# 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 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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### Doubt on base point for a linear system

I'm studying divisors on Riemann surfaces and I got stuck on a thing said by the professor at lesson. Probably I'm getting lost in a glass of water, or there is something wrong in my notes: if this is ...
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### Multiplicity of Weil divisor on affine cone

Let $D=(x, y)$ be a Weil divisor in the cone $A = k[x,y,z,w]/(xz-yw)$. I want to show that no multiplicity of $D$ is a Cartier divisor. This appears in Vakhil's book as an excersize, but he invokes ...
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### Reference request: Calabi-Yau toric manifold condition in algebraic geometry

I'm currently studying toric Calabi-Yau manifolds, and in particular, am looking at how we can construct them from fans. A fact keeps coming up in the papers I am reading, for instance in https://...
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### canonical divisor from canonical sheaf

Let $X$ be a complete intersection of $m$ hypersurfaces in $P^n$ over some field $k$. I have computed that the canonical sheaf is $\omega_X=\mathcal{O}_X(\sum d_i-n-1)$, where $d_i$ are the degrees of ...
I know that if I have a divisor $D$ on a Riemann surface $X$, there is a line bundle associated to $D$, that I write as $[D]$ following the terminology in Griffiths & Harris (Principles of ...