# Questions tagged [divisors-algebraic-geometry]

For questions involving Cartier and Weil divisors, the Riemann-Roch theorem and related topics (e.g. Chern classes and line bundles) on algebraic varieties.

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### Example II.3.5 in Arithmetic of Elliptic curves

The example is Let $C$ be a smooth curve, let $f \in \overline{K}(C)$ be a nonconstant function, and let $f:C\rightarrow \mathbb{P}^1$ be the corresponding map (II.2.2). Then directly from the ...
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### Genus of function fields, Riemann-Roch and base field extension

Let $K$ be a one variable function field over any base field $k_0$. By that, I mean a field extension of transcendantal degree 1. Let $k$ denote the constant field of $K$. By that I mean the field ...
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### Degree of divisors of projective algebraic varieties

Let $k$ be any commutative field, not necessarily algebraically closed. I have some question on the definition of the degree of divisor on algebraic varieties over $k$. Since the degree is a linear ...
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### Is this sheaf cohomology group trivial?

Let $X$ be a smooth irreducible curve (i.e. quasi-projective algebraic variety over $k$ such that all its irreducible components have dimension $1$), and in fact let $X$ be projective. Let $D$ be an ...
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### Irreducible vertical divisor on a (normal) fibered surface has dimension $1$ ? ( Liu's Algebraic Geometry )

I am reading the book Liu's Algebraic Geometry and arithmetic curves and some question arises. Let $S$ be a Dedekind scheme. We call an integral, projective, flat $S$-scheme $\pi : X \to S$ of ...
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### Help with Harshorne Example II 6.5.2 on the Weil divisor class group of the affine cone

The example is about computing the Weil divisor class group of the affine cone Spec$k[x,y,z]/ \langle xy-z^2\rangle$. I can show that the group is cyclic generated by the divisor $V(y,z )$ and that ...
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### The plane in the quadric 3-fold is not a (set-theoretic) hypersurface

I think my question has a top-bottom answer, but as of yet I am not familiar enough with divisors and class groups to be sure of what I am claiming. I also include an "elementary answer" to ...
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### Bijection between effective Cartier divisors isomorphic to a line bundle $\mathscr{L}$ and regular sections of $\mathscr{L}\mod\mathscr{O}_X^\times$

$\def\sK{\mathscr{K}} \def\sO{\mathscr{O}} \def\sL{\mathscr{L}}$(All definitions I will be using are explained in detail in Görtz, Wedhorn, Algebraic Geometry I, Ch. 11, Divisors.) Let $X$ be a scheme....
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### If the degree of a divisor on a Riemann surface is $\deg(D) \geq 2g$, then $L(D-(p)) \subsetneq L(D)$ for any point $p$

Let $S$ be a compact connected Riemann surface, $D$ a divisor on $S$, and $p\in S$ a point. I want to show that if $\deg(D) \geq 2g$ (where $g$ is the genus of $S$), then we have a strict inclusion ...
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### The sheaf of modules coming from a Cartier divisor is a line bundle (claim in Görtz, Wedhorn, Algebraic Geometry I)

$\def\sK{\mathcal{K}}\def\sO{\mathcal{O}}$Let $X$ be a integral scheme, denote $K(X)$ to the function field of $X$ and $\sK_X$ to the $\sO_X$-module constantly $K(X)$. In Görtz, Wedhorn, Algebraic ...
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### Linearly equivariant prime divisors on a curve [duplicate]

Let $C$ be an algebraic curve, and $P_1,P_2$ be prime divisors on $C$. I would like to prove : $$P_1\sim P_2\Rightarrow C \text{ is rational.}$$ I tried to use Riemann-Roch theorem to prove that the ...
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### Example of a family of ample divisors $\{A_m\}_{m\geq 1}$ on a smooth projective variety $X$ such that $mA_m$ has a basepoint?

I know this famous example due to Kollár: Take $E$ an elliptic curve, and on $E\times E$ consider a horizontal fiber $F_1$, a vertical fiber $F_2$ and the diagonal $\Delta$. Let $X$ be a triple cover ...
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### Kodaira dimension on curve is $0$ implies elliptic curve

Consider $C$ a smooth projective curve, suppose its Kodaira dimension is $0$. What this means for me is that $\max \{ n\in \mathbb N(K_C)|\dim \overline{\phi_n(C)}\subseteq \mathbb P(H^0(C,nK_C)) \}=0$...
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### Linearly equivalent divisors are numerically equivalent

Let $X$ be a projective variety over a field. Is there a direct way of seeing why every pair of linearly equivalent divisors $D_1$ and $D_2$ is numerically equivalent? I simply found myself unable to ...
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### Relation between the sheaf of relative differentials and the canonical divisor

Let $\hspace{0.2cm}f:$ $X\longrightarrow Y \hspace{0.2cm}$ be a finite morphism of curves over $K$. Consider $\hspace{0.2cm}\Omega_{X/K}\hspace{0.2cm}$ and $\hspace{0.2cm}\Omega_{Y/K}\hspace{0.2cm}$ ...
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### Computing the p-rank of Divisor class group for function field

In the context of my work, I am trying to develop an algorithm to factorize some operators on algebraic function fields of positive characteristic $p$. To this end I need to be able to compute ...
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### Krull dimension of the local ring at the generic point of a divisor is 1.

Let $X$ be a nonetherian integral separated scheme which is regular in codimension one, i.e. every local ring $\mathscr{O}_x$ of $X$ of dimension one is regular. Let $Y$ be a prime divisor, i.e. a ...
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### For what schemes $X$ are Cartier divisors the same thing as invertible subsheaves of $\mathcal{K}_X$?

Let $X$ be a scheme, and let $\mathcal{K}$ be the sheaf of total quotient rings of $X$. Is the data of a Cartier divisor on $X$ equivalent to the data of an invertible subsheaf of $\mathcal{K}$ for ...
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### Pullback of a translation map of a divisor in Birkenhake-Lange's book "Complex Abelian Varieties"

I'm currently studying the book 'Complex Abelian Varieties' by Birkenhake and Lange. On page 74, after lemma 1.5, the authors make the following statement: 'Another observation, which will prove ...
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### For $D$ very ample on a threefold $X$, is it true that $D^2.S\geq(\operatorname{mult}_x D)^2\cdot\operatorname{mult}_x S$?

Let $X$ be a smooth projective variety over an algebraically closed field $k$, and suppose $\dim X=3$. Let $D$ be an effective, very ample divisor and let $x\in X$ be a point. Then is it true that for ...
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### Using Riemann-Roch Theorem to show every elliptic curve can be written as a plane cubic

I've been studying how to show that every elliptic curve can be written as a plane cubic through the book of Joseph H. Silverman "Arithmetic Elliptic Curves", the proof of proposition III.3....
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### Confusion with automorphisms of a projective variety with ample or antiample canonical class

Question. Suppose $X$ is a smooth irreducible projective variety over a field $k$. Let $\omega_{X/k}=\bigwedge^{\dim X} \Omega_{X/k}$ be the canonical sheaf with associated divisor $K_X$. Suppose ...
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I am reading the section on Weil divisors in Vakil's FOAG, where he defined a sheaf $\mathcal{O}(D)$ for a Weil divisor $D$ on a normal integral Noetherian scheme $X$ that is regular in codimension 1 ...