Questions tagged [picard-group]

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Compactification of Picard variety over normal, projective varieties

Let $X$ be a normal, projective, integral variety (over $\mathbb{C}$) and $P$ be the Picard variety parametrizing invertible sheaves on $X$. Does there exist a compactification $\overline{P}$ of $P$ ...
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2answers
355 views

Picard group of symplectic group modulo orthogonal group

Let $Sp(2n)$ be the group of complex symplectic $2n\times 2n$ matrices, and $O(2n)$ the group of complex orthogonal $2n\times 2n$ matrices. Consider $Sp(2n)\cap O(2n)\subset Sp(2n)$ and the quotient $...
6
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105 views

Picard group of resolution

Let $X$ be a normal variety and $f:Y\rightarrow X$ a birational morphism, contracting exceptional divisors $E_1,\dots,E_k$ onto the singular locus of $X$, with $Y$ smooth. In this situation is $Pic(Y)...
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115 views

Lefschetz type theorems for linear sections

Let $X\subset\mathbb{P}^n$ be e normal variety, $L\subset\mathbb{P}^n$ a linear subspace, and $Y = X\cap L$ a linear section. Assume that $Y$ is also normal. In particular, we have that $Sing(X)$ has ...
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387 views

Postnikov invariants of the Brauer 3-group

Given a commutative ring $k$ there is a bicategory with algebras over $k$ as objects, bimodules as morphisms, bimodule homomorphisms as 2-morphisms. This is a monoidal bicategory, since we can ...
2
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1answer
167 views

Picard group modulo codimension 2

Let $X$ be a normal (possibly singular) projective surface over $\mathbb{C}$. Consider the set $M_X$ of all coherent sheaves $F$ on $X$ such that there exists a finite subset $Y\subset X$ such that $F$...
3
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1answer
131 views

Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion?

Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-...
5
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137 views

Picard group of a normal crossing scheme

I would like to know if someone has an explicit example for the rank of the Neron-Severi group of a normal crossing scheme (proper over a field) being different from the rank of the kernel of $\...
8
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1answer
233 views

Bézout ring with non-trivial Picard group?

[I asked this on stackexchange here a few weeks ago to no response] A ring is called Bézout when its finitely generated ideals are principal. Q: Is there a nice example of a Bézout ring $R$ with ...
2
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0answers
246 views

Picard group of blowup

Let $Y$ be a nonsingular subvariety of a normal, Cohen-Macaulay variety $X$. Further, let $\pi:X'\to X$ be the blowup of $X$ along $Y$. Question: Is there a formula for the Picard group of $X'$ ...
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114 views

Hypersurfaces with maximal Picard rank

Is it true that for any $d \ge 4$, there exists a smooth, degree $d$ surface $X$ in $\mathbb{P}^3$ with maximal Picard rank i.e., Picard rank of $X$ equals $h^{1,1}(X)$?
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460 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)...
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355 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 ...
6
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1answer
363 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
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1answer
304 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 $L$...
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128 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) ...
8
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1answer
373 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
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1answer
178 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 $M$ ...
10
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1answer
472 views

Picard group and reduced schemes

If $A$ is a ring, then we know that $Pic(A)=Pic(A_{red})$, but for a scheme $X$ it is false in general. On the other hand, we have that $Pic(X)=H^{1}_{et}(X,\mathbb{G}_m)$ and étale cohomology doesn'...
4
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1answer
252 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 ...
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141 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
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2answers
640 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
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1answer
318 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 ...
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2answers
668 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)} \...
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0answers
157 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 ...
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343 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 $X$ ...
9
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1answer
435 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 ...
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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)/(\...
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0answers
231 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
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671 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
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1answer
179 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
135 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
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1answer
361 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
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1answer
257 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
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1answer
480 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 ...
3
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1answer
434 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
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1answer
495 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 $T$, $...
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0answers
43 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}M_L(...
6
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120 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
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1answer
690 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
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1answer
494 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
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1answer
581 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 $\mathcal X$ ...
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0answers
309 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
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3answers
449 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 ...
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2answers
595 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, ...
4
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1answer
537 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
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2answers
244 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 ...
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0answers
196 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
224 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
572 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 \...