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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.

STUB: someone more knowledgeable please edit!

References:

Springer EOM

nLab

46 questions
11
votes
2 answers

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{Pic}(\mathbb A^1) = \{\mathcal O_{\mathbb…
JoS
  • 235
8
votes
0 answers

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 $\Lambda(\mathcal L):= \mu^{\star}\mathcal L\otimes p_1^{\star}\mathcal L^{-1}\otimes p_2^{\star}\mathcal L^{-1}$ on…
Suzet
  • 6,055
6
votes
1 answer

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 $S = \operatorname{Spec}(k)$, $T$ a scheme over $k$ and let $\mathcal{L}$ be a…
bjck
  • 81
5
votes
1 answer

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?id=M1IT0J_sPr8C&redir_esc=y). They define an…
user40276
  • 5,543
5
votes
1 answer

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 $R^1\pi_*(\mathcal{O}_X) \in \text{Pic}(C)$ has…
RobertMuller
  • 771
5
votes
1 answer

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’ over $k$ (I.e Three copies of $\mathbb{A}^1_k$ glued…
User20354
  • 1,070
5
votes
1 answer

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 $\Lambda (L) = m^*L \otimes p_1 ^* L^{-1} \otimes p_2 ^* L^{-1}$ on $X \times X$…
Future
  • 4,582
4
votes
0 answers

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 constructed a Picard scheme in the non-reduced…
Robert Hanson
  • 91
4
votes
3 answers

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 $\mathbf{K}_{\mathbf{N L}}(\mathbf{X})$ represents the nonlinear stiffness matrix which is…
omgtheykilledkenny
  • 43
4
votes
1 answer

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 $E$? can we say something about it?
Z.A.Z.Z
  • 577
4
votes
0 answers

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 $\mathrm{Pic}^0_{X/k}$ for the connected component of the identity.…
Louis
  • 3,499
4
votes
0 answers

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 $CH_i(W):= Z_i(W)/\sim$ be the Chow group of $i$-cycles on $W$, where…
Jamie
  • 41
3
votes
1 answer

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 scheme $X/k$ such that$\overline{X}= X \times…
user267839
  • 9,217
3
votes
2 answers

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}e_0 \oplus {\Bbb Z}e_1 \cdots \oplus {\Bbb…
Pierre MATSUMI
  • 923
3
votes
0 answers

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 smooth and irreducible generic fiber implies that…
hungnk1998
  • 31
1
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