Questions tagged [divisors-algebraic-geometry]

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

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48 views

Fulton problem 8.32: a projective variety of curves

I am trying to understand the following problem of Fulton's "Algebraic Curves": Notation: let $d \in \mathbb{N}$, $p_1, \cdots p_m \in \mathbb{P}^2$ distinct points and $r_1, \cdots, r_m \in ...
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62 views

Pencil of divisors in algebraic geometry

Let $X \subset \mathbb{P}^n$ be projective variety over alg closed field of char $0$ and $C = V(F), D= V(G) \subset X$ two distinct divisors (e.g. two quadrics, curves or lines lying in a surface,...) ...
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23 views

How to devide an linear object to calculate how many of transverse smaller objects would be there and to be simetrical

I have started new work, in which I have to use some math. It is my firs job in engineer environment so I have to get used to it and remind myself some math from school. I have an linear object, ...
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34 views

The complement of a neighborhood of a divisor.

Given a projective variety $X$ and a hyperplane section $H$ (intersection of some hyperplane in the ambient projective space with $X$). Is it true that complement of any neighborhood of $H$ is a ...
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56 views

Which is canonical $|D|\cong \mathbb{P}(\Gamma(X,\mathscr{O}_X(D))^{\vee}$) or $|D|\cong \mathbb{P}(\Gamma(X,\mathscr{O}_X(D))$)?

Sorry for my bad English. Let $X$ be a smooth projective curve over an algebraically closed field $k$, and $D$ is divisor on $X$. By The stacks project, $|D|\cong \mathbb{P}(\Gamma(X,\mathscr{O}_X(D))^...
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38 views

Prove that rational map associated to divisor is birational

My question arises from the proof of theorem 1.4.13, $ii)\Rightarrow iv)$ on the first paragraph of page 38 in the set of notes: http://homepages.math.uic.edu/~ein/DFEM.pdf Let $X$ be a projective ...
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30 views

Ample divisors with fixed part and movable cone

Let $X$ be a normal complex projective variety. In some papers a divisor $D$ is said to be movable if its base locus has codimension $\geq 2$, in others if there exists a positive multiple $mD$ of $D$,...
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63 views

Example of a Cartier divisor that cannot be written as the difference of effective divisors.

Suppose $X$ is a variety and $D$ is a Cartier divisor on $X$. Fulton argues in his Intersection theory, that if $\pi \colon \tilde X \to X$ is the blow-up of $X$ with respect to the ideal of ...
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53 views

Question about algebraically equivalent divisors

I'm trying to do exercise V.1.1.7 in Hartshorne, about algebraic equivalence of divisors, and the first problem is to show that the divisors algebraically equivalent to zero form a subgroup of $Div(X)$...
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62 views

Proving that intersection number of divisor is 0

My question arises from proposition 8.3 in the algebraic geometry text by Shigeru Iitaka: Let $f:W\rightarrow S$ be a surjective morphism between nonsingular projective surfaces. Let $E$ be a divisor ...
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29 views

Are the numbers calculated from a log resolution birational invariants?

Let $C = V(x^2 - y^3)$ be the cuspidate cubic which sits inside $\mathbb{C}^2$. Let $\pi: \text{Bl}_0(\mathbb{C}) \to \mathbb{C}^2$ be the blow up of the origin. The reduction of the total transform ...
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163 views

Liu's “Algebraic Geometry and Arithmetic Curves” - Proposition 4.4. (Ch. 7) and Exercises 5.1.29 and 5.1.30

In Liu's book, Chapter 7, Proposition 4.4, the question is about closed embeddings (or: closed immersions) coming from very ample divisors. The theorems are quite well-known when the ground field $k$ ...
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42 views

inverse image of fiber product of scheme

I asked this question and I couldn't understand the following result. https://math.stackexchange.com/questions/3956704/hartshorne-Ⅱ-prop-6-6-irreducible Let $X$ be a noetherian integral separated ...
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61 views

Hartshorne Ⅱ prop 6.6, irreducible [duplicate]

I'm trying to understand the following proposition, which is Hartshorne Ⅱ prop 6.6. Let $X$ be a noetherian integral separated scheme which is regular in codimension one, then $X \times_{\operatorname{...
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112 views

When/why is a point on a curve an effective Cartier divisor?

The definition of effective Cartier divisor that I'm using is: a closed immersion whose corresponding quasicoherent sheaf of ideals is an invertible sheaf. Let $X$ be a scheme that is dimension 1 and ...
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34 views

Primary decompositions and divisorial fractional ideals

Consider a divisorial (fractional) ideal $\mathfrak{a}$ over some normal Noetherian domain $A$ with fraction field $K$. Then it can be shown that $\mathfrak{a}$ is a product of integer powers of ...
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20 views

Conditions for a dlt pair $(X,\Delta)$.

I am referring to the book ‘Birational Geometry of Algebraic Geometry’ by Janos Kollár and Shigefumi Mori. I would like to refer to proposition 2.42 on page 60 of the text. Proposition 2.42 states the ...
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21 views

Commutativity of Intersection product

I am reading the proof of the commutativity of intersection product from Ravi Vakil's notes: http://virtualmath1.stanford.edu/~vakil/245/245class8.pdf This theorem states that if $D$ and $D'$ are ...
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1answer
82 views

Preimage of Prime Divisors is finite union of Prime divisors

I am reading the book ‘An Introduction to Birational Geometry of Algebraic Varieties’ by Shigeru Iitaka. In section 2.11 we attempt to define the pullback of a prime divisor. In particular, we have ...
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72 views

very ample divisors on curves using Riemann Roch Spaces

Let $C$ be a projective non-singular irreducible curve, let $D$ be a divisor on $C$. Suppose the Riemann-Roch Space is $L(D)=\langle f_1,...,f_n\rangle$. Define $\phi_D:C\to \mathbb{P}^{n-1}$ by $\...
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82 views

Multiples of an effective degree 2 divisor on a hyperelliptic curve.

Let $L$ denote the Riemann-Roch space, and $l$ its dimension. We are given the following definition: A curve $C$ of genus $g$ is hyperelliptic if and only if there is a degree 2 divisor $D$ such that $...
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29 views

Is the positive part of the Zariski decomposition of a big $\mathbb{R}$-divisor big?

I can't understand why the positive part of the Zariski decomposition of a big class is itself big. More concretely: let $X$ be a smooth projective surface over $\mathbb{C}$. Let $N^1_{\mathbb{R}}(X)$...
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34 views

any divisor of a projective non-singular irreducible curve is equivalent to an effective divisor minus a multiple of a point

Here is a problem I need help getting started with. I am brand new to algebraic geometry. Let $C$ be a projective non-singular irreducible curve of genus $g$. Let $P_0\in C$. Prove that any divisor $D ...
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28 views

Principal Divisors on a projective non-singular irreducible curve if and only if degree is 0

Let $C$ be a a projective non-singular irreducible curve under a ground field $k$. Let $D$ be a divisor on $C$ with degree 0. Does this necessarily imply that $D$ is principal? That is, $D=\text{div}f$...
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38 views

How to compute the canonical bundle of a generalized flag bundle?

If $X:=Gr_k(V)$ denotes the Grassmannian of $k$-dimensional subspaces of an $n$-dimensional vector space $V$, it is well-known that the canonical bundle $\omega_X$ is given by $$\omega_X:= \det \...
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1answer
62 views

genus$=2$ implies Hyperelliptic.

In the book of Rick Miranda (Algebraic Curves and Riemann Surfaces), in Proposition 1.10 of Chapter VII (p. 198), the claim is that every compact Riemann Surfaces of genus $2$ is hyperelliptic. The ...
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35 views

Birational Transform of a Divisor

Let $f: X\rightarrow Y$ be a birational morphism between normal varieties. Let $V\subset X$ be a prime divisor of $X$, i.e. irreducible closed subvariety of codimension 1. Choosing an open subset $U$ ...
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57 views

Why are plurigenera birational invariants?

Let $X$ be a variety over an algebraically closed field $k$ and let $\Omega_X^n=\bigwedge_{i=1}^n\Omega_X$, where $\Omega_X$ is the canonical bundle. I'm trying to understand the well-known fact that ...
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9 views

Zariski density of a matrix semigroup generated by Jordan blocks

In the field of random matrix products, it seems a lot of theorems which give nice statistical properties (central limit theorem, large deviation, etc.) assume—among other things—a Zariski density ...
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37 views

$\text{Cl}(X)\cong \text{Cl}(X\times \mathbb{A^1})$

This question has been asked on this site already, but in every related topic I only find references to Hartshorne, where this fact is proven for schemes. I would like to learn about this isomorphism ...
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1answer
67 views

Finding ramification points of a map to $\mathbb{P}^1$ given by a divisor

Let $X$ be a elliptic curve over $k$ ("curve" in the sense of Hartshorne Chapter IV). A closed point $P_0\in X$ gives rise to a base-point free linear system $|2P_0|$ of dimension 1, which ...
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29 views

Ample divisor can be represented by an effective cycle class

My question arises from the proof of theorem 5.11 page 18 ~ 19 of the notes by Mihnea Popa: https://sites.math.northwestern.edu/~mpopa/483-3/notes/notes.pdf#page16 We wish to prove Kleinman’s theorem: ...
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1answer
105 views

Is the degree homomorphism $\text{deg}: \text{Pic}(X)\to \mathbb{Z}$ surjective?

Let $k$ be a field, $X$ a curve over $k$, $\operatorname{Div}(X)$ the divisor group of $X$, and $\operatorname{Pic}(X)$ the divisor class group (the Picard group) of $X$. Consider the degree ...
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12 views

Equivalent dividers on Riemann Surfaces

Let $X$ be a compact Riemann surface of genus $g$, it's true that, if $D$ is a effective divisor with degree equal to $g$ then there is $p_1,...,p_g\in X$ such that $D\sim p_1+\cdots+p_g$, i.e there ...
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38 views

Why divisors determined by points of depth 1?

Let $X$ be a Noetherian scheme, $D_1,D_2$ are Cartier divisors on $X$. I'm confused with following two questions: If $D_1$ has local equation $f_x$ which are all non-zerodivisor at points $x\in X$ ...
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39 views

Simplifying a rational function on an algebraic curve

Does a rational function $\phi$ on a smooth projective algebraic curve $F$ over a algebraically closed field $K$ always have a representative $\frac{f}{g}$, where $f$ and $g$ are polynomials without ...
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32 views

Very ample divisor and hyperplane sections

I already searched on the site and there is several topics which deal with this question, but actually it doesn't make it clear as crysal to me. For the context, we take $X$ a good variety (let say ...
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46 views

Intersection of a divisor with a principal divisor

I am new in the study of surfaces in the algebraic geometry point of view. I am studying chapter V of Hartshorne. At some point while studying I came accross with the following thought which I was ...
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37 views

Is this effective divisor $E$ linearly equivalent to zero?

Let $E$ be an effective divisor such that $E \cdot E=0$. If $-E$ is linearly equivalent to an effective divisor, can I say that $E$ is linearly equivalent to $0$? Or are there any counterexamples?
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31 views

Free Abelian Group Generated by Codimension 1 Subvarieties: A Line.

i) A Weil divisor is a sum over the codimension 1 subvarieties, does that mean any point is generating the other points or are they all equally considered generators? ii) This paper says that the sum'...
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47 views

Showing that $\mathcal{O}(D)$ is coherent

Let $X=\operatorname{Spec}A$ be a normal affine variety and $D$ be a divisor on $X$; we would like to show that the sheaf $\mathcal{F}:=\mathcal{O}(D)$ is coherent. The sheaf $\mathcal{F}$ is defined ...
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77 views

Divisor of meromorphic functions on Riemann surfaces

Let $C$ be the Riemann surface $y^2=x^3+1$ defined over $\mathbb{C}$, then I want to calculate the divisor of the meromorphic function of $g=\frac{x^2}{y}$. In the class of Riemann surface, we know ...
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44 views

Normal sheaf of a nonsingular subvariety

I would like a reference to a proof or a proof of the following fact. Let $ X $ be a nonsingular variety over an algebraically closed field $ k $ and $ Y $ a nonsingular subvariety of $ X $ of ...
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1answer
76 views

A proof and explanation for why divisor classes pull back along flat morphisms

Let $\pi: X \rightarrow Y$ be a finite type morphism of noetherian normal schemes and let $D$ be a prime Weil divisor in $Y$. It seems to be a well known fact that if $\pi$ is flat, then $\pi$ pulls ...
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77 views

For curves $A,B$ in $X\cap Y$, the intersection numbers $(A\cdot B)_X$ and $(A\cdot B)_Y$ are the same

Let $X,Y\subset\Bbb{P}^3$ be a smooth surface of degree $d$ and a plane respectively. If $X\cap Y=A+B$ for curves $A,B$, then $(A\cdot B)_X=(A\cdot B)_Y$ (intersection numbers in $X$ and $Y$ ...
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115 views

Divisor of a rational section in Ravi Vakil's notes

The following is from Exercise 14.2.A of Ravi Vakil's algebraic geometry notes (page 401 here). The exercise asks us to consider the rational section $\frac{x^2}{x+y}$ of the sheaf $\mathcal{O}(1)$ on ...
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60 views

Preimage of prime Weil divisor is prime Weil divisor in Hartshorne

Let $X$ be a noetherian normal scheme, let $\mathbb{A}^1$ be the affine line over $\mathbb{Z}$, and let $X \times \mathbb{A}^1$ be the fibre product over $\mathbb{Z}$ with projection $$ \pi: X \times ...
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1answer
66 views

The degree of vector bundle on integral projective curve is the degree of its determinant bundle.

This is exercise 18.4.J in Vakil's "The Rising Sea Foundations of Algebraic Geometry". Here the degree of a coherent sheaf $\mathscr{F} $ on integral projective curve $ C $ is defined to be ...
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96 views

Doubly periodic meromorphic function with prescribed poles and zeros

The field of the meromorphic functions on a complex torus $\mathbb{C} \mathbin{/} \Lambda$ is $\mathbb{C}(\wp, \wp')$, where $\wp$ is the weierstrass p-function to the lattice $\Lambda$. Furthermore, ...
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1answer
49 views

When are curves smooth in a pencil $C_{(s:t)}=\{sF+tG\}$

Let $C_1=V(F),C_2=V(G)$ be smooth curves in $\Bbb{P}_{\Bbb{C}}^2$ and consider the pencil of curves $C_{(s:t)}=sF+tG$ for $(s:t)\in\Bbb{P}^1$. I'm trying to prove/disprove that $C_{(s:t)}$ is smooth ...

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