The picard-group tag has no wiki summary.

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### Picard sequence for sujective morphisms

Given $\phi:X\rightarrow Y$ a surjective morphism of $k$-algebraic varieties ($k$ separably closed), I wanted to find how the write an exact sequence involving Pic(X) and Pic(Y). We can use the long ...

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### Severi's theorem of base and Hilbert polynomial

Let $X$ be a smooth projective variety over $\mathbb{C}$ satisfying $H^1(\mathcal{O}_X)=0$. Fix $i:X \to \mathbb{P}^n$ a closed immersion and let $\mathcal{O}_X(1)$ be the corresponding very ample ...

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### Relative Picard functor for the Zariski topology

I'm trying to understand better the relative Picard functor, as defined, for example, in Kleiman's article.
Let $X \to S$ be a smooth projective morphism of schemes whose geometric fibres are ...

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### Picard group of classifying stack

Suppose $S$ is a scheme, and $G$ a smooth $S$-group scheme.
Then there exists an algebraic stack BG called the classifying stack of $G$, defined as the quotient stack $[S/G]$ where $G$ acts trivially ...

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### Does a line bundle on a normal Noetherian algebraic space come from a Weil divisor?

Let $X$ be a normal Noetherian algebraic space and $\mathscr{L}$ a line bundle on $X$. If $X$ is a scheme, then there is locally principal Weil divisor on $X$ that gives rise to $\mathscr{L}$. Is the ...

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### Does a semistable curve descend to a regular base?

Let $f\colon X \rightarrow S$ be a semistable curve of genus $g \ge 0$. Being a semistable curve means that $f$ is a morphism of schemes such that
$f$ is proper, flat, and of finite presentation;
...

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### Jacobian of a semistable curve

My question is about the proof of Example 8 in section 9.2 of the book "Neron models." There we have a semistable curve $X$ over an algebraically closed field $K$ and we let $\pi\colon \widetilde{X} ...

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### Fiberwise vanishing of $H^2$ and formal smoothness of the Picard functor

My question is about the proof of 8.4/2 in "Neron models." The claim is that if $f\colon X \rightarrow S$ is a proper flat morphism of finite presentation such that $H^2(X_s, \mathscr{O}_{X_s}) = 0$ ...

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### Picard functor of an algebraic group

Let $K$ be a field, $G$ a $K$-group scheme of finite type, and $X$ a $G$-torsor. Is the Picard functor $\mathrm{Pic}_{X/K}$ representable? I recall that $\mathrm{Pic}_{X/K}$ is the fppf sheafification ...

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### Does every relative curve have a Picard scheme?

More precisely:
Let $X \to S$ be a smooth proper morphism of schemes such that the geometric fibers
are integral curves of genus $g$. Must the fppf relative Picard functor
$\operatorname{\bf ...

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### Picard scheme of varieties over imperfect fields

Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite ...

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### On a property of the Grothendieck group of a smooth projective curve

Let $K$ be a complete DVR of characteristic $0$, $X$ a smooth projective curve over $K$. Denote by $K^0(X)$ the Grothendieck group of locally free sheaves on $X$ and by $\mbox{det}$ the natural group ...

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### Jacobian of a curve and field extension

Let $K$ be a field of characteristic zero and $X_K$ a smooth projective curve on $K$. Denote by $\bar{K}$ the algebraic closure of $K$ and $X_{\bar{K}}$ the base change of $X_K$ to $\bar{K}$. Under ...

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### Injectivity under flat base change of the Picard group on smooth projective curves

Let $K$ be a field of characteristic $0$, $X_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X_{\bar{K}} \to X_K$, induces via the ...

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### Picard group generated by effective divisors: counterexample?

Let $X$ be an integral variety defined over an algebraically closed field $k$ of characteristic 0 with finitely generated Picard group $Pic(X)$ and such that $k[X]^\times=k^\times$ (i.e. the only ...

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### adjacency matrix of a graph and lines on quartic surfaces

Suppose you are given a smooth quartic surface $X$ in $\mathbb P^3$. I would like to find an upper bound for the number of lines on $X$ in the case that there is no plane intersecting the curves in ...

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### Picard group of infinite direct product of DVRs trivial

Let $R = \prod_{n\in\mathbf{N}}R_n$ be an infinite direct product of discrete valuation rings $R_n$. Why is $\mathrm{Pic}(R) = 0$?

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### Picard group of Drinfeld upper half space

Let $K$ be a $p$-adic field and $\Omega^{(n)}_K$ the $n$-dimensional Drinfeld upper half space over $K$ (which is a rigid analytic space over $K$).
Is the Picard group of $\Omega^{(n)}_K$ known? ...

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### divisors on $\overline{\mathcal{M}}_{g,n}$ that are trivial on certain $F$-curves

Inside the moduli space of curves $\overline{\mathcal{M}}_{g,n}$ one can distinguish two classes of $F$-curves isomorphic to $\mathbb{P}^1$: those of type $\overline{\mathcal{M}}_{0,4}$, and those of ...

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### Elegant definition for the scheme parametrizing $g_d^r$'s on a curve

Let $X$ be a smooth projective curve over $k=\bar{k}$, and $Pic^d$ the $d$-part of the Picard group of $X$ (isomorphism classes of line bundles of degree $d$ on $X$).
I'd like to define a scheme ...

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### Hypersurfaces with Picard group generated by classes of lines on the same plane

For which values of $d$ is the following possible: There exist a smooth hypersurface $X$ in $\mathbb{P}^3$ of degree $d$ with Picard number $d$, containing $d-1$ lines $l_1,...,l_{d-1}$ on the same ...

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### $Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space

There's a natural map $f:\overline{\mathcal{M}}_{1,1}\to \overline{M}_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence ...

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### Relatively numerically trivial divisor

Hi,
Let $f : X \rightarrow Y$ a projective morphism of quasi-projective algebraic varieties over $\mathbb{C}$. Assume that $X$ is smooth, that $Y$ is normal and that:
$$\textbf{R} f_* \mathcal{O}_X ...

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### Definition of relative Picard functor

Let $X \to S$ be a morphism of schemes. The relative Picard functor from schemes over $S$ to abelian groups is usually defined by the formula $T \mapsto \text{Pic}(X \times_S T)/\text{Pic}(T)$ (e.g. ...

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### Line bundles on a pointless curve

Let $X$ be a smooth projective curve over a field $k$. In chapter 8 of the book Neron Models by Bosch et al., there is a general result (namely Proposition 4) which implies that if $X$ admits a ...

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### $Pic(X)/l=0$ in terms of $H^*_{et}(X,\mu_{l^n})$?

I would like to calculate Picard groups of certain schemes over fields; I'm mostly interested in the question whether $Pic(X)$ is infinitely $l$-divisible, i.e. whether $Pic(X)/l=0$, $l$ is a prime ...

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### Blowing down -1 curves

After a good look for anything in the way of an explicit example, and harassing the algebraic geometers in the department, I still am unable to find an answer to my question, so any light will be very ...

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### Picard group of a singular projective curve

Let $X$ be a singular irreducible projective curve over an algebraically closed field and $\pi : \widetilde{X} \to X$ the normalization morphism. In the book on Neron models by Bosch et al. (I have ...

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### Picard/cohomology lattice of surfaces of low degree in $\mathbb P^3$

Let $S_{d>3}\subset\mathbb{P}^3_{\mathbb{C}}$ be a smooth surface of degree $d$. What is known (where to read?) about the Picard/cohomology lattice for small d?
e.g. for $d=4$ the cohomology ...

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### Is Pic( G((z)) ) = $\mathbb{Z}$?

There are a fair number of papers by Beauville, Laszlo, Sorger, Kumar and others on the geometry of $LG/L^+G = G((z))/G[[z]]$ where $G$ is a simply connected and simple group over $\mathbb{C}$. In ...

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### $\psi$ class in $\overline{M}_{0,n}$

Basic question, but I found no reference.
Is the $\psi$ class the only one which is not a boundary class in the PIcard group of the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$? Or can it ...

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### How can one check that two line bundles on $\overline{M}_{0,n}$ coincide?

Let $X$ be the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$. Suppose I have two (big) line bundles $L$ and $L'$ on $X$ and that I want to show that they are the same element of $Pic(X)$. Of ...

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### Picard group of $\mathcal{M}_{0,n}$

Let $\mathcal{M}_{0,n}$ be the complement of the boundary of the Mumford-Knudsen compactification of the moduli space of genus zero, n-pointed curves.
Is $Pic(\mathcal{M}_{0,n})$ trivial?

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### On generators of the Picard group of a projective smooth surface over a finite field

Let $X$ be a smooth projective surface over a finite field $k=\mathbb{F}_q$. Let us first review the proof of the finite generation of $Pic(X)$ (notice that the proof is valid for any smooth ...

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### What is the Picard group of a Schubert variety in the affine Grassmannian?

I'm not sure I have a lot more to say than the title. Let $G$ be your favorite simple algebraic group over $\mathbb{C}$, and let $$\overline {\mathrm{Gr}}_\lambda= \overline{G(\mathbb{C}[[t]])\cdot ...

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### Picard group, Fundamental group, and deformation

One of the most elementary theorems about Picard group is probably $\mathrm{Pic} (X \times \mathbb{A}^n) \cong \mathrm{Pic} X$ and $\mathrm{Pic} (X \times \mathbb{P}^n) \cong \mathrm{Pic} X \times ...

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### $G_m$-cohomology of a motif (that corresponds to a stack?)

As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety
I am in the following situation: $G$ is an algerbraic group, and X is a smooth ...

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### A motivic complex

By definition, Voevodsky's motivic complex (an object of his $DM^{eff}_-$) is a complex of sheaves with transfers whose cohomology sheaves are homotopy invariant. Now, I consider the complex (of ...

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### Is the ring of all cyclotomic integers a Bezout domain?

My previous question about the theorem (apparently due to Dedekind -- thanks, Arturo Magidin!) that the ring $\overline{\mathbb{Z}}$ of all algebraic integers is a Bezout domain got me thinking about ...

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### What is ample generator of a Picard group?

First a note of caution: I am a physicist with a rudimentary knowledge of algebraic geometry picked up here and there. So don't assume I know anything besides basic properties of sheaves and try to ...

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### An example where $Pic(X) = H^0(k,Pic(\overline{X}))$?

Let $X$ be a geometrically integral smooth projective variety over a number field $k$. Then if $X$ is everywhere locally soluble, we have $Pic(X) = H^0(k,Pic (\overline{X}))$, where $\overline{X}=X ...

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### What does the Riemann-Hurwitz formula tell us on the Picard variety

Let $f:X\longrightarrow Y$ be a finite separable morphism of smooth projective integral curves over an algebraically closed field.
Then we have a linear equivalence of Weil divisors on $X$: $$ ...

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### Two questions about finiteness of ideal classes in abstract number rings

Let us say that an abstract number ring is an integral domain $R$ which is not a field, and which has the "finite norms" property: for any nonzero ideal $I$ of $R$, the quotient $R/I$ is finite.
(I ...

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### Abel-Jacobi map for regular fibered surfaces.

Let $f:C\to S$ be a regular fibered surface where $S=Spec(R)$, $R=dvr$. Assume $C$ has smooth geometrically integral generic fibre $C_K$. We also assume the existence of a section $x\in C(S)$. Let ...

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### Picard groups of non-projective varieties

As far as I know, the main representability result for the relative Picard functor $Pic_{X/k}$, for a noeth. sep. scheme of finite type over a field $k$ is:
If $X$ is proper then $Pic_{X/k}$ is ...

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### Torsion-freeness of Picard group

Let $X$ be a complex normal projective variety.
Is there any sufficient condition to guarantee the torsion-freeness of Picard group of $X$?
One technique I sometimes use is following:
If $X$ can be ...

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### Seeking Noetherian normal domain with vanishing Picard group but not a UFD

Once again, the question says it all.
My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...

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### Picard Groups of Moduli Problems

First, yes, I've seen Mumford's paper of this title. I'm actually interested in specific ones, and looking for really the most elementary/elegant proof possible.
I'm told that for $g\geq 2$ it is ...

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### For a line bundle L on a smooth projective variety X, what is meant by Pic^L(X)

Hi everyone,
Let $X$ be a smooth projective variety over a field $k$ and let $L$ be a line bundle on $X$. I'm reading the article Heights for line bundles on arithmetic varieties and there one ...

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### What is the right definition of the Picard group of a commutative ring?

This is a rather technical question with no particular importance in any case of actual interest to me, but I've been writing up some notes on commutative algebra and flailing on this point for some ...