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.
32
questions
1
vote
0answers
33 views
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}...
2
votes
1answer
75 views
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 ...
4
votes
2answers
146 views
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 $\mathbf{K}_{\mathbf{N L}}(\mathbf{X})$ ...
1
vote
2answers
70 views
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 ...
0
votes
1answer
32 views
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, ...
1
vote
1answer
61 views
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 ...
0
votes
0answers
15 views
Picard-Iteration for SDEs: why are the $X_n$ progressively measurable?
I need an argument why the $X_n$, and especially $\sigma(s,X_s^n)$, in the standard Picard-Iteration are progressively measurable:
$$X_t^n=X_0 + \int_0^t \mu(s,X_s^{n-1})ds + \int_0^t \sigma (s,X_s^{n-...
9
votes
2answers
237 views
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{...
8
votes
0answers
173 views
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 ...
2
votes
0answers
50 views
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 $\...
0
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0answers
49 views
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 ...
2
votes
0answers
61 views
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 ...
5
votes
1answer
251 views
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’ ...
0
votes
1answer
53 views
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 ...
1
vote
0answers
40 views
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 ...
1
vote
0answers
75 views
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 ...
2
votes
2answers
263 views
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$$
Now I want to obtain an approximation of $x(t)$ by using Picard's method. Then the ...
1
vote
1answer
148 views
Definition of $\operatorname{Pic}^0(V)$ for $V$ a singular variety
How does one define the $\operatorname{Pic}^0(V)$ for $V$ being a singular, not necessarily normal variety?
Until now the approach I found by searching Google is to prove that the Picard functor is ...
0
votes
1answer
138 views
Convergence of the Picard sequence
Consider the Cauchy problem
$$
\begin{cases}
y'= \cos(y)=f(x,y)\\
y(0)=0
\end{cases}
$$
The question is: Does the Picard sequence converge?
My attempt: We have that
$y_0=0$, $y_1(x)= x$, $y_2(x)=...
0
votes
1answer
269 views
Picard group of a fibration
Assume that $X$ is a projective variety. Let $Pic(X)$ be its Picard group. Let $E$ be a vector bundle over $X$ say of rank $r$ (for example TX).
What is the picard group of the total space of $E$? ...
1
vote
0answers
53 views
Picard Iteration: Convergence of system [duplicate]
I want to prove that, for any $t$, a solution exists in the interval $[0,T]$, when $T>0$.
$x'(t)=A(t)x(t)$
My question is quite similar to this one Picard iteration (general), but with one small ...
3
votes
1answer
159 views
First Order Time-Variant System : Picard Method
I looking at a time variant first order system. I am trying to prove that a sequence of functions $x^{[k]}(t)$ generated using Picard iterations converges uniformly on some interval $[0,T]$.
Given:
$...
2
votes
0answers
411 views
Solution of differential equation $x'=\cos(x),$ with condition $x(0)=0$
Write out the Picard iteration scheme. If possible, find the solution.
$x'=cos(x),x(0)=0$
I did the picard iterations but I don't know how to get the solution from this
$$u_0=0,...,u_3=\int_0^t\cos(\...
5
votes
1answer
158 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 ^...
3
votes
0answers
171 views
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/...
0
votes
2answers
88 views
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?
4
votes
0answers
362 views
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 $CH_i(W):= Z_i(W)/\sim$ ...
2
votes
1answer
679 views
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:
$$y_1 = \frac{x^3}{3}$$
...
1
vote
1answer
49 views
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:
$...
0
votes
1answer
70 views
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 $A-$...
0
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0answers
202 views
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 $...
4
votes
1answer
324 views
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?...
