Questions tagged [divisors-algebraic-geometry]

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

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Show that function gives a bijection from curve to projective line but $\textrm{div}(f) = [P] - [Q] \implies P = Q$?

Washington's book on Elliptic Curves Chapter 11 on Divisors, page 380, question 11.3 says: Suppose $f$ is a function on an algebraic curve $C$ such that $\textrm{div}(f) = [P] - [Q]$ for points $P$ ...
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Linear System Attached To Projective Embedding

I've been doing some reading on divisors (Chapter 4 of Murty's book on Abelian Varieties) and I have a fairly elementary question I've spent far too much time trying to figure out. Many authors treat ...
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Notions of "locally principal" for closed codimension 1 subschemes and divisors.

Suppose $X$ is an irreducible Noetherian normal scheme. Let $D$ be an irreducible closed codimension $1$ subset/subscheme of $X$. Recall that the irreducible Weil divisor $[D]$ is locally principal if ...
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Possibilities for a divisor in a degree $10$ K3 surface?

Let $X$ be a Picard rank $1$, index $1$, degree $10$ Fano threefold with polarization $H$. Let $S \subset X$ be a hypersurface with polarization $H_S$ (so $S$ is a degree $10$ K3 surface). Let $D \...
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Blowing Up the Indeterminancy Locus - Why is this sheaf invertible?

The following is explained in Hartshorne, chapter 2.7. I will be considering varieties instead of schemes. Let $X$ be a variety over $k$, and let $L$ be a line bundle on $X$. Let $s_0,\dots,s_n$ be ...
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What is the fiber class of a fibred surface

Let $\Pi:X\to S$ be a fibred surface, i.e., $X,B$ are integral regular complete schemes of respective dimensions 2 and 1. Let $\operatorname{Num}(X)$ be the quotient of $\operatorname{Pic}(X)$ by the ...
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Desingularization of the standard Cremona involution of $\mathbb P^2$.

Consider the birational map $\chi$ given by the blow up of the points $(1:0:0)$, $(0:1:0)$, and $(0:0:1)$ of $\mathbb P^2$, followed by the contraction of the strict transforms of the three lines ...
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Divisors: function composition with multiplication by n map on elliptic curve

Washington's Weil pairing proof gives a divisor in section 11.2 $$\textrm{div}(f) = n[T] - n[\infty]$$ But then he says let $f \circ n$ denote applying the multiplication by n map, then applying f. ...
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Divisors and line bundles

Let $X$ be a complex manifold and $D\subset X$ a smooth hypersurface, that is there exists an open covering $\{U_{\alpha}\}$ of $X$ and assign each $U_{\alpha}$ a holomorphic submersion $f_{\alpha}$, ...
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Are Weil divisors defined over any algebraic scheme?

I notice that: In fulton's book "Intersection Theory", $r$-cycles are defined in any algebraic scheme (see page 10). In particular, Weil divisors (i.e. cycles of codimension 1) are well ...
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$D_1,D_2$ are effective divisors, prove $h^0(D_1)+h^0(D_2)\leq h^0(D_1+D_2)+1$.

Suppose $D_1, D_2$ are two effective divisor on a compact Riemann surface $\Sigma$ (i.e. $D_i\geq 0$). By convertion, define $$ h^0(D):=\dim\{f\in\mathfrak{M}(\Sigma):f\equiv 0\ or \ (f)+D\geq 0\}. $$ ...
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Intersection Number vs Intersection Multiplicity

I am studying from https://userpage.fu-berlin.de/aconstant/Alg2/Bib/Shafarevich.pdf. Let $X$ be a smooth irreducible quasi-projective variety of dimension $n$ (over an algebraically closed field $k$) ...
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Why is the generator of the Picard group of $\mathbb{P}^1$ isomorphic to $\mathcal{O}(1)$?

Assume $\mathbb{P}^1_k$ is the projective line over an algebraically closed field. In Hartshorne, Chapter II.6, Corollary 6.17, Hartshorne claims that the generator of $\text{Pic}(\mathbb{P}^1_k)$ is ...
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An Exact Sequence of Chow Groups: Fulton Proposition 1.8

Proposition 1.8 in Fulton's book on Intersection Theory is as follows: Proposition 1.8: Let $Y$ be a closed subscheme of a scheme $X$, and let $U$ be $X - Y$. Let $i: Y\to X$ and $j: U\to X$ be the ...
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Structural sheaf of a submanifold

I am currently studying some topics in Algebraic Geometry, Algebraic Surfaces to be precise, and I have some doubts about the identity which I am going to write down. To fix notations, let $X$ be some ...
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Why is the number of poles of this function on a curve not the same as the number of zeros?

This is from the following video - https://www.youtube.com/watch?v=fDdzg83z6Xc The function is $f = \frac {y}{z}$ The curve is ${y^2}z - x^3 - x{z^2}$ The poles are $(0:0:1)$, $(1:0:i)$, $(1:0:-i)$ ...
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For Divisors, do we use order of the point or multiplicity of the point?

This is from the Book Mathematical Cryptography by Silverman, Pipher etc. From the book where they describe Rational Functions & their Divisors on Elliptic Curves 5.8.2 Rational functions and ...
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Fibers of the Abel Jacobi map over curves

I am studying the Abel Jacobi map $$\mathrm{Div}_{X/k} \to \mathrm{Pic}_{X/k}$$ for projective, smooth, irreducible curve $X/k$ where $k$ is algebraically closed. Let $S = \operatorname{Spec}(k)$, $T$ ...
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Decomposing a big and nef divisor into ample + effective

Let $\pi: X \to \mathbf{P}^2$ be the blow-up of the projective plane at one point. Write $H$ for a hyperplane divisor of $\mathbf{P}^2$. The pullback $\pi^* H$ is big and nef, so it can be written in ...
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Is a base point free nonspecial invertible sheaf generated by two global sections?

Let $X$ be a projective nonsingular integral curve of genus $g$ over an aglebraically closed field. In Hartshorne Chapter IV exercise 6.8, we know that: If $d\ge g+1$ then there is an effective ...
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Calculating divisor of function on elliptic curve

I read Pairings for Beginners by Craig Costello. In the example 3.1.1 at 37-th page we consider $ E/F_{103} : y^2 = x^3 + 20x + 20$, with points $ P = (26, 20), Q = (63, 78), R = (59, 95), T = (77, 84)...
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Can $\nu_Y (f) = 0$ for every prime divisor containing $Z = \overline{\{z\}}$, where $f \notin \mathcal{O}_z$?

This is about the argument in Hartshorne exercise III.6.8(a), which is supposed to show $X_s$ form a base for the topology of a noetherian, integral, separated, locally factorial scheme $X$, where $s$ ...
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How is it that the quotitient of two rationals is a common divisor for both?

When $2$ number's ratio can not be expressed as a rational number then we call these number incommensurable. But I also read that incommensurable also has the meaning of being able to measure two ...
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Why do we have $\mu_* \mathcal{O}_Y(K_{Y/X}) = \mathcal{O}_X$ for a resolution?

Let $X$ be a normal complex variety and $\mu : Y \to X$ a resolution of singularities. We define the relative canonical divisor to be $K_{Y/X} := K_Y - \mu^* K_X$. In his book Positivity in Algebraic ...
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Why consider $dx/x$ on a complex curve?

In a paper I'm reading, the author considers a compact Riemann surface -- or smooth algebraic curve, you pick -- $X$ given by the equation $y^d=x^n-1$ for some natural numbers $n,d$. My understanding ...
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What is a Cartier divisor on an affine scheme ring-theoretically?

A Cartier divisor is usually defined to be a section of the sheaf $\mathscr{K}^\times/\mathscr{O}^\times$. For an affine scheme, does a Cartier divisor on $\mathrm{Spec}(A)$ have a simple description ...
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Definition of $\nu_C(f)$ where $C$ is a subvariety of codimension 1 and $f$ is rational.

I am studying divisors from Shafarevich. The book is available as a free download at http://userpage.fu-berlin.de/aconstant/Alg2/Bib/Shafarevich.pdf. My issue comes at page 164 of the pdf. Assume we ...
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Is a degree zero divisor on a curve always basepoint-free?

Let $X$ be a smooth projective curve and $D$ a divisor on $X$ of degree zero. Is it always the case that $D$ is basepoint-free? If not, then is there always some (positive) power $mD$ which is?
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Grothendieck Group of a Nonsingular Curve (Hartshorne Exercise II.6.11).

I have copied the exercise below for reference. I was able to figure out how to do (a) and (d), so let me focus on (b) and (c). Please do not provide me with full solutions, but hints (and if ...
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Nef divisor on a surface and intersection number

Let $X$ be a complex surface and $D$ a nef divisor that is not numerically trivial. Then for any $n\in \Bbb N$, can we choose a smooth curve $C$ such that $D\cdot C\geq n$? Certainly there is a ...
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On a particular exact sequence in cohomology

The setup for my question is as follows (from page 9 of Deschamps' expository notes on the Artin-Winters proof of semi-stable reduction here). We want to prove that for $X$ the special fiber of a ...
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$h^0(S^nF(nD_1+D_2))=O(n^2)$ for a rank 2 vector bundle $F$ on a smooth curve

The following proposition and proof are given in Lemma 2.5 of https://mathscinet.ams.org/mathscinet-getitem?mr=1272710, and I have some questions about it. Proposition. Let $F$ be a rank 2 vector ...
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Applying Riemann-Roch to a rank 2 vector bundle on a complex surface

Let $X$ be a complex surface, $F$ a rank 2 vector bundle on $X$, $n$ a positive integer, $S^nF$ the $n$-th symmetric product of $F$, $P$ a rational divisor, and $a$ a rational number such that $naP$ ...
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Projective surface has an ample divisor

Let $S$ be a projective surface. In https://mathoverflow.net/questions/63999/nef-divisor-on-surface, there is an argument proving that any nef divisor $D$ on $S$ has $D^2\geq 0$. But the argument ...
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2 votes
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About quotient singularities on a complex projective surface

Let $S$ be a normal (complex) projective surface with quotient singularities (locally analytically isomorphic to $\Bbb C^2/G$ where $G\subset \text{GL}(2,\Bbb C)$ is a finite group whose action is ...
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1 vote
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Either $\pm c_1(S)$ is ample or $c_1(S)=0$ for a normal projective surface with quotient singularities with $b_2(S)=1$

According to this paper: https://arxiv.org/pdf/math/0602562.pdf, in p.2 (below Theorem 1), it is written that if $S$ is a normal projective surface (so there are only finitely many isolated ...
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Canonical effective divisors on a degree 4 curve

I am trying the following problem from Hartshorne: Original Problem (IV.3.2.i): Let $X$ be a plane curve of degree 4. Show that the effective canonical divisors on $X$ are exactly the divisors $X \...
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Sections of pullback bundle

Let $X$ be a genus 3 curve canonically embedded in $\mathbb{CP}^2$. Why is it that the line bundle $L$ obtained by pulling back the hyperplane bundle $\mathcal{O}_{\mathbb{P}^2}(1)$ has 3 independent ...
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Embedding of complex torus into $\mathbb{P}^3$

I'm dealing with Riemann Surfaces and I saw how a basis for the space $L(D)$ (where $D\in Div(X)$ is a divisor for the complex torus $T=\mathbb{C}/\Lambda$) of D-bounded meromorphic functions give ...
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What is a non-effective holomorphic line bundle $L$ of degree 0 such that $L \otimes L$ is effective?

I am trying to solve an exercise that asks for a non-effective holomorphic line bundle $L$ of degree 0 such that $L \otimes L$ is effective. I think I have mostly solved it but since I am a bit shaky ...
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How to define the intersection product of a curve with itself: $(C\cdot C)$?

I'm reading through Positivity in Algebraic Geometry by Lazarsfeld, and it defines the intersection product for bundles $L$ and (Cartier) divisors $D$ on a complete, irreducible complex variety $X$, ...
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Rank of normal sheaf in singular varieties

Let $X$ be a $n$-dimensional normal, singular projective variety (over the field of complex numbers), and let $Y$ be a subvariety of $X$ of dimension $k$. I know that, if $X$ was non-singular, than ...
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Possible degrees of nonconstant map $f:C\rightarrow \mathbb{P}^1$ for a plane curve $C$

I am looking for the possible degrees of nonconstant map $f:C\rightarrow \mathbb{P}^1$ for a plane curve $C$. By combining the Brill-Noether theorem with the equality $g={d-1\choose 2}$ for a plane ...
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$D-P-Q$ is nonspecial for all $P, Q \in X$ implies $D$ is nonspecial

(Sorry for my bad English) For the proof of Halphen's theorem, in Hartshorne p.349 I need this: A divisor $D$ on a curve $X$ is nonspecial and very ample if and only if $D-P-Q$ is nonspecial for all $...
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Hom set between curves on a surface

Let $S$ be a smooth projective surface and $i:C\subset S,j:D\subset S$ be two smooth curves on $S$ intersecting each other transversely. How can we say about the set $\text{Hom}_S(i_*\mathcal{O}_C,j_*\...
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1 vote
1 answer
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Exact sequence in Hartshorne's proof of Clifford's theorem (Theorem IV.5.4)

In Theorem 5.4 (Clifford) of chapter IV of Hartshorne we have an exact sequence: $$0\rightarrow \mathscr{L}(D')\rightarrow \mathscr{L}(D)\oplus \mathscr{L}(E)\rightarrow \mathscr{L}(D+E-D')\rightarrow ...
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3 votes
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What does linear equivalence geometrically mean for varieties?

Suppose we have a sufficiently nice scheme or say we are working with an abstract nonsingular variety $X$ over an algebraically close field. In this setting, one can study (Weil) divisors. It is then ...
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First Chern class coincides with degree of divisor without poincare duality or de rham cohomology

I know there are a lot of references (e.g. Griffiths-Harris page 141), but the issue is that these references always prove the proposition in arbitrary dimensions, using a somewhat contrived ...
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1 vote
1 answer
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Can effective divisors generate all Cartier divisors?

Let $X$ be a Noetherian scheme. My question is that: for any Cartier divisor $D$, can we write it as $D_1-D_2$ where $D_1$,$D_2$ are effective? What about further assume $X$ is integral? I can see $D$ ...
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General way of determine where a rational function $\phi\in k(X)$ is not regular

So, if you have an irreducible variety $X$ over $k=\bar{k}$ and you consider a rational function $\phi=f/g\in k(X)$, is there a general way to determine where $\phi$ is not regular? At this point I ...
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