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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Linearly equivalent divisors on compact complex manifold
Recently I am reading Griffiths' Principle of Algebraic Geometry. On page 136 he mentioned this:
Now suppose $M$ is compact. For every $D' \in |D|$ there exists $f \in \mathfrak{L}(D)$ such that
$$ D'...
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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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Confusion about line bundles and the intersection product
Let $X = \mathbb{A}_{\mathbb{C}}^{2}$ and let $Y$ be the blowup of $X$ at the origin. Let $E \cong \mathbb{P}^{1}$ the exceptional divisor.
I think that we have a canonical inclusion $\mathcal{O}_{Y} \...
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Structure sheaf of the projectivization of a bundle
Let $0\to V'\to V\to M \to 0$ be an exact sequence with $V$ a vector bundle over a scheme $B$, $M \in \operatorname{Pic}(B) $. Let $H:=\mathbb{P}(V')$ be the Cartier divisor, let $p: \mathbb{P}(V) \to ...
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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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A divisor $D$ that satisfies $D|_{U_i}=(f)$ on complements of prime divisors $Z_i=X-U_i$ is principle, and $D=(f)$.
I have stumbled upon the following Lemma in Hartshorne exercise II.6.3(a).
Let $X$ be a noetherian, integral, separated scheme which is regular in codim 1. Let $Z_i$ be prime divisor of $X$, s.t. $\...
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Reference request: ramification divisor of simple perverse sheaf
I'm reading Beilinson's paper "Constructible sheaves are holonomic"
In 4.6 of this paper, he used the following fact:
Consider sheaf of $\Lambda$-modules, where $\Lambda=\mathbb{Z}/l^N\...
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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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Theorem 7.3.17 in (Qingliu) Algebraic Geometry and Arithmetic curve
I don't understand the logic [As $F$ is a finite scheme, $\mathcal{O}_X(E)|_F\cong \mathcal{O}_F$].
Let $i:F\to X$ be the closed immersion, then the isomorphism really is $\mathcal{O}_X(E)\otimes_{\...
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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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A basic question on the sheaf associated with a Weil divosor
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 ...
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Is there a name for the following important group of line bundles with rational sections?
Is there a name for the following important group of line bundles with rational sections, given by Vakil in FOAG 15.4.3, page 436?
15.4.3. The important group of "line bundles with rational ...
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Understanding Pullback of Cartier Divisors
Let $g\colon X\to Y$ be a morphism of schemes, and let $D=(U_i,f_i)$ be a Cartier Divisor on $Y$. I've seen the following definition of the pullback of $D$: $g^*D\colon=(g^{-1}(U_i),f_i\circ g)$, as ...
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Intersection of divisor and curve on a subvariety
Let $X$ be a normal $\mathbb Q$-factorial variety (irreducible) over an algebraically closed field $k$ of characteristic $0$. Let $D\subseteq X$ be an irreducible divisor (which must be $\mathbb Q$-...
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Cartier Divisor can have torsion element?
I was reading Lazarsfeld's Positivity in Algebraic Geometry, and I realize that Cartier divisor can have torsion elements, but I can't find such an example.
I know for the Noetherian scheme that is ...
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Divisors on a non-singular non-hyperelliptic curve of genus 5
Now $X$ is a non-singular and non-hyperelliptic curve of genus 5. Then if $X$ has no $g^1_3$, prove that there is a divisor of degree 6 such that $\mathcal{O}_X(D)$ is base point free.
I am confused ...
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Multiplicity on strict tranform and exceptional divisor : $m_x(\hat{C}\cap E)\geq m_x(\hat{C})$ for $x\in \hat{C}\cap E$
Let $S$ be a surface, $C$ an irreducible curve on $S$ and $p\in C$ a point. Then consider the blow-up at $p$ and write $E$ the exceptional divisor. I want to show that if $x\in \hat{C}\cap E$, then $...
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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 ...
user1055643
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What does it mean to take the first Chern class of a sheaf?
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 ...
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Complete Intersection of Hypersurfaces are Fano varieties
I am currently studying Algebraic Geometry, and the wikipedia page of Fano Variety says the following "a smooth complete intersection of hypersurfaces in n-dimensional projective space is Fano if ...
user1055643
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Equivalent definitions of hyperelliptic Riemann surfaces
Let $X$ be a connected Riemann surface. I’m trying to understand the equivalence about these two definitions (I’m still new to this subject):
$X$ is hyperelliptic iff there is a holomorphic degree $2$...
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When is a hypersurface in $\mathbb{CP}^n$ irreducible?
Let $f$ be a homogenous polynomial in $z_0, ..., z_n$ of degree $d$ and consider the divisor $D$ with defining data $(U_i, f(z_0/z_i, ..., z_n/z_i))$ where $U_i$ is the affine space $\{z_i \neq 0\}$. ...
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When is $\dfrac{k[x,y,z]}{(x^a +y^b+z^c)} $ a unique factorization domain?
Let $k$ be an algebraically closed field of characteristic zero. Let $a,b,c$ be relatively prime positive integers. Then, is $\dfrac{k[x,y,z]}{(x^a +y^b+z^c)} $ a UFD?
This question is motivated from $...
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Question about divisors on a compact Riemann surface
I am trying to prove the following question from here without relying on the Riemann-Roch theorem.
Let $X$ be a compact Riemann surface, and let $D$ be a divisor on $X$.
(i) If $\mathrm{deg}(D) = 0$, ...
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A question of proposition 3.1, chapter4 in hartshorne
As in the picture says, hartshorne says there exists a exact sequece of sheaves
$$0 \to \mathcal{L}(D-P) \to \mathcal{L}(D) \to k(p) \to 0 $$ where $\mathcal{L}(D)$ means associated invertible sheaves ...
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Complete linear system induce an immersion iff the associated sheaf is very ample
Let $X$ be a geometrically integral projective scheme over a field $k$, let $D$ be a Cartier divisor over $X$, and let $\mathcal L=\mathcal O_X(D)$ be the associated invertible sheaf. Then $H^0(X,\...
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When determinant bundle is very ample
For a vector bundle $V$ on a projective variety $X$, let $\Bbb P(V) $ be the projective bundle of hyperplanes. Call $V$ a very ample if $\mathcal O_{\Bbb P(V)}(1)$ is very ample on $\Bbb P(V)$.
Let $...
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Show that divisor in $\mathbb P^2\times \mathbb P^2$ is very ample.
Let $H$ denote a quadric hypersurface in $\mathbb P^2\times \mathbb P^2$. In the Chow ring of $\mathbb P^2\times \mathbb P^2$, we have $H\equiv 2H_1+2H_2$, where $H_1,H_2$ are the classes of linear ...
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Geometric interpretation of the ampleness of the canonical class of a normal algebraic surface
Let $X$ be a minimal, smooth and projective algebraic surface of general type over the complex numbers. Then the ampleness of $K_X$ has a very geometric interpretation: $K_X$ is ample if and only if $...
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Blowup of points in the plane and linear systems
Let $p_1,\dots,p_r\in \mathbb P^2$ be points in general position, $r\leq 6$. Let $S$ be the blowup of these $r$ points, then the anticanonical divisor $K_S$ is very ample and it gives an embedding, in ...
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Canonical embedding for divisors?
Let us consider a nonhyperelliptic curve $X$ of genus $g \geq 2$, canonically embedded in $\mathbb{P}^{g-1}$ and $D = P_1+ \dots+P_d$, a positive divisor on $X$. One could take an hyperplane $H \...
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Weil divisor classes containing prime divisors
In algebraic number theory, the distribution of prime ideals in the ideal classes is a well-studied topic. It is, for instance, a fundamental theorem that every ideal class of a global field contains ...
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