# 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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### 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 ...
1 vote
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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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### 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 ...
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 ...
1 vote