Questions tagged [picard-scheme]

In algebraic geometry, the scheme that represents the Picard functor and the natural generalisation of the Picard variety for a given algebraic variety to the theory of schemes. If your question is not about algebraic or arithmetic geometry, then this is likely not the right tag to use.

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"Universal" line bundle over the Picard variety $\operatorname{Pic}^0(X)$.

$\DeclareMathOperator{\Pic}{Pic}$This question is inspired by my attempt to answer another question. At the end of my answer, there is a missing step, which I don't know how to fill. Let $X$ be a ...
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Picard's theorem and Cauchy sequences in different norms

I understand that in order to prove the existence and uniqueness of a solution to a first order non-linear ODE, we first convert it into an integral equation: $y_{n+1} := Ty_{n} = y_{0} + \int_{0}^{t}...
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Iterative scheme of succesive approximations for a particular integral equation.

Consider the integral equation $$ y(x) = x + \int_0^x (t-x)y(t) \, dt, \quad x \in [0,1].$$ My goal is to use the iterative scheme of succesive approximations to find the unique solution of the ...
xyz's user avatar
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On the Picard group of the affine n-space over a normal affine base scheme [duplicate]

Question: Let $S=Spec(\Lambda)$ be a normal affine scheme. Consider the morphism of schemes $\mathbb{A}^{n}_{S}=Spec(\Lambda[x_{1}, \ldots, x_{n}]) \rightarrow S$. Is it true that the induced map $Pic(...
PIELEO13's user avatar
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Embedding of Picard functor into $\text{Hom}_k(-,\text{Pic}(X/k))$

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\pr{pr}$ Let $X$ be an algebraic variety $X$, that is proper over $k$ (here a variety is a ...
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Global sections of algebraically trivial line bundle

Let $\pi \colon X \rightarrow C$ be a smooth minimal elliptic surface over an algebraically closed ground field $k$. Furthermore assume that $\pi$ has a section and that the fundamental line bundle $R^...
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What is the union of picard variety?

This question comes from a defnition of FGA explained, the chapter of Picard scheme, wirtten by Professor Dr. Steven L. Kleiman. Suppose $X/S$ is an abelian scheme over $S$, my question is how to ...
Vector's user avatar
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Basic questions on Picard group (references)

I want a reference/explanation for things as follows. For nice enough commutative ring $A$, we have $Pic(A[[x]]) \cong Pic(A)$. Also, I want to know about comparing $Pic(A((x)))$ and $Pic(A)$, for ...
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Fibers of the Abel Jacobi map over curves

I am studying the Abel Jacobi map $$\mathrm{Div}_{X/k} \to \mathrm{Pic}_{X/k}$$ for projective, smooth, irreducible curve $X/k$ where $k$ is algebraically closed. Let $S = \operatorname{Spec}(k)$, $T$ ...
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Assumptions on a scheme X needed to construct the Picard scheme $\operatorname{Pic}(X)$

Let $X$ be a scheme. I believe that Grothendieck was the first to put a scheme structure on the Picard group $\operatorname{Pic}(X)$, where he assumed that X is projective and reduced. Then Mumford ...
Robert Hanson's user avatar
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Structure ring of constant group scheme.

For the finite abelian group $G$, the group scheme $G_{\mathrm{Spec}\,{\Bbb Z}} = {\mathrm{Spec}}\,{\cal O}_G$ over ${\mathrm{Spec}}\,{\Bbb Z}$ is defined as follows$\colon$ $$ {\cal O}_G = {\Bbb Z}...
Pierre MATSUMI's user avatar
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A condition that implies $f_* \mathcal{O}_X \cong \mathcal{O}_S$

I am reading the lecture note of Dori Bejleri about Picard schemes: https://people.math.harvard.edu/~bejleri/teaching/math259xfa19/math259x_lecture12.pdf In Example 12.8, I don't understand why the ...
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Interpreting some cohomology groups

Let $C$ be a smooth geometrically integral curve over a number field $k$, we do not assume $C$ to be proper, i.e., $C$ is not projective. Under the spectral sequence $H^p(k,H^q_{\mathrm{et}}(C_{\bar{k}...
oleout's user avatar
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How to prove $\operatorname{Pic}(\mathbb{P}_X^n)\cong\operatorname{Pic}(X)\times\mathbb{Z}$?

Let $X$ be a Noetherian regular scheme. Then how can one prove $\operatorname{Pic}(\mathbb{P}_X^n)\cong\operatorname{Pic}(X)\times\mathbb{Z}$? I want to use this specific case for the more general ...
Redundant Aunt's user avatar
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very slow convergence of Picard method for solving nonlinear system of equations

I have a nonlinear system of equations as $$ \left(\mathbf{K}_{\mathbf{L}}+\mathbf{K}_{\mathbf{N L}}(\mathbf{X})\right) \mathbf{X}=\mathbf{F} $$ in which $\mathbf{K}_{\mathbf{N L}}(\mathbf{X})$ ...
omgtheykilledkenny's user avatar
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Example of Picard number in family of smooth variety jumping

For a scheme or formal scheme $X$, let $\mathrm{Pic} X$ be its Picard group. If $X$ is a smooth proper variety over an algebraically closed field, let $\mathrm{Pic}^{0}(X)$ be the subgroup consisting ...
Mike's user avatar
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Using Picards theorem to show that the initial value problem has a unique solution

I am trying to show that the IVP $$x'=\sqrt{x(t)}+1, t\in[0,1],\\x(0)=0, (t_0=0)$$ has a unique solution and show whether the initial value problem satisfies the assumptions of Picard’s Theorem, ...
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Morphism of curves and Jacobian

Let $k$ be a finite field, and let $X$ and $Y$ be some curves ($k$-variety of dimension $1$ with all the goods properties we want), and $\pi : X \rightarrow Y$ be a morphism. Then, it induces a ...
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Coarse moduli space of relative Picard functor for affine line

Consider the relative Picard functor $\mathrm{Pic}_{\mathbb A^1/\mathrm{Spec}(\mathbb C)}$ sending a complex scheme $X$ to $\mathrm{Pic}(X \times \mathbb A^1)/\pi_X^* \mathrm{Pic}(X)$. Since $\mathrm{...
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When $\phi_{\mathcal L}=0$ for $\mathcal L$ a line bundle over an abelian scheme $X/S$

Let $X\rightarrow S$ be a projective abelian scheme. To a line bundle $\mathcal L$ on $X$, we associate its Mumford line bundle $\Lambda(\mathcal L):= \mu^{\star}\mathcal L\otimes p_1^{\star}\mathcal ...
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Is the Poincaré sheaf symmetric?

The following discussion is based on the content of FGA explained about the Picard scheme. This is mostly formal: I am trying to find a good way to think about the Poincaré sheaf. Let us consider $\...
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Divisorial sheaves as sheaves on $\operatorname{Pic}(X)\times X$

I'm not very skilled in sheaf theory, but my question is the following: Consider a family of sheaves $\{\mathcal{O}(D)\}$ where $[D]\in\operatorname{Pic}(X).$ Is it possible to define some scheme (I ...
Grisha Taroyan's user avatar
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Polarization of Picard variety

Let $X$ be a complex projective manifold of dimension $m$ with positive closed $(1,1)$ form $\omega$ induced by a projective embedding and let $Pic^0(X)$ be the associated Picard torus. This is a ...
Tommaso Scognamiglio's user avatar
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Computing the Picard group of ${\rm Spec}\left(\frac{k[x,y]}{xy(x+y+1)}\right).$

I’m trying to compute the Picard group of ${\rm Spec}\left(\frac{k[x,y]}{xy(x+y+1)}\right)$ where $k$ is a field. The question came up when I was trying to compute the Picard group of a ‘triangle’ ...
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The sheafification of the relative Picard functor

In the proof for the second part of theorem 2.5 found in Kleiman's paper https://arxiv.org/pdf/math/0504020.pdf, he claimed that any $\lambda\in \text{Pic}_{(X/S)(\text{fppf})}(T)$ can be represented ...
chi-yu Cheng's user avatar
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Motivation for Construction of Jacobian

$\DeclareMathOperator{\Pic}{Pic}$ Hello everybody, in a course on the Jacobians of Curves, the lecturer gave the following Motivation for the Construction of the Jacobian: Let $D$ be a divisor on a ...
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Irreducibility of the Jacobians of a curves.

I'm studying Jacobian varieties.I assume that the existence of the Jacobian variety for a curve and attempt to show irreducibility of the Jacobian for a curve according to Remark:IV.4.10.9 of ...
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2 answers
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Picard's method does not solve first order differential equation?

I have the following first order differential equation $$x^\prime(t)=-(x(t))^2+2x(t),\quad t\geq 0,\quad x(0)=1$$ Now I want to obtain an approximation of $x(t)$ by using Picard's method. Then the ...
Wim Verboom's user avatar
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1 answer
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Definition of $\operatorname{Pic}^0(V)$ for $V$ a singular variety

How does one define the $\operatorname{Pic}^0(V)$ for $V$ being a singular, not necessarily normal variety? Until now the approach I found by searching Google is to prove that the Picard functor is ...
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Convergence of the Picard sequence

Consider the Cauchy problem $$ \begin{cases} y'= \cos(y)=f(x,y)\\ y(0)=0 \end{cases} $$ The question is: Does the Picard sequence converge? My attempt: We have that $y_0=0$, $y_1(x)= x$, $y_2(x)=...
rosy's user avatar
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1 answer
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Picard group of a fibration

Assume that $X$ is a projective variety. Let $Pic(X)$ be its Picard group. Let $E$ be a vector bundle over $X$ say of rank $r$ (for example TX). What is the picard group of the total space of $E$? ...
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Picard Iteration: Convergence of system [duplicate]

I want to prove that, for any $t$, a solution exists in the interval $[0,T]$, when $T>0$. $x'(t)=A(t)x(t)$ My question is quite similar to this one Picard iteration (general), but with one small ...
Chemical Engineer's user avatar
3 votes
1 answer
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First Order Time-Variant System : Picard Method

I looking at a time variant first order system. I am trying to prove that a sequence of functions $x^{[k]}(t)$ generated using Picard iterations converges uniformly on some interval $[0,T]$. Given: $...
Chemical Engineer's user avatar
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Solution of differential equation $x'=\cos(x),$ with condition $x(0)=0$

Write out the Picard iteration scheme. If possible, find the solution. $x'=cos(x),x(0)=0$ I did the picard iterations but I don't know how to get the solution from this $$u_0=0,...,u_3=\int_0^t\cos(\...
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5 votes
1 answer
288 views

The Mumford line bundle of $(-1)^* L$

Let $X$ be an abelian variety over a field $k$, $L$ a line bundle on $X$. Let $\varphi_L : X \to X^t$ be the morphism obtained by considering the Mumford line bundle $\Lambda (L) = m^*L \otimes p_1 ^...
Future's user avatar
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3 votes
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Questions about the connected component of a relative Picard Scheme.

Let $X$ be a smooth, projective surface (i.e. $2$-dimensional connected variety) over $k=\mathbb{C}$. Denote by $\mathrm{Pic}_{X/k}$ the associated relative Picard scheme. We write $\mathrm{Pic}^0_{X/...
Louis's user avatar
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why are picard groups called picard groups

I am looking for an origin of the picard groups. Unfortunately I cann't find a reference where this is mentionned. Does anybody know where the name comes from?
user28082's user avatar
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Pushforward of algebraic cycles

Let $f: X \to Y$ be a proper morphism of smooth integral projective varieties over an algebraically closed field $k$ of characteristic 0, where $dim X = dim Y = n$. Denote by $CH_i(W):= Z_i(W)/\sim$ ...
Jamie's user avatar
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2 votes
1 answer
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Picard's iteration method

I want to find a series of functions converging to the solution of $$\frac{dy}{dx}=\frac{x^2}{y^2+1},y(0)=0$$. I am stuck using picard's iteration method First iteration: $$y_1 = \frac{x^3}{3}$$ ...
Waseem Khan's user avatar
1 vote
1 answer
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Using Picards theorem to find unique interval

Consider the initial value problem: $\frac{dy}{dx}$= $xy - x^2 + 1$ with $y(0) = 0$ In order the find the unique interval we first find that $f(x,y)$ and $f_y$ are continuous in the rectangle: $...
user107783's user avatar
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1 answer
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Inversion of an element in Picard group over commutative ring

I'm having some troubles understanding a proof in Commutative Algebra Chapter I - VII of N. Bourbaki. It's on pag 114 of the book. Here's what it says: Theorem 3 ... (ii) Conversely, if $M$ is an $A-$...
user49685's user avatar
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Picard group schemes of degree d

Let $C$ be a smooth curve. I know that $Pic^0(C)$, i.e. the Picard group of degree 0 line bundles on $C$, is isomorphic to the jacobian $J(C)$, so it is an abelian variety. My question is, what about $...
Larry Harvey's user avatar
5 votes
1 answer
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Group law for an elliptic curve using schemes

I was trying to understand better the definition of the group law for an elliptic curve given in Katz and Mazur's book (http://books.google.com.br/books/about/Arithmetic_Moduli_of_Elliptic_Curves.html?...
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