Questions tagged [picard-group]

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9
votes
2answers
361 views

On a morphism from the Brauer group to the Picard group

Suppose that $k$ is a commutative ring and that $A$ is an Azumaya $k$-algebra. Then there is a well-known morphism from $Aut_{Alg_k}(A)$, the group of algebra automorphisms, to the Picard group $Pic(k)...
2
votes
0answers
327 views

Finite Picard group

Does there exist a connected scheme, smooth, proper, and positive-dimensional over $\mathbb{C}$ with finite Picard group? Note that Picard group has cardinality$>1$. Also note that this can not ...
4
votes
1answer
226 views

Proper scheme such that every vector bundle is trivial

It is claimed here that there exist proper schemes (probably over a field but not explicitly stated) with trivial Picard group. This means that every locally free $O_X$-module of rank 1 is trivial. ...
2
votes
1answer
195 views

Extension of line bundle defined over an open subscheme

Let $X$ be a normal projective (or, quasi-projective) variety over $\mathbb{C}$. Let $U \subset X$ be an open subscheme whose complement $Z = X \setminus U$ has codimension at least $2$ in $X$. Let $L$...
2
votes
0answers
126 views

The growth of class number in $\mathbb{Z}_p$-extensions of function fields

Let $X$ be a curve (proper, smooth, ...) over a finite field $\mathbb F_q$ where $q$. Suppose also that $\mathbb F_q$ contains the $p$-th roots of unity, in this case we have the following (unique) ...
7
votes
1answer
263 views

Del Pezzo surfaces and Picard-Lefschetz theory

Let $X$ be a smooth compact del Pezzo surface. For instance, one can consider the most classical case of a cubic surface. It is well known that the Picard lattice of $X$ is related to a root system (...
3
votes
1answer
151 views

Invertible bimodules which are isomorphic in the stable module category

I'm in the following situation. I have a self-injective finite-dimensional basic algebra $\Lambda$ (hence Frobenius) over a perfect field and two finite-dimensional invertible $\Lambda$-bimodules $M$ ...
4
votes
1answer
240 views

Examples of smooth projective varieties with “nice” Picard group

I am looking for examples of smooth projective varieties $(X,H)$ with $H$ a polarization on $X$, $\dim \mbox{Pic}^0(X)=0$, $\mbox{Pic}(X) \not= \mathbb{Z}$ satisfying the property: for any two line ...
8
votes
0answers
129 views

On a smooth curve $C$, when is $K_C \sim_\mathbb{Q} (2g-2)P$?

Let $C$ be a smooth curve of genus $g$ over $\mathbb{C}$. I am interested in the following property: There exists a point $P \in C$ such that $K_C \sim_\mathbb{Q} (2g-2)P$. Equivalently, $K_C - (2g-2)...
8
votes
2answers
528 views

Picard group of a finite type $\mathbb{Z}$-algebra

Let $A$ be a finitely generated $\mathbb{Z}$-algebra. Is $\operatorname{Pic}(A)$ finitely generated (as an abelian group)? Thoughts: We may assume that $A$ is reduced since $\operatorname{Pic}(A) = \...
9
votes
1answer
296 views

Why is the theorem of the base mostly cited only for smooth proper varieties

This is a very soft question, and I'm not sure what I expect as an answer. In SGA6, Expose XIII, Theoreme 5.1 it is proven that, if $X$ is a proper scheme over a field $k$, then $NS(X)$ is finitely ...
9
votes
2answers
554 views

Galois invariant Picard group elements

Let $X$ be a smooth variety over a perfect field $k$ with $X(k) \neq \emptyset$. Then is the natural map \begin{equation} \mathrm{Pic}(X) \to (\mathrm{Pic}(X_{\bar{k}}))^{\mathrm{Gal}(\bar{k}/k)} \...
3
votes
0answers
151 views

Explicit algebraic cycles

Fix a smooth sextic curve curve $C = \{f_6(x,y,z) = 0\}$ in $\mathbb{P}^2$, and consider the double cover $X_{f_6}$ defined by $z^2 = f_6$ in the appropriate weighted projective space. This is known ...
5
votes
0answers
246 views

Picard group of normalization

Let $X$ be a projective variety with at worst (analytic) normal crossings singularities and $\pi:\tilde{X} \to X$ be the normalisation. Is there a "nice" description relating the picard group of $X$ ...
8
votes
1answer
384 views

Pic^0 and H^0(K,Pic^0)

Let $K$ be a field and $C$ a smooth and projective curve over $K$. Then the kernel $Pic^0(C)$ of the degree map injects into $H^0(K,Pic^0_C)$, where $Pic_C^0$ is the connected component of the Picard ...
1
vote
0answers
47 views

Connection on line bundle over general simplicial toric variety

In https://arxiv.org/pdf/hep-th/0005247.pdf, on page 60 and 61, it is mentioned that the connection of $\mathcal{O}(-n)$ over a (simplicial) toric variety of the form $$ (\mathbb{C}^N \backslash U)/(\...
2
votes
0answers
216 views

Surjectivity of map of Picard schemes implies abelian

Note: This question was asked on MSE first, but got zero reactions. So I deleted it there, and am now posting it here. I am looking for a reference or explanation of the fact that is used in Mumford'...
6
votes
0answers
493 views

Picard group of toric varieties

I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf . Here, a toric variety has ...
4
votes
1answer
154 views

Is the class (resp. Picard) group of a $G$-variety generated by invariant divisors?

Let's work over the complex numbers. Let $S$ be a normal surface, $\mathrm{A}^1(S)$ the class group of divisors on $S$ and $\mathrm{Pic}(S)$ its Picard group. Let $G$ be a reductive group acting on $S$...
3
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0answers
134 views

Given an embedding of $X$ into $\mathbb{P}^n_K$, do you get an induced embedding of any twist of it into $\mathbb{P}^n_K$?

Let $X$ be a projective algebraic curve over some number field $K$, and let $\varphi:X\hookrightarrow \mathbb{P}^n_K$ be an embedding of it (defined over $K$) into some projective space. Now let $X'$ ...
7
votes
1answer
348 views

Galois invariant line bundles on a product of varieties

Let $k$ be a field with separable algebraic closure $k^{\rm s}$ and corresponding absolute Galois group $\varGamma={\rm Gal}(k^{\rm s}/\,k)$ and let $X$ and $Y$ be geometrically connected and ...
6
votes
1answer
234 views

Picard group of derived category of sheaves

Let $X$ be a topological space and $R$ be a commutative ring with unit, $D(X,R)$ is the derived category of unbounded complexes of sheaves of $R$-modules. Moreover we suppose that $X$ is a stratified ...
7
votes
1answer
414 views

Picard groups and birational morphisms

Let $f:X\rightarrow Y$ be a birational morphism of projective varieties. Assume that $Pic(X)$ is a free abelian group generated by $n$ divisors $D_1,...,D_n$. Under which hypothesis on $X$ and $Y$ is ...
2
votes
1answer
399 views

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

I have been studying Chapter 8 of Neron models by Bosch et al. The first part deals with the relative Picard functor. A lot of work is done to make it representable. My question would be why this work ...
5
votes
1answer
465 views

Cohomological interpretation of G-equivariant line bundles

In the theory of reductive algebraic groups, there is the following map (notation: $G$ reductive over an algebraically closed field, $T$ a maximal torus, $B$ a Borel, $X(T)$ the characters of $T$, $...
1
vote
0answers
40 views

Reference needed for exact sequence of ACM curves with homogeneous coordinate ring.

Let $C\subset\Bbb{P}^n$ be an ACM (arithmetically Cohen-Macaulay) curve with homogeneous coordinate ring $R$. Then there is an exact sequence $$0\to \text{Tor}_i(R,\Bbb{C})_k\to H^1(C,\wedge^{i+1}...
6
votes
0answers
117 views

Center Picard group non-commutative algebra

I am wondering if there is a way to describe the center of the Picard group of a non-commutative algebra. Namely, let $A$ be a finitely generated algebra over a field $k$. Denote by $\mathrm{Pic}(A)$...
9
votes
1answer
590 views

Restriction of the Picard group of a surface to a curve

In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious: For a general (smooth) surface $S$ in $\mathbb{P}^3$ ...
3
votes
1answer
402 views

Proper pushforward of algebraic cycles

Let $f:X\to Y$ be a finite surjective morphism of smooth integral projective varieties over an algebraically closed field $k$ of characteristic 0. Denote by $CH_i(W):=Z_i(W)/\sim$ the Chow group of $i$...
6
votes
1answer
516 views

Are Picard stacks group objects in the category of algebraic stacks

I've been wondering about what a "group algebraic stack" should be, and ran into the notion of a Picard stack. I'm slightly confused by the terminology here. Given an algebraic stack $\mathcal X$ ...
1
vote
0answers
224 views

Fiber of the specialization map of Picard groups

Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective ...
3
votes
3answers
406 views

Is $Pic^0(X)$ of a curve of genus $\geq 1$ over a non-algebraically closed field still non-finitely generated?

Qing Liu's "Algebraic Geometry and Arithmetic Curves" page 299 COrollary 7.4.41 gives the following result. Let $X$ be a smooth, connected, projective curve over an algebraically closed field $k$, of ...
8
votes
2answers
549 views

Are there varieties with non finitely generated Picard group and vanishing irregularity?

Let $X$ be a smooth projective variety over an algebraically closed field $k$. Can it happen that $q(X) := \dim H^1(X,\mathcal O_X) =0$ and $\textrm{Pic} \,X$ is not finitely generated? Certainly, ...
3
votes
1answer
454 views

Picard groups, ample cones, and proper birational maps

Let $f:Y\to X$ be a proper birational map of normal varieties over an algebraically closed field which is an isomorphism over the regular locus. Q1: Is it the case that the pullback $f^*\...
2
votes
2answers
219 views

Normalization of a Noetherian local domain and line bundles on the punctured spectrum

Let $A$ be a Noetherian local domain ($2$-dimensional if needed) such that its punctured spectrum $U$ is regular, and let $A'$ be the normalization of $A$. 1) Is it possible for $A'$ to have ...
1
vote
0answers
186 views

Twisting locally free sheaves in characteristic $p$

Let $X$ be an irreducible nodal projective curve over an algebraically closed field of characteristic $p>0$. Denote by $\pi:\tilde{X} \to X$ the normalization of $X$. Recall, the short exact ...
8
votes
1answer
202 views

Obstructions to Picard-graded groups of maps

Suppose $(C,\odot,\Bbb I)$ is an additive category with a compatible symmetric monoidal structure and $Pic(C)$ is the group of isomorphism classes of objects which have an inverse under $\odot$. For $\...
4
votes
1answer
516 views

Picard groups of Fano varieties in positive characteristic

Let $k$ be an algebraically closed field of characteristic $p \geq 0$. Let $X$ be a smooth Fano variety over $k$ and let $\ell \neq p$ be a prime. Is the natural morphism $\mathrm{Pic}(X) \otimes \...
1
vote
0answers
86 views

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 ...
5
votes
1answer
323 views

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 ...
14
votes
1answer
499 views

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 ...
2
votes
1answer
480 views

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 ...
1
vote
1answer
174 views

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 ...
9
votes
1answer
421 views

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; The ...
4
votes
1answer
289 views

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} \...
3
votes
1answer
593 views

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 $H^2(X_s, \mathscr{O}_{X_s}) = 0$ ...
1
vote
0answers
754 views

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 ...
20
votes
1answer
766 views

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 ...
7
votes
0answers
210 views

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 ...
2
votes
1answer
363 views

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 ...