# Tag Info

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### Relative Picard functor for the Zariski topology

Let $Y$ be Cayley's nodal cubic surface over the complex numbers, given in $\mathbb{P}^3$ by $X_0X_1X_2 + X_0X_1X_3 + X_0X_2X_3 + X_1X_2X_3 = 0.$ This surface has four simple double points. It also ...

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### Picard group of connected linear algebraic group

$\DeclareMathOperator\Pic{Pic}$The statement is false over most imperfect fields, even for smooth affine group schemes. In particular, it is false over any separably closed imperfect field $k$. I will ...
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### p-torsion in the Picard group of a regular projective curve

Take $p=3$ and $C\subset \mathbb{P}^2$ with equation $y^2 z=x^3 - t z^3$ where $t\in K$ is not a cube. Then $C$ is regular but, putting $L:=K(t^{1/3})$, $C_L$ is isomorphic to the usual cuspidal cubic ...
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### Reference request: Generic k3 surface has Picard number 1

Welcome new contributor. I am just writing my comment as an answer, and expanding on the observation of Prof. Arapura. For a smooth, projective scheme $X$ over a field $k$, the space of first order ...
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### Does every relative curve have a Picard scheme?

The answer is yes: $\mathbf{Pic}_{X/S}$ is representable by a scheme. I will argue that this follows from the SGA 3 result mentioned by user27920 and from Theorem 2 (c) in section 6.6 of Neron models (...
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### Are there varieties with non finitely generated Picard group and vanishing irregularity?

As pointed out by Jason Starr, the answer to your last question is yes, so the answer to the question in your title is no. Let me give a quick (and straightforward) proof in the case $k= \mathbb{C}$. ...
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### Is $Pic^0(X)$ of a curve of genus $\geq 1$ over a non-algebraically closed field still non-finitely generated?

As Will Sawin says, the key phrase here is Mordell-Weil theorem. Also, this isn't really a theorem about Picard groups of curves, it's a theorem about abelian varieties. Here is a fairly general ...
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### On a morphism from the Brauer group to the Picard group

It seems to me that the involution of $Q \otimes Q$ that exchanges $a \otimes b$ and $b\otimes a$ is inner, which means that the homomorphism you describe should always be trivial. Here is the ...
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### Picard-surjectivity and Morita-equivalence

Yes, the basic algebra of $A$ will be Picard-surjective. The basic algebra is the endomorphism algebra $\operatorname{End}_A(\bigoplus_{i=1}^{n}P_i)$ of the direct sum of indecomposable projective (...

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

1) About the second definition: $\alpha$) It is not true that for an arbitrary ring a) is equivalent to c): Indeed Bourbaki in Algèbre commutative, Chapitre II, Exercices §5, 12) c) exhibits a ring ...
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### Injectivity under flat base change of the Picard group on smooth projective curves

This map is injective. There is a Hochschild-Serre spectral sequence with $E^{pq}_2=H^p(\mathrm{Gal}(\bar{K}/K), H^q(X_{\bar{K}},\mathbb{G}_m))$ converging towards $H^{p+q}(X_{K},\mathbb{G}_m)$. This ...

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

Yes this is true. It can be proved using the Hochschild-Serre spectral sequence plus Hilbert's theorem 90 (it is true more generally for any geometrically connected projective variety $X$). The ...

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### Why is it useful for the (relative) Picard functor to be representable?

Let me give one cute example: the Theorem of the Cube (cf. e.g. Mumford's "Abelian varieties") can be proved quite easily if we have Picard schemes at our disposal. In contrast, the proof in Mumford's ...
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Thanks to Drew Heard's comment I was able to find answers to my questions. In his paper "Picard groups of derived categories" H. Fausk proves the following theorem (see Theorem 4.2). Theorem: Let $(\... 6 votes Accepted ### Galois invariant Picard group elements Updated. The example by @Lucifer is completely correct. Thanks to @Count Dracula who explained the example proposed by @Lucifer. That example is fine. I am keeping the counterexamples below, since ... 6 votes ### Examples of smooth projective varieties with "nice" Picard group Here is a simple example. Let$X = P^1 \times Q^3$and$H = O(1,1)$. The Hilbert polynomial of the line bundle$O(a,b)$is $$P(t) = (t+a+1)(t+b+1)(t+b+2)(t+b+3/2)/3.$$ It has three integral roots$-...
Let $f : X \to Y$ be a universal homeomorphism of schemes. Then as you noted above, the pullback functor on (small) étale sites \begin{eqnarray} Y_{\text{ét}} &\to& X_{\text{ét}}\\ U &\...
The group $G_X$ can be identified with the group of rank 1 reflexive sheaves on $X$ ($F$ is reflexive if the canonical morphism $F \to F^{\vee\vee}$ is an isomorphism) by taking a sheaf $F$ to the ...