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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1answer
109 views

Pull back of twisted sheaf under a regular map associated to a base point free linear system

Let $D$ be a divisor on a normal projective variety $X$ and $V$ be a subspace of the global section of $\mathscr O_X(D),$ L is a base point free linear system and $\phi_L:X\overset{(g_0:\cdots:g_n)}\...
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1answer
88 views

Different definitions of Cartier divisor and when they agree

On a scheme $X$, the most general definition of Cartier divisor is a global section in $\Gamma(X, \mathcal{K}^{*}/\mathcal{O}^{*})$, where $\mathcal{K}^{*}$ is the sheaf of invertible elements of the ...
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48 views

“Pullback” of a divisor in Weil's/Lefschetz's trace formula

In Lei Fu's "Etale Cohomology Theory", theorem 8.6.7 is Lefschetz's Trace formula for curves, it is stated as follow: Theorem: Let $X$ be a smooth projective curve over an algebraically closed ...
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1answer
69 views

On the product of divisors

I am currently studying this paper, and I am not being able to prove one argument on the proof of the Theorem $5.1$, namely, the last part where I have to show that a product of divisors is bigger ...
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33 views

$D$ a divisor numerically 1-connected implies that $H^0(\mathcal{O}_D)$=constants

I'm trying to understand the following: Let $X$ be a nonsingular complex projective surface and $D$ an effective divisor. Then $D$ numerically 1-connected implies that $H^0(\mathcal{O}_D)=...
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1answer
75 views

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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45 views

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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23 views

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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69 views

Picard group of curves

I suppose the Picard group of a complete curve $C$ in $\mathbb P^n$ of degree $d>2$ is complicated. So, if we remove a general point $x\in C$ and denote $C'=C-x$, I think we have $\mathcal O_{C'}(1)...
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34 views

What goes wrong in the proof that $Spec \mathbb{R}[x,y]/(x^2+y^2)$ is a principal divisor in $Proj \mathbb{R}[x,y,z]/(x^2+y^2+z^2)$

Let $C=\text{Proj}\mathbb{R}[x,y,z]/(x^2+y^2+z^2)$. Consider a closed subset $Z=\{z\neq 0\}$. I claim that $Z=(f)$ with $f=x^2+y^2\in K(C)$. Indeed, if $z=0$ then the we get exactly $Z$. If $...
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1answer
27 views

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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1answer
24 views

Any nonzero meromorphic $1$-form on a compact Riemann surface has degree $2g-2$

I am reading "Compact Riemann Surfaces" by Raghavan Narashimhan. Say X be a compact Riemann surface; after proving that the degree of the canonical bundle $K_X$ is $2g-2$ (using Riemann-Roch), where $...
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22 views

Fibre of morphism to projective line connected

Let $S$ be an algebraic complex surface and $C_{\lambda}$ a pencil of divisors on it without base points,assuming the generic element of it is irreducible and smooth. I would like to prove that every ...
2
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1answer
38 views

Canonical Divisors versus Principal Divisors

Perhaps this is so obvious a question that no one has asked it before, but can someone provide a simple example of a canonical divisor which is not a principal divisor or conversely on a Riemann ...
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1answer
38 views

Definition of degree of invertible sheaf?

In Hartshorne's, he states that #$C\cap D=deg_C(\mathcal{F}(D)\otimes\mathcal{O}_C)$, where $deg_C$ denotes the degree of the invertible sheaf $\mathcal{F}(D)\otimes\mathcal{O}_C$. I couldn't find ...
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37 views

Divisors, line bundles and group homomorphisms

Let $X$ be a normal, integral, Noetherian scheme. In the Wikipedia article about divisors it is claimed that the class group $Cl(X)$ of Weil divisors modulo linear equivalence is isomorphic, as a ...
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94 views

Relative Bertini theorem

Setup of the problem : Let $f : X \rightarrow Y$ be a flat(smooth) projective morphism with relative dimension $d \geq 1$ and relative very ample sheaf $\mathcal{O}(1)$. Q : Is it possible to find a ...
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35 views

Divisors on the complex plane

I am trying to understand the notion of $L(D) = \{f \text{ is meromorphic}| D + (f) \geq 0\}$ on $C\mathbb{P}^1$. I have come across an exercise where it is asked to prove that $dim(L(D))=\max{(0,1+...
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30 views

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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39 views

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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28 views

Zeroes of global section and degree of a line bundle

Let $X$ be a Riemann surface and $K$ be the canonical bundle. Consider $D = p_1 + p_2 + .... + p_n $ a divisor on $X$. Is it true that number of zeroes of a global section of $K(D)$ is equal to deg$K(...
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1answer
41 views

How can two copies of $\mathbb{P}^2$ intersect?

Let $C$ be a curve in $\mathbb{P}^2$. I am wondering if we can find a number $n$ and two embeddings $$ \varphi_i:\mathbb{P}^2\hookrightarrow \mathbb{P}^n, i\in\{1,2\} $$ such that $\varphi_{i}^{-1}(\...
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31 views

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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27 views

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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1answer
24 views

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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42 views

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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30 views

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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45 views

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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19 views

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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1answer
65 views

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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41 views

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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0answers
39 views

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 ...
2
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1answer
66 views

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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1answer
94 views

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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1answer
38 views

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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18 views

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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1answer
50 views

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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45 views

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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19 views

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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110 views

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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1answer
65 views

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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1answer
59 views

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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33 views

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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0answers
72 views

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 ...
3
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2answers
109 views

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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27 views

Divisor over ellitptic curves

I struggle to prove the following theorem : Let $E$ be an elliptic curve over a field $K$. Let $D=\sum n_p P$ be a divisor on $E$. Then $D \sim 0$ if and only if $\sum [n_p]P=\mathcal{O}$ where $\...
0
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1answer
38 views

Curves and divisors in weighted projective planes

Let us consider the weighted projective plane $\mathbb{P}(q_0,q_1,q_2)=\mathrm{Proj}(\mathbb{C}[x_0,x_1,x_2])$ where $x_i$ has weight $q_i$ for every $i\in \{0,1,2\}$. Let $f\in \mathbb{C}[x_0,x_1,x_2]...
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1answer
47 views

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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1answer
28 views

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 ...