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.
604 questions
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Generalisation of the inequality $D.C\geq\operatorname{mult}_x(D)\cdot\operatorname{mult}_x(C)$.
Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field, $x\in X$ a closed point, $C\subset X$ an integral curve containing $x$, $Y\subset X$ a prime divisor ...
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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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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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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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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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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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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 ...
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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, \...
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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 ...
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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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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 ...
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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 ...
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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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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(...
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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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