mapsto \langle x, L \rangle \cap L' $$ where $L'$ is a line such that $...
fish_monster's user avatar
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86 views

Computing divisors of elliptic curves

I recently have a tough time trying to compute divisors of functions on elliptic curves. This is a part of exercise 11.1. from Washington: Find the divisor of $g(x,y)=\frac{y^4}{(x^2+1)^3}$ over $\...
HyperPro's user avatar
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Degree of divisors of projective algebraic varieties

Let $k$ be any commutative field, not necessarily algebraically closed. I have some question on the definition of the degree of divisor on algebraic varieties over $k$. Since the degree is a linear ...
Flöven's user avatar
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Holomorphic sections of line bundles over ruled surfaces $\mathbb {CP}^1\times \Sigma_g$

Let $\Sigma$ be a Riemann surface of genus $g$, and consider the ruled surface $X = \mathbb P^1 \times \Sigma$. Consider an holomorphic line bundle $\mathcal L \to X$ with $$P.D.(c_1(\mathcal L )) = n ...
Overflowian's user avatar
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Why does the MMP terminates when the canonical divisor is nef?

I have recently taken interest in Mori's Minimal Model Program (MMP) and I struggle to figure out why it stops when the canonical divisor $K_X$ of our variety $X$ is nef. For now, I have understood ...
user1319604's user avatar
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Positivity of self intersection of indeterminacy locus of 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
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Pair of torsors and divisors and Galois cohomology

Let $E/\Bbb{Q}$ be an elliptic curve. $H^1(G_{\Bbb{Q}},E[2])$ is bijection with the set of pair of torsors and divisor of degree $2$. Let call the latter set $WCD(E/\Bbb{Q})$. I tried to construct a ...
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Divisor of a differential form

Let $k$ such that $\operatorname{char}(k)=0$ and $C\subset \mathbb{P}^2$ be the smooth projective curve defined by $$X^4+Y^4+Z^4=0$$ I want to compute the divisor of $d\left(\frac{X}{Z}\right)$. My ...
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Calculation regarding divisors and resolution of singularities

I'm trying to understand a calculation on the second page of https://sma.epfl.ch/~filipazz/notes/adjunction_and_inversion_of_adjunction.pdf, and I have a couple questions. Here is the setup. $X$ is ...
EJAS's user avatar
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Normal bundle of transverse intersection of two irreducible components

Let $X$ be an equidimensional reduced scheme of finite type over an algebraically closed field $k$. Assume that $X$ has two irreducible components $X_1$ and $X_2$. Assume also that $X_1$ and $X_2$ are ...
Suzet's user avatar
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2.3.A Zariski's construction (PAG1 - R. Lazarsfeld), big and nef divisor which is not finitely generated

I'm reading Lazarfeld's book «Positivity in Algebraic Geometry I» and I'm stuck on the construction of a big and nef divisor on a variety $X$ such that its canonical ring/algebra $R(X,D) = \bigoplus_{...
NaNoS's user avatar
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Understanding the claim that holomorphic divisors contained in a divisor class are in bijection with the projectivisation of the space of sections

Let $V$ be a complex manifold. I am trying to approach the subject of divisors through complex geometry since I am not familiar enough with algebraic geometry to come that way. A divisor $D$ on $V$ is ...
rosecabbage's user avatar
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Big and base point free divisor on surfaces

Let $X$ be a complex algebraic surface. Let $D$ be a divisor. Assume that $D$ is base point free. Then the linear system $|D|$ defines a morphism $$\varphi:X\to \mathbb{P}^N.$$ I wonder if the ...
Display Name's user avatar
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Computing $\operatorname{div}(f)$ and $\operatorname{div}(df)$ for $f=\frac{xs}{yt}$

I am trying to do the following exercise: Consider the algebraic curve $C=\left\{((x: y),(s: t)) \mid x^2 s=y^2 t\right\} \subseteq \mathbb{P}^1 \times \mathbb{P}^1$ and consider the rational ...
kubo's user avatar
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115 views

Dimension of sheaf cohomology of divisor on $\mathbb{P}^1$ depends only on the degree of the divisor

I am struggling with the following exercise, which I have to do without Riemann-Roch: Show that the dimensions of $H^0(\mathbb{P}^1, D)$ and $H^1(\mathbb{P}^1, D)$ depend only on the degree of D. ...
Anakhand's user avatar
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Néron–Tate height with respect to a positive divisor

In section 3 of the paper Canonical heights on varieties with morphisms, page 180 after equation (26), the writers say: “The height function with respect to a positive divisor is bounded below off of ...
Or Shahar's user avatar
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Quotient of Elliptic Curve $E$ homeomorphic to $\mathbb{P}^1_{\mathbb{C}}$

Let $E$ be a complex elliptic curve and let $-1$ be the involution on the group structure over $E$. I would like to prove that $E/{-1}$ is homeomorphic to $\mathbb{P}^1$. I am following two main ideas:...
WindUpBird's user avatar
2 votes
1 answer
57 views

Correspondence, cycle class map and Bloch's decomposition of the diagonal

I'm studying Weil cohomology theories, in particular étale $\ell$-adic cohomology, and I have found some problems related to the cycle class map. Let $X$ be a scheme of dimension $d$, my ingredients ...
FreeFunctor's user avatar
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Homogenizing rational function in projective field to find poles gives different poles depending on which $X_i$ is used for homogenizing

Lets say we have a wierstrauss normal form elliptic curve $C : y^2 = x^3 + Ax^2 + B$. Then we look at the vertical line in the function field $f/g = (x - a)/1 \in K(C)$ which intersects at points $(a, ...
Ignatio Mobius's user avatar
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Linearly equivalent divisors on compact complex manifold

Recently I am reading Griffiths' Principle of Algebraic Geometry. On page 136 he mentioned this: Now suppose $M$ is compact. For every $D' \in |D|$ there exists $f \in \mathfrak{L}(D)$ such that $$ D'...
Kimoji's user avatar
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Is this sheaf cohomology group trivial?

Let $X$ be a smooth irreducible curve (i.e. quasi-projective algebraic variety over $k$ such that all its irreducible components have dimension $1$), and in fact let $X$ be projective. Let $D$ be an ...
kubo's user avatar
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Irreducible vertical divisor on a (normal) fibered surface has dimension $1$ ? ( Liu's Algebraic Geometry )

I am reading the book Liu's Algebraic Geometry and arithmetic curves and some question arises. Let $S$ be a Dedekind scheme. We call an integral, projective, flat $S$-scheme $\pi : X \to S$ of ...
Plantation's user avatar
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1 answer
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Help with Harshorne Example II 6.5.2 on the Weil divisor class group of the affine cone

The example is about computing the Weil divisor class group of the affine cone Spec$k[x,y,z]/ \langle xy-z^2\rangle$. I can show that the group is cyclic generated by the divisor $V(y,z )$ and that ...
Luke's user avatar
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The plane in the quadric 3-fold is not a (set-theoretic) hypersurface

I think my question has a top-bottom answer, but as of yet I am not familiar enough with divisors and class groups to be sure of what I am claiming. I also include an "elementary answer" to ...
Andrei.B's user avatar
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1 vote
1 answer
85 views

Bijection between effective Cartier divisors isomorphic to a line bundle $\mathscr{L}$ and regular sections of $\mathscr{L}\mod\mathscr{O}_X^\times$

$\def\sK{\mathscr{K}} \def\sO{\mathscr{O}} \def\sL{\mathscr{L}}$(All definitions I will be using are explained in detail in Görtz, Wedhorn, Algebraic Geometry I, Ch. 11, Divisors.) Let $X$ be a scheme....
Elías Guisado Villalgordo's user avatar
2 votes
1 answer
84 views

If the degree of a divisor on a Riemann surface is $\deg(D) \geq 2g$, then $L(D-(p)) \subsetneq L(D)$ for any point $p$

Let $S$ be a compact connected Riemann surface, $D$ a divisor on $S$, and $p\in S$ a point. I want to show that if $\deg(D) \geq 2g$ (where $g$ is the genus of $S$), then we have a strict inclusion ...
dahemar's user avatar
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The sheaf of modules coming from a Cartier divisor is a line bundle (claim in Görtz, Wedhorn, Algebraic Geometry I)

$\def\sK{\mathcal{K}}\def\sO{\mathcal{O}}$Let $X$ be a integral scheme, denote $K(X)$ to the function field of $X$ and $\sK_X$ to the $\sO_X$-module constantly $K(X)$. In Görtz, Wedhorn, Algebraic ...
Elías Guisado Villalgordo's user avatar
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1 answer
33 views

Linearly equivariant prime divisors on a curve [duplicate]

Let $C$ be an algebraic curve, and $P_1,P_2$ be prime divisors on $C$. I would like to prove : $$P_1\sim P_2\Rightarrow C \text{ is rational.}$$ I tried to use Riemann-Roch theorem to prove that the ...
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Example of a family of ample divisors $\{A_m\}_{m\geq 1}$ on a smooth projective variety $X$ such that $mA_m$ has a basepoint?

I know this famous example due to Kollár: Take $E$ an elliptic curve, and on $E\times E$ consider a horizontal fiber $F_1$, a vertical fiber $F_2$ and the diagonal $\Delta$. Let $X$ be a triple cover ...
imtrying46's user avatar
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2 votes
1 answer
79 views

Kodaira dimension on curve is $0$ implies elliptic curve

Consider $C$ a smooth projective curve, suppose its Kodaira dimension is $0$. What this means for me is that $\max \{ n\in \mathbb N(K_C)|\dim \overline{\phi_n(C)}\subseteq \mathbb P(H^0(C,nK_C)) \}=0$...
raisinsec's user avatar
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1 answer
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Confusion about line bundles and the intersection product

Let $X = \mathbb{A}_{\mathbb{C}}^{2}$ and let $Y$ be the blowup of $X$ at the origin. Let $E \cong \mathbb{P}^{1}$ the exceptional divisor. I think that we have a canonical inclusion $\mathcal{O}_{Y} \...
Fraktale Fatalität's user avatar
2 votes
1 answer
83 views

Structure sheaf of the projectivization of a bundle

Let $0\to V'\to V\to M \to 0$ be an exact sequence with $V$ a vector bundle over a scheme $B$, $M \in \operatorname{Pic}(B) $. Let $H:=\mathbb{P}(V')$ be the Cartier divisor, let $p: \mathbb{P}(V) \to ...
Conjecture's user avatar
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0 votes
2 answers
406 views

Linearly equivalent divisors are numerically equivalent

Let $X$ be a projective variety over a field. Is there a direct way of seeing why every pair of linearly equivalent divisors $D_1$ and $D_2$ is numerically equivalent? I simply found myself unable to ...
mathfan24's user avatar
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0 answers
69 views

Relation between the sheaf of relative differentials and the canonical divisor

Let $\hspace{0.2cm}f:$ $X\longrightarrow Y \hspace{0.2cm}$ be a finite morphism of curves over $K$. Consider $\hspace{0.2cm}\Omega_{X/K}\hspace{0.2cm}$ and $\hspace{0.2cm}\Omega_{Y/K}\hspace{0.2cm}$ ...
Giuseppe's user avatar
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Computing the p-rank of Divisor class group for function field

In the context of my work, I am trying to develop an algorithm to factorize some operators on algebraic function fields of positive characteristic $p$. To this end I need to be able to compute ...
raphitek's user avatar
1 vote
1 answer
133 views

Krull dimension of the local ring at the generic point of a divisor is 1.

Let $X$ be a nonetherian integral separated scheme which is regular in codimension one, i.e. every local ring $\mathscr{O}_x$ of $X$ of dimension one is regular. Let $Y$ be a prime divisor, i.e. a ...
Degenerate D's user avatar
3 votes
1 answer
215 views

For what schemes $X$ are Cartier divisors the same thing as invertible subsheaves of $\mathcal{K}_X$?

Let $X$ be a scheme, and let $\mathcal{K}$ be the sheaf of total quotient rings of $X$. Is the data of a Cartier divisor on $X$ equivalent to the data of an invertible subsheaf of $\mathcal{K}$ for ...
Bun's user avatar
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Pullback of a translation map of a divisor in Birkenhake-Lange's book "Complex Abelian Varieties"

I'm currently studying the book 'Complex Abelian Varieties' by Birkenhake and Lange. On page 74, after lemma 1.5, the authors make the following statement: 'Another observation, which will prove ...
William Gibson's user avatar
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0 answers
26 views

For $D$ very ample on a threefold $X$, is it true that $D^2.S\geq(\operatorname{mult}_x D)^2\cdot\operatorname{mult}_x S$?

Let $X$ be a smooth projective variety over an algebraically closed field $k$, and suppose $\dim X=3$. Let $D$ be an effective, very ample divisor and let $x\in X$ be a point. Then is it true that for ...
imtrying46's user avatar
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2 votes
1 answer
67 views

A divisor $D$ that satisfies $D|_{U_i}=(f)$ on complements of prime divisors $Z_i=X-U_i$ is principle, and $D=(f)$.

I have stumbled upon the following Lemma in Hartshorne exercise II.6.3(a). Let $X$ be a noetherian, integral, separated scheme which is regular in codim 1. Let $Z_i$ be prime divisor of $X$, s.t. $\...
Elad's user avatar
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3 votes
2 answers
216 views

Doubt on base point for a linear system

I'm studying divisors on Riemann surfaces and I got stuck on a thing said by the professor at lesson. Probably I'm getting lost in a glass of water, or there is something wrong in my notes: if this is ...
PS48725's user avatar
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0 answers
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Reference request: Calabi-Yau toric manifold condition in algebraic geometry

I'm currently studying toric Calabi-Yau manifolds, and in particular, am looking at how we can construct them from fans. A fact keeps coming up in the papers I am reading, for instance in https://...
anna's user avatar
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1 answer
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Theorem 7.3.17 in (Qingliu) Algebraic Geometry and Arithmetic curve

I don't understand the logic [As $F$ is a finite scheme, $\mathcal{O}_X(E)|_F\cong \mathcal{O}_F$]. Let $i:F\to X$ be the closed immersion, then the isomorphism really is $\mathcal{O}_X(E)\otimes_{\...
Z Wu's user avatar
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217 views

Using Riemann-Roch Theorem to show every elliptic curve can be written as a plane cubic

I've been studying how to show that every elliptic curve can be written as a plane cubic through the book of Joseph H. Silverman "Arithmetic Elliptic Curves", the proof of proposition III.3....
Taken Spark's user avatar
1 vote
1 answer
256 views

Confusion with automorphisms of a projective variety with ample or antiample canonical class

Question. Suppose $X$ is a smooth irreducible projective variety over a field $k$. Let $\omega_{X/k}=\bigwedge^{\dim X} \Omega_{X/k}$ be the canonical sheaf with associated divisor $K_X$. Suppose ...
Hank Scorpio's user avatar
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1 answer
61 views

A basic question on the sheaf associated with a Weil divosor

I am reading the section on Weil divisors in Vakil's FOAG, where he defined a sheaf $\mathcal{O}(D)$ for a Weil divisor $D$ on a normal integral Noetherian scheme $X$ that is regular in codimension 1 ...
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