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.
603 questions
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Why is the sequence $0\rightarrow\mathscr{O}_C(C)^\ast\rightarrow k(C)^\ast\rightarrow\rm Div(C) \rightarrow Pic(C)\rightarrow 0$ exact?
Definition
A curve over a field $k$ is a separated scheme $C$ of finite type over $k$ which is integral of dimension 1.
Let $C$ be a normal curve over a field $k$. A divisor is an element of the free ...
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0
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64
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Is there any relation between group law of smooth projective Elliptic curve and divisor class group of completion of coordinate ring?
Let $f$ be a homogeneous polynomial in $\mathbb C[x,y,z]$ defining an Elliptic curve in $\mathbb P^2_{\mathbb C}$. In general, is there a relation between the Elliptic group law of this Elliptic curve ...
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1
answer
114
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Calculating the self-intersection number of a plane curve on a smooth surface in $\mathbb{P}^3.$
I am working through the final chapter of Shafarevich's first AG book and I am stuck on the following exercise:
Suppose that a nonsingular plane curve $C$ of degree $r$ lies on a nonsingular surface ...
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0
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55
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The form of a rank 0 torsion sheaf on a Calabi-Yau threefold with nontrivial $c_1$
This is a question regarding the paper "Bogomolov-Gieseker Type Inequality and Counting Invariants" by Y. Toda.
Let $X$ be a smooth projective Calabi-Yau 3-fold and $H \in H^2(X)$ an ample ...
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3
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1
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170
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What is the relation between the Chern character of a coherent ideal sheaf and the fundamental class of its closed subvariety?
This is on pages 2-3 in the paper Bogomolov-Gieseker Type Inequality and Counting Invariants.
Context:
First, we set up some notation:
Let $X$ be a Calabi-Yau threefold.
Given
$$
(R, d, \beta, n) \...
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1
answer
101
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Properties of left and right adjoint functors to pushforward functor from a divisor
Let $X$ be a Noetherian variety, and $D$ a Cartier divisor. Let $i:D\hookrightarrow X$ be the inclusion. Let $i_* : Coh(D)\to Coh(X)$ be the functor between derived category of bounded coherent ...
2
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0
answers
60
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The identification of the hyperplane bundle with $\mathcal{O}(1)$
I was reviewing my notes when I suddenly had a bit of confusion regarding the identification of the hyperplane bundle with $\mathcal{O}(1)$.
Let $\mathbb{P}^n = \operatorname{Proj} \mathbb{C}[Z_0, \...
- 177
3
votes
1
answer
84
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Correspondence between divisors and line bundles
I want to understand how the correspondence between divisors and holomorphic line bundles on a compact Riemann surface $S$ works. Griffiths and Harris describe this correspondence in detail in their ...
- 311
2
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1
answer
137
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Can effective divisors be numerically trivial?
I'm working on problems related to divisor theory and rational connectedness of algebraic varieties, specifically focusing on the behavior of prime divisors. I'm trying to deepen my understanding of ...
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1
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1
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117
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Is a birational morphism a composition of blow-downs?
My question is:
Let $f:X\to Y$ be a birational morphism between normal projective varieties such that the exceptional locus of $f$ has codimension $1$ in $X$ ($f$ is a divisorial contraction). Is it ...
- 93
1
vote
0
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65
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Calculation of pole of order for the two multiplied function
Assume we have two lines.
$y = \lambda_1 x + v_1$ and $y = \lambda_2 x + v_2$
We also have an elliptic curve given by $y^2 = x^3 + ax + b$.
If we separately want to figure out pole of order for each ...
- 79
2
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0
answers
94
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Degree of a line bundle is equal to the degree of the corresponding divisor
Let $X$ be a smooth projective irreducible curve and let $\mathcal{L}$ be a line bundle on $X$. We denote by $D=\sum_{i=1}^n m_iP_i$ the divisor on $X$ corresponding to $\mathcal{L}$, i.e. such that $\...
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0
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89
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Using the divisor $(Q) - (O )$ to compute Tate pairing
There is the following task:
Consider a curve $y^2 = x^3 + 2$ over $\mathbb{F}_{11}(i)$ and $P = (9,4), Q = (0, 3i).$ Find Tate pairing value $\tau_3(P, Q).$
Hint 1: Find a divisor $(D_Q)$ with $sum(...
2
votes
1
answer
109
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Globally generated sheaf
In Forster's book Riemann surfaces for an arbitrary divisor $D$ on a Riemann surface $X$ the sheaf $\mathcal{O}_D$ is defined as $\mathcal{O}_D(U):=\{f\in\mathcal{M}(U):ord_x(f)\geq-D(x)$ for all $x\...
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1
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109
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Can't understand the multiplicity of poles in divisors
I am studying the divisors in relation to elliptic curve cryptography.
Assume that elliptic curve is defined by $x^3 = y^2 + b$ and let's assume our rational function is $f(x,y) = x - P_x$. Let's fix ...
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0
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0
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111
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Intersecting divisors
I want to ask whether the proof of (2.4) in Fulton can be simplified under better assumptions. Instead of copying all the definitions and constructions from that specific book, let me state things a ...
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2
votes
1
answer
109
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Pushforward of Line Bundles under Birational Morphisms
I'm considering a birational morphism $f:X \to Y$ between smooth projective varieties, and a divisor $D$ on $X$.
I'm trying to study $f_*(\mathcal O_X(D))$. Its structure depends on $D$ and on how $f$ ...
- 21
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1
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99
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Relationship between the complete linear system of a line bundle and the Proj of its section ring
Let $X$ be a projective variety over $\mathbf C$ and assume it has all the good hypotheses one can wish for, and let $ \mathcal{L} $ be a line bundle on $ X $. One can consider the complete linear ...
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1
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0
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103
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Action of the absolute Galois group on Weil divisors and fixed part
Let $K$ be a field of characteristic $0$ and let $X$ be a normal projective variety over $K$ (i.e. a projective, geometrically integral, separated scheme of finite type over $K$). By fixing an ...
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0
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30
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Computing number of n with no functions having zeros of order n on smooth curve
My past paper question is,
Let X be a smooth projective curve of genus g over the complex numbers C.
For p ∈ X let
G(p) = {n ∈ N | there is no f ∈ k(X) with $v_p(f) = n$, and $v_q(f)\neq0$ for all $q\...
- 137
1
vote
1
answer
109
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What's the relation between the degree of (basepoint free) $g^r_d$'s on curves and the degree of induced maps $C\to \mathbb P^r_k$?
Let $C$ be a smooth (integral, projective) curve over some field $k$. In my case, I suspect that $k$ can be taken arbitrarily, but one might suppose $k=\mathbb C$ for simplicity.
A $g^r_d$ in $C$ is a ...
- 2,958
0
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0
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129
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Locally principal Weil divisor that is not associated to a Cartier divisor
[The question was crossposted to mathoverflow, where it was satisfactorily answered.]
If $X$ is an integral separated Noetherian scheme that is regular in codimension 1, then there is a natural map $\...
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-3
votes
1
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56
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Question about a statement on Debarre's book. [closed]
The following statement is on page 192 of Debarre's Higher-Dimensional Algebraic Geometry.
Let $X$ be a projective algebraic variety of dimension $n$, and $C$ an irreducible curve on X. For any ample ...
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0
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0
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110
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Canonical Divisor of $y^7=x^2(x-1)$
This comes from an exercise in ACGH's Geometry of Algebraic Curves. Let $\Gamma_0\subseteq\mathbb{C}^2$ be the plane curve $y^7=x^2(x-1)$. Let $\Gamma$ be its projectivization (which includes one ...
- 1,173
1
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1
answer
87
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Divisor class group of $V(Y^2Z-X^3-Z^3)$
I am trying to do some divisor computations on a curve from ACGH exercise A.3(ii).
For context, the affine equation is $y^2=x^3+1$. It's projectivization, $C$, is the projective curve $Y^2Z=X^3+Z^3$ ...
- 1,173
0
votes
2
answers
77
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Locally principal divisors over a curve
I am aware of the following fact:
A divisor over a scheme defines a line bundle if it is locally principal.
Somehow, this seems to implies the following fact:
A point on a curve $C$ (over a field $...
- 419
0
votes
1
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150
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Canonical divisor of a normal projective variety
I encountered some difficulties when defining the canonical divisor of a normal projective variety.
Let $X$ be a normal projective variety over $\mathbb{C}$, and let $X^{*}$ denote its smooth locus. I ...
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1
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0
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94
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Pullback of divisor along blow up
Let $X$ be a smooth surface over a characteristic $0$ field $k$. Let $\pi:\widetilde{X}\rightarrow X$ be the blowing-up of $X$ along finite many closed points. We assume $\widetilde{X}$ is also smooth....
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0
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81
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Do complex submanifolds in Calabi-Yau threefolds necessarily have global sections?
I am a physicist with a limited mathematics background so please be gentle with me.
I am aware that effective divisors in Calabi-Yau threefolds are calibrated by the Kähler form. Let me be precise. A ...
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1
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0
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142
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Maps to Projective Space induced by Line Bundles
Let $X$ be a quasi compact quasi separated scheme over field $k$ and $\mathcal{L}$ a line bundle on it. Assume that the base locus of $L$ is empty, so $L$ semiample.
This gives rise to induced map $$...
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$l(D)$ for Elliptic Curves
This exercise is from Gathmann's Plane Algebraic Curves class notes (Exercise 8.19) and a follow-up question to another.
EDIT: Notice that this is different question from Exercise 8.19 of his ...
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1
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118
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Anormal curve on a surface implies singularity (?)
This question is derived from this post but I think the question has its own interest so I'd like to ask separately.
General question (or context): we know that a smooth cubic curve is (isomorphic to)...
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0
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113
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Pushforwards of principal divisors
Suppose we have a proper morphism of algebraic $k$-schemes $f : X\rightarrow Y$. We know that for an integral subvariety $V\subset X$, the reduced closed subscheme $W$ of $f(V)$ is integral, and that ...
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1
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93
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Indeterminacy locus birational map big divisor
Let $Y$ be a normal, $\mathbb{Q}$-factorial projective complex variety, let $X=\mathbb{P}(\mathcal{O}_Y(D)\oplus \mathcal{O}_Y(D'))$ be a $\mathbb{P}^1$-bundle over $Y$, and suppose that $D,D'$ are ...
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1
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Restriction of a torus invariant divisor to an affine toric subvariety
Using the notation from Cox, let $\Sigma$ be a fan and $X_\Sigma$ the correcponding toric variety. Let $D=\displaystyle\sum_{\rho\in\Sigma(1)}a_\rho D_\rho$ be a Weil divisor. For every open subset $U\...
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The concept of zeroes and poles of a function on an algebraic curve
I'm going to understanding the Weil pairing and other elliptic-curve targeted stuff. And now I'm stuck on understanding the concept of a divisor, which is defined as a formal list of poles and zeroes. ...
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2
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220
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Learning Weil divisor, divisor class group ( Understanding Hartshorne Example 6.5.2. more rigorously )
I am reading the Hartshorne, Algebraic Geometry, Example 6.5.2. and stuck at some statement.
Let $k$ be a field with $A:= k[x,y,z]/(xy-z^2)$. Let $X:= \operatorname{Spec}A$. Let $Y:=V(( \bar{y}, \bar{...
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Cassels's proof that isogeny of elliptic curves preserves group law
I'm reading Cassels's Lectures on Elliptic Curves, namely his proof that 2-isogeny $\phi:C \to D$, where $C : y^2 = x(x^2 + ax + b)$ and $D: \mu^2=\lambda(\lambda^2 - 2a\lambda +(a^2 - 4b))$ ...
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2
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If two prime divisors on a Riemann surface $X$ are equivlant, then $X$ is the Riemann sphere
Given a connected compact Riemann Surface $X$, show that $X$ is the Riemann sphere if and only if there exists two points $P,Q\in X$ that are linearly equivalent.
If $X$ is the sphere, then given a ...
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$D$ is Pseudo-effective if $f^*D$ is Pseudo-effective?
Let $f:X\rightarrow Y$ be a surjective morphism with connected fibers between normal projective varieties, and let $D\subset Y$ be an $\mathbb{R}$-Cartier divisor. Suppose that $f^*D$ is pseudo-...
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262
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Intersection pairings on divisors on Calabi-Yau threefolds
Let $X$ be a Calabi-Yau threefold, i.e. a compact Kähler manifold with trivial canonical bundle of complex dimension three, and $D$ a (Weil) divisor on $X$. My question is:
What can be said about the ...
- 53
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0
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45
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Limit of irreducible (rational) curves is connected
Let $S$ be an algebraic surface, $C\subset S$ an irreducible rational curve. Suppose $\dim |C|\geq 1$ so that we can choose a pencil $\{C_\lambda\}_{\lambda \in \mathbb{P}^1}\subset |C|$ containing $C$...
- 2,366
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99
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Computing the inverses of two linear maps
I have two linear maps $X: \mathcal{L}(G) \rightarrow \mathbb{F}_q^k$ and $Y: \mathcal{L}(2G) \rightarrow \mathbb{F}_{q^n}$, where $G$ is a divisor of the rational function field $F_q(x)$ over $\...
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If $D$ is a divisor on a curve and $P$ is not a base point of $D$, then $\ell(D-P)=\ell(D)-1$
In Hartshorne's Algebraic Geometry, Chapter IV, remark 4.10.9, $X$ is a curve and $K$ is a canonical divisor. He states that $l(K) = g$ and $l(K-P_1) = g-1$ if we take $P_1$ not a base point of $K$.
I ...
- 1,757
1
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0
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42
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Global sections of a line bundle associated to a non-effective divisor
The following is a proof of the index theorem for divisors given in pp.472-473 of Griffiths-Harris.
Let $M$ be a projective surface, and $E$ a positive divisor, meaning that the Chern class of the ...
- 2,366
1
vote
1
answer
79
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Positive divisors on a projective surface
Let $M$ be an (projective) algebraic (nonsingular) surface and $L$ a line bundle on $M$.
In p.472 of Griffiths-Harris, the formula $$\chi(L)=\chi(\mathcal{O}_M)+\frac{L^2-LK}{2} $$
is proved. The ...
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2
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1
answer
84
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Intersection of all divisors in a linear system
I am reading Griffiths-Harris algebraic geometry, and in p.137 it is written that if $E=\{D_\lambda\}_{\lambda \in \mathbb{P}^n}$ is a linear system, then for any $\lambda_0,\dots,\lambda_n$ linearly ...
- 2,366
2
votes
1
answer
96
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algebraic Cartier divisors vs. analytic Cartier divisors
Suppose $(X, \mathcal{O}_X)$ is a smooth $\mathbb{C}$-variety. Let $X^h$ denote its analytification, so that $(X^h, \mathcal{O}_{X^h})$ is considered as a complex manifold with its sheaf of analytic ...
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1
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0
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66
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Cartier divisors on reduced Noetherian schemes “described” by integral open sets
I'm currently reading Olivier Debarre's book “Higher-Dimensional Algebraic Geometry” and I'm stuck on a small remark made in the definition of Cartier divisors (it's on page 3 if you have the book). ...
- 968
0
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0
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83
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Divisor class group of non-reduced schemes
Let $X$ be an irreducible scheme, separated of finite type over a field. Let $i:X_{\mathrm{red}}\to X$ be the associated reduced closed subscheme. Is the Weil divisor class group (i.e., the quotient ...
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