# Questions tagged [divisors-algebraic-geometry]

For questions involving divisors, invertible sheaves and/or line bundles on varieties and schemes.

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### Van der Waerden's Purity Theorem (Liu"s AGAC)

I have some questions about the arguments used in the proof of Theorem 7.2.22 (Waerden's Purity Theorem) from Liu's "Algebraic Geometry and Arithmetic Curves" (page 273): Rmk: Lemma 2.20 and Def 2.21 ...
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### Properties of a linear system on a K3 surface

Here i assume to work on $\mathbb{C}$. Let $X$ be a projective K3 surface. On Reid's "Chapters on algebraic surfaces" there is a theorem (page 69) whose point (b) says: If $D>0$ is nef and $D^2=0$ ...
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### Global sections of a line bundle associated to a smooth irreducible curve on a K3 surface.

I would like to know it the following proof is correct: Lemma: Let $C$ be a smooth, irreducible curve of genus $g$ over a complex projective K3 surface $X$. Let $L:=\mathcal{O}_X(C)$. Then $C^2=2g-2$...
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### Global section of a closed point as a divisor for a degree 2 curve

This is part of Ex1.7 in Chapter IV of Hartshorne's Algebraic Geometry. Let $X$ be a curve of degree $2$ and genus $2$ over $\mathbb{P}^1$. Show that the canonical divisor defines a complete linear ...
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### Riemann Roch theorem for surfaces

Hi am a student of Maths at university; I am studying the theorem of Riemann-Roch for curves. I am interested in understanding what happens in the case of surfaces. I do not want to look for the whole ...
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### Question about the proof of Hodge Index Theorem

In the proof of Hodge Index Theorem in Hartshorne's, it says that given an ample divisor $H$ on surface $X$, and if $mD$ is effective for all $m>>0$, we will have $mD.H>0$. The book hints ...
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### How to define a Weil divisor from a Cartier divisor on a variety?

Let $X$ be an irreducible variety. Here are the definitions I'm working with (from Shafarevich). A Weil divisor on $X$ is a formal finite sum of irreducible closed codimension 1 subvarieties of $X$. ...
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### Degree of a morphism is equal to the degree of a divisor?

A morphism $f:X\to P^n$ can be determined by a linear system $|D|$ for a given divisor. It seems that degree of morphism $degf$ is equal to degree of divisor $degD$. I guess the divisor may be ...
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### Divisor class group of vector bundle over an integral Noetherian scheme

Ler $X$ be an integral Noetherian scheme. Then one can show that taking the inverse image induces an isomorphism of Weil divisor class groups $\operatorname{Cl}(X)\to \operatorname{Cl}(\mathbb A^n_X)$....
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### On this particular elliptic curve, how can I construct a function with a prescribed set of poles and zeros?

Consider the elliptic curve given by $E: Y^2 = X^3-X$ over the field $\overline{\mathbb{F}}_5$. I have computed the $\mathbb{F}_5$-rational points (in projective space, where $(0:1:0)$ is taken as the ...
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### Degree of the intersection of subvarieties from different branches

Let $k$ be an algebraically closed field. Let $X$ be an irreducible hypersurface of $\mathbb{P}_k^n$, where $n\geq 4$. Let $Y$ be an irreducible subvariety of $X$ of codimension one. Let $d$ be the ...
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### Explicit description of sheaf of differentials on $\mathbb{P}^1_k$ using affine charts, and corresponding Cartier Divisor

Let $k$ be a field. We define the scheme $X = \mathbb{P}^1_k$ to be the gluing of the affine schemes $\text{spec}(k[T])$ and $\text{spec}(k[U])$ via the isomorphism \begin{align*} \phi: k[T,T^{-1}] &...
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### Line bundles correspoding to a hyperplane

Assume we have a smooth projective variety $X$ over a field and a hyperplane section $H$ on it. For each Weil divisor on $X$ you can construct a line bundle on $X$. For $H$ this line bundle which is ...
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### Morse Theory on a complement of a Semi-ample divisor

I am studying some topological properties of the complement of divisors on algebraic varieties X, and for that, I want to know if there exists some Morse theory on the complement of a semi-ample ...
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### Cartier divisor example in Harthsorne

This question is about example 6.11.4 in Chapter II of Hartshorne. The example is about computing the Cartier divisor class group of the cuspidal cubic curve $y^2z = x^3$ in $\mathbb{P}_{k}^{2}$. He ...
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### A function being “finite” over a point on non-normal schemes?

I recently came across a remark about Cartier divisors in a textbook on algebraic geometry. I'm not sure how to interpret the remark. I've attached the previous paragraph as well for context. The ...
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### Cartier divisor corresponds to Weil divisor on a curve.

Let $E= \{y^{2}z = x^{3} - xz^{2}\}\subset \mathbb{P}^{2}_{k}$ be an elliptic curve over an algebraically closed field $k$. Let $P = [0:0:1]$ be a point and let $D$ be a Weil divisor correspons to the ...
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### Weil divisors on Noetherian local ring of dimension $1$

Ler $A$ be a Noetherian local ring of dimension $1$, with maximal ideal $\mathfrak m$ and minimal prime ideals $\mathfrak p_1,\dots, \mathfrak p_r$. In exercise 11.18 of the book "Algebraic Geometry 1"...
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### Why is this Weil divisor not a Cartier divisor

I'm reading "Introduction to toric varieties" by "William Fulton", here is an example in page 61 illustrating the difference between Weil divisors and Cartier divisors. Here my question is only about ...
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### Divisors in a pair of planes intersecting in a line

Let $\pi_1, \pi_2$ be two planes in $\mathbb{P}^3$ intersecting in a line $L$. Let us denote its union by $X$. I would like to understand better the Picard group of $X$. I would also like to know ...
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### to which intrinsic object corresponds connection's hypersurface

Given a complex manifold $M$ and an hypersurface $S$, and some connection on the line bundle associated to $S$, to which intrinsic object of $S$ corresponds the connection ? (more specifically, same ...
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### Definition of codimension of variety

Let $X$ be a variety over field $k$. A Weil divisor on $X$ is an integral linear combination of irreducible subvarieties of $X$ of codimension $1$. So I want to know the definition of codimension of a ...
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### Weak solutions for divisors

I have a question on the following definition in the Forster: I don't get the part where it says "Clearly a weak wolution $f$ is a proper, i.e., meromorphic function, solution precisely if $f$ is ...
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### Question on divisors of meromorphic functions on Riemann surfaces

I have a question on the following definition I don't quite get why the definition says " if f is identically zero in a neighborhood of a", I mean there's nothing wrong with that I just don't get why ...
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### What is a rational section of an invertible sheaf?

I am studying Cartier divisors, and I am confused about exactly how they correspond to rational sections of a line bundle, or what a rational section of a line bundle even is. Let $X$ be an integral ...
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### Linearly Equivalence of pullback of divisors

Let $X_0=\mathbb{P}^2$ and $\eta: X_r \mapsto X_0$ be the blow-up of $p_1,\cdots, p_r$, where $p_1 \in X_0$. In a paper I am reading, the author states the following: If $C\subset X_r$ is an ...
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### Restriction section of a sheaf to a closed set?

I am doing an exercise from Hartshorne (II Ex 6.2) on divisors and I have come across an abuse of notation that I am not entire sure how to interpret. Let $X \hookrightarrow \mathbb{P}_{k}^{n}$ be a ...
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### Discrete valuation of a rational function composed with an automorphism

The answer to my question might be trivial, although I can not see it. Details: let $k = \mathbb{F}_q$, $\bar{k}$ the algebraic closure of $k$, $C \subset \mathbb{P}^n(\bar{k})$ a smooth projective ...
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### Constructing an invertible sheaf from a Cartier divisor?

Let $X$ be a scheme. By definition, a Cartier divisor $D$ is a global section of the sheaf $\mathcal{M}_X^{\times}/\mathcal{O}_X^{\times}$, where $\mathcal{M}_X^{\times}$ is the sheaf of ...
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### Derive Group Law on Elliptic Curve with Riemann Roch

Consider $E$ be an elliptic curve and $k$ a field. I read that one way to show that $E(k)$ has an abelian group structure can be derived using Riemann Roch. Could anybody explain how it concretely ...
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### Computation of $L(p+q+r)$ on a smooth projective curve

Let $X$ be a smooth projective curve in $\mathbb{P}^2(\mathbb{C})$ of degree $4$ and $p,q,r \in X$. What's $L(p+q+r)$? With a standard computation, the genus of $X$ is $3$, so applying Riemann-Roch ...
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### Prove that any divisor of order 0 on non-singular projective curve of genus $g$ is equivalent to other

Could you please check whether the solution below is ok? There is an exercise from Shafarevich's Basic Algebraic Geometry, vol. 1, ex. 7.7.21. Let $o$ be a point of an smooth algebraic curve $X$ of ...