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Questions tagged [divisors-algebraic-geometry]

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

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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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18 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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35 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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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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38 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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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
44 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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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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27 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 ...
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59 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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41 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
36 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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15 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
37 views

Definition of codimension of variety

Let $X$ be an 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 ...
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39 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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18 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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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
42 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
50 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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30 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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55 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 ...
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2answers
83 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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25 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 $\...
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32 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
45 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
27 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 ...
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1answer
80 views

Divisor on cubic curve $y^2z=x^3-xz^2$

Let $X$ be the nonsingular cubic curve $y^2z=x^3-xz^2$. Let $P_0=(0,1,0)$. Then the line bundle associated to $3P_0$ is $O_X(1)$. I already know that $3P_0$ is produced by cut the curve with $z=0$. ...
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Does $\deg_k F(-D)/F = \deg_k D$ hold for effective divisors $D$ and coherent, torsion-free $\mathcal{O}_X$-modules $F$?

$\DeclareMathOperator{\F}{\mathcal{F}}\DeclareMathOperator{\o}{\mathcal{O}}$Let $X$ be a reduced, pure dimensional, projective curve over some field $k$. Let $\F$ be a coherent and torsion-free $\o_X$...
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Type I pencils on Riemann surfaces (complex curves)

I was reading the paper https://www.researchgate.net/publication/265872523_On_the_number_of_pencils_of_minimal_degree_on_curves_with_small_gonality and found the statement in the second paragraph to ...
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Pulling-back functions that vanish of order one respectively two in $x$ yields a commutative diagram

Let $X$ be a complex manifold, $\sigma:\hat{X}\to X$ is the blow up of $X$ at $x$. Define $E:=\sigma^{-1}(x)$. $\mathcal{I}_{\{x\}}$ is the ideal sheaf at $x$. we compare the two exact sequences $0\...
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38 views

Efficently find divisors of numbers up to 20 digits

I am aware that finding divisors of large numbers is a well known mathematical problem (which is one of the reasons why cryptography works). Most solutions I've stumbled across used prime ...
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1answer
77 views

Regular global sections of invertible sheaves

$\newcommand{\L}{\mathcal{L}}$ $\newcommand{\ox}{\mathcal{O}_X}$ Let $X$ be a projective scheme of dimension one over a field $k$ and let $\L$ be an invertible sheaf on $X$. What are sufficient ...
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1answer
129 views

Question on Hartshorne II.6.1

I found the proof of Hartshorne II.Lemma 6.1 very confusing. I will break down places I'm confused by. First, (*) means we have a noetherian, integral, separated scheme which is regular in ...
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1answer
56 views

Sufficient conditions for two ideals with the same zero set to be equal

Let $R$ be a one-dimensional noetherian ring finite over $k[x]$. Let $I$, $J$ be two integral ideals of $R$ with $J \subseteq I$ and $I$ invertible. Moreover, assume that $V(I) = V(J)$. What are ...
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275 views

Example of nef and big, not ample

What would be a common, simple example of a nef and big divisor that is not ample? Are there any common, less simple examples? Are there any common strategies for finding examples?
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Effective Cartier divisors defined by a section

Let $S$ be a scheme, and $C$ be a smooth curve over $S$. It is known that any section $s\in C(S)$ of the curve defines a (relative) effective Cartier divisor of degree $1$, often denoted $[s]$ (see ...
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50 views

Canonical Divisors On Projective Curves

I am working on an expository paper and need help with a problem from Algebraic Geometry: A Problem Solving Approach by Thomas Garrity (and many more authors). Suppose $x$ and $y$ are both local ...
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121 views

Is there one canonical principal polarization of a Jacobian per nonisomorphic curve?

Why is there only one canonical principal polarization per Jacobian? I don't yet see why it is true, but I have seen "the canonical polarization" stated many times. This is perhaps a naive question, ...
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40 views

Effective divisor on curve of genus zero

I have been working on Exercise IV.1.5 in Hartshorne's Algebraic Geometry, which says For an effective divisor $D$ on a curve $X$ of genus $g$, show that $\operatorname{dim} |D| \le \operatorname{...
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17 views

Meromorphic function with poles along a hypersurface

Let $X$ be a complex variety and $D \subset X$ an hyperusrface. We say that a function $$f: X-D \to \mathbb{C}$$ is meromorphic along $D$ if for every $p \in D$ there exixsts $V_p \ni p $ open subset ...
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163 views

Divisor class group of the quadric cone in $\mathbb{P}^3$

I would like to compute the divisor class group of the projective quadric cone $$ Q=\mathrm{Proj}(\mathbb{C}[X_0,X_1,X_2,X_3]/(X_1X_2-X_3^2)). $$ It has as an open subset the quadric cone $U$ in $\...
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Pull-back of the sheaf associated to a divisor on a curve

Let $\nu\colon Y\rightarrow B$ be a finite morpism of projective curves, $E$ and $F$ very ample locally free sheaves on $B$ of rank $n+1$ and $n$, respectively. Denote with $X=\mathbb{P}(\nu^*E)$ the ...
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Equivalent Divisor and Evaluation

I've been reading the "Pairing for Beginners" tutorial from Craig Costello and have been asking myself about the example given in section 3.3 (Weil Reciprocity page 44, example 3.3.2). It states that ...
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1answer
130 views

Group Law on Elliptic Curves in terms of Divisors

There is a question in Hartshorne which asks for the group law on the elliptic curve defined by the equation $$y^2+y= x^3-x,$$ namely to find an expression for $P+Q$, where $P = (0,0)$ and $Q=(a,b)$ ...
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1answer
67 views

How to compute divisor degrees over non-algebraically closed fields

I've read in several places a statement like "a principle divisor on a projective curve has degree zero" (roughly). A closer examination of some proofs (Fulton, Hartshorne) shows that this is based ...
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1answer
42 views

Pullback of divisor under the map $z\mapsto z^p$

I'm a little confused right but I think this question can easily be answered. Let $X\subset \mathbb C^3$ be the (affine) surface defined by $z=x^ay^b$, where $(x,y,z)$ are the coordinates on $\mathbb ...
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1answer
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Rational function on elliptic curve giving $x$-coordinate of translate can be written as ratio of linear functions

I'm working through exercise 3.29 from Silverman's Arithmetic of Elliptic Curves: Let $E$ be an elliptic curve over an algebraically closed field $K$ with Weierstrass coordinates $x$ and $y$, and ...
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25 views

Lifting of a holomorphic map from the unit disk

Let $X,Y$ be complex manifold and $f : X \to Y$ an holomorphic proper map, such that $f : X-D \to Y-D'$ is biholomorphic, with $D,D'$ hypersurfaces and $f^{-1}(D')=D$. Let $B$ be the unit disk in the ...
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30 views

Induced map on divisor class group

Suppose $C \to Y$ is a closed immersion of schemes (not necessarily of codimension 1), and suppose that every prime divisor $C\subset Z \subset Y$ is equivalent to a prime divisor which does not ...
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63 views

Volume of a divisor of a Calabi-Yau manifold

Let $X$ denote a (compact) Calabi-Yau 3-fold, and suppose $D_I$ denotes a basis of divisors on $X$ (these are classes in $H_4(X, \mathbb{Z})$) and $\omega_I$ denotes a basis of $H^2(X, \mathbb{Z})$ (...