Questions tagged [picard-group]
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148 questions
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Computing with the Picard group of non-integral curves
Let $C$ be a curve over $\mathbb{Q}_p$ and let $\mathcal{C}$ be a regular model of $C$ over $\mathbb{Z}_p$, with $\mathcal{J}$ the Neron model of the Jacobian of $C$. Raynaud's theorem asserts that $\...
- 111
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When does isomorphism on singular cohomology imply isomorphism on Picard and Brauer groups?
Assume that $f:X\to Y$ is a morphism of complex varieties, and the homomorphisms $H^i_\text{sing}(f)$ are bijective for $0\le i\le 3$ (though possibly $3$ is too much here:)). Under which ...
- 16.9k
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The importance of the Balmer spectrum
Why are Balmer spectra important? Can someone give examples of reconstruction a category by its spectrum (in some sense)?
It would also be interesting to see applications of Balmer spectra to the ...
- 87
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Picard group of the category of numerical motives
Is anything known about the Picard group of $Chow_{Num}(k, \mathbb{F}_{p})$ (numerical Chow motives with $\mathbb{F}_{p}-$coefficients)?
Perhaps the Picard groups of some other categories of pure ...
- 87
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Nice blowups are universal algebraic fiber spaces?
We say that a proper (maybe projective) morphism $f:X \to Y$ is a universal algebraic fiber space if $f_* O_X = O_Y$ holds universally. (This means: for any morphism $Y' \to Y$, if $X' = Y' \times_Y X$...
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Characterizing principal polarizations of abelian surfaces
Suppose $X$ is a complex abelian variety of dimension 2. Then I believe the ring of endomorphisms $\mathrm{End}(X)$, tensored with $\mathbb{C}$, is isomorphic to a subalgebra $M_2(\mathbb{C})$ of $2 \...
- 22.3k
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Picard group of almost module category
I am very new to the world of almost mathematics and I am curious about the following:
Fix an almost mathematics situation $(R,I)$ throughout. Very generally, the almost module category comes with a ...
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Finite étale cover of factorial ring
Let $A$ be a regular factorial ring.
Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?
- 3,472
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Computing basis of $\mathrm{Pic}(\bar{X})$ for a Del Pezzo surface
Say we are given a degree 2 del Pezzo $X$ given by $w^2=Q(x,y,z)$ where $Q(x,y,z)$ is degree 4. We can compute the exceptional lines by computing the 28 bitangent lines of $Q$ and look at the ...
- 31
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Why are ordinary spheres not strictly invertible?
Introduction
This question is about Picard spectra for the symmetric monoidal $\infty$-category of spectra. We say that a spectrum $X$ is invertible if there is another spectrum $Y$ such that $X\...
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When $R $ is a cusp then $K_0(R) \ncong K_0(R[s])$
Quillen's classical result shows that if $R$ is a regular ring then $K_0(R) \cong K_0(R[t_1,...,t_m])$ for all $m \in \mathbb{N}$. So I wanted to construct some elementary examples where $K_0(R)$ ...
- 167
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A question on "Ample subvarieties of algebraic varieties"
Corollary 3.3 in Chapter IV of "Ample subvarieties of algebraic varieties" by R. Hartshorne asserts the following:
Let $X$ be a smooth projective variety and $Y\subset X$ a smooth subvariety ...
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$K_0((k[x]/(x^2))[y])$
Let $K_0(R):= K_0(P(R))$ where $P(R)$ is the category of finitely generated projective $R$-modules, where $R$ is a commutative ring with unity. Now if $R = k[x]/(x^2)$, $R$ is a local ring thus all ...
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Determine the class of a non-isomorphic projection of a rational normal scroll as a divisor in a higher dimensional scroll
This is a generalized problem of Theorem 1.1 of Park's and Theorem 1.4 of Nagel's. Consider the vector bundle $E=\mathcal{O}(1)\oplus\mathcal{O}(1)\oplus\mathcal{O}(1)\oplus\mathcal{O}(1)$ on $\mathbb{...
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Picard number of Hilbert modular surfaces
Hilbert modular surfaces are discussed in various papers by Hirzebruch. Following [HZ] (and their notations), one obtains Hilbert modular surfaces by the action of Hilbert modular group on $\mathcal{H}...
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Computing Picard groups of arbitrary quadric hyperplane
I know the Picard group of a smooth two dimensional quadric surface is $\mathbb Z^2$, but I am wondering if the computation can be generalized to higher dimension? In particular, is the Picard group ...
- 231
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Class numbers of functions fields and spanning trees
In Discrete groups, expanding graphs, and invariant measures (in the notes at the end of Chapter 7), Lubotzky mentions that certain estimates for the number of spanning trees $\kappa(G)$ of a $k$-...
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Picard group of a normal conical affine variety
Let $k$ be an algebraically closed field. Let $X\subset \mathbb{A}^n_{k}$ be a conical closed subvariety. In other words,
$\mathcal{O}(X)=k[x_1,\cdots, x_n]/I$, where $I$ is generated by homogeneous ...
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Proposition 1.5 in Mumford's Geometric Invariant Theory
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\pr{pr}$I have some problems to understand the proof of Proposition 1.5 from Mumford's ...
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Picard group of a cusp [duplicate]
$\DeclareMathOperator\Pic{Pic}$For a commutative ring with unity $R$, if $R_{\mathrm{red}}$ is semi-normal then we have an equivalent criterion which states that $\Pic(R) \cong \Pic(R[t])$. Here $\Pic(...
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Reference for torsion-freeness of the group of correspondences on a smooth projective variety
In Beauville's "Variétés de Prym et jacobiennes intermédiaires", Proposition 3.5, it is claimed that $\textrm{Corr}(T)$ is torsion-free for a smooth projective variety $T$. Here $$\textrm{...
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Automorphisms of matrix algebras and Picard group
This is a repost of https://math.stackexchange.com/questions/4692364/automorphisms-of-matrix-algebras-and-picard-group (asked on MSE).
Notation. In what follows, $R$ is a commutative ring with $1$, $n\...
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Reference request: Generic k3 surface has Picard number 1
I keep running into the statement that "the generic k3 surface has Picard rank 1".
For instance the answer of this question (end) and this paper (following Example 1.1) or this paper (proof ...
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Semi-continuity of the Picard number
Let $f:X\rightarrow S$ be a family of smooth projective varieties. For $s\in S$ set $X_s := f^{-1}(s)$, and let $\rho(X_{s})$ be the Picard number of the fiber over $s\in S$. Fix a point $s_0\in S$.
...
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Does the Grothendieck group detect the Picard group?
Let $C$ be a curve (=smooth projective curve) of genus $g$ over an algebraic closed field $\mathbb{k}$.
It is well known that the Grothendieck group $K_0(\operatorname{coh} C)$ of the category of ...
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Stable Picard group of the tensor product of two Hopf algebras
Suppose we know the stable Picard groups (=Picard group of the stable module category) of two cocommunicative Hopf $k$-algebras $G$ and $H$. Is it possible to deduce the stable Picard group of $G\...
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Homogeneous components of Cox RIngs
Let $X$ be an irreducible smooth projective variety over a field $k$ (algebraically closed and of characteristic zero if needed). Let $U \subseteq X$ an affine open such that $O_X(U)$ is factorial and ...
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A map between Brauer groups
Let $R$ be a henselian dvr over $\mathbb{C}$ and $A$ be a flat $R$-algebra of finite type. Suppose $\hat{R}$ is the completion of $R$ and $\hat{A}:=A\otimes_R \hat{R}$. For an ideal $I\subseteq A$, ...
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Picard group of a cubic hypersurface
Consider the following cubic hypersurface in $\mathbb{P}^5$:
$$
X = \{z_0z_3z_5-z_1^2z_5-z_0z_4^2+2z_1z_2z_4-z_2^2z_3 = 0\}\subset\mathbb{P}^5
$$
The singular locus of $X$ is the Veronese surface $V\...
user168611
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Picard group of connected linear algebraic group
Here's a statement:
Suppose $G$ is a connected linear algebraic group over a field $k$, then $Pic(G)$ is a finite group.
I know this is true when $k=\mathbb{C}$. My question is does this true for ...
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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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p-torsion in the Picard group of a regular projective curve
Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$.
If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of ...
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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 ...
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Galois stable elements of the Picard group of a curve and the rational divisors
Let $C$ be a (smooth,proper) curve over a field $k$. Let $\operatorname{Div}_C(k)$ be the free abelian group generated by the closed points of $C/k$ and $k(C)^\times$ be the group of rational ...
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A question on effective divisors
Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. Then ...
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Relation between $\mathrm{Pic}^\natural_{X/S}$ and two notions of rigidification
Let $X/S$ be a relative curve (perhaps with more adjectives). I have come across a few instances of rigidifying and rigidificators, which I would like to understand better.
In Liu-Lorenzini-Raynaud (...
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Specializing p-torsion in a family of elliptic surfaces
Let $R$ be a DVR of mixed characteristic, with algebraically closed residue field of characteristic $p$ and fraction field $K$. Let $Y\longrightarrow \operatorname{Spec} R$ be a smooth projective ...
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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 $...
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Map to study $K(n)$-local Picard Group
Let $R$ be an $E_{\infty}$-ring. There's a fiber/cofiber sequence $S$: $gl_1 R \to \text{Pic}(R) \to H(\text{Pic}(R))$, where $\text{Pic}(R) =\pi_0 \text{Pic}(R)$ is the Picard group of $R$. Rotating ...
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Picard group vs class group
The question.
Let $R$ be a commutative ring. Let $M$ be an $R$-module with the property that there exists an $R$-module $N$ such that $M\otimes_R N\cong R$. Does there always exist an ideal $I$ of $R$ ...
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Picard-surjectivity and Morita-equivalence
Let us say that an algebra $A$ over a field $k$ is Picard-surjective if the canonical map
$$ \mathrm{Aut}(A) \rightarrow \mathrm{Pic}(A)$$
is surjective. Here $\mathrm{Pic}(A)$ denotes the group of ...
- 9,863
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Detecting non-principal Weil divisors on normal varieties using curves
Let $X$ be a normal projective variety over an algebraically closed field $k$. Given any morphism $f:Y\to X$, there is a pullback homomorphism $f^*:\text{Cl}(X)\to\text{Cl}(Y)$, where $\text{Cl}(X)$ ...
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Picard group of moduli of principal bundles
I am looking for the Picard group of the moduli space of principal $G$-bundles for a connected reductive complex algebraic group $G$.
Is it isomorphic to $\mathbb{Z}$? If not, what can we say when $G=\...
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Postnikov invariants of the Brauer 3-group
Given a commutative ring $k$ there is a bicategory with
algebras over $k$ as objects,
bimodules as morphisms,
bimodule homomorphisms as 2-morphisms.
This is a monoidal bicategory, since we can ...
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What is the right definition of the Picard group of a commutative ring?
This is a rather technical question with no particular importance in any case of actual interest to me, but I've been writing up some notes on commutative algebra and flailing on this point for some ...
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Picard group of $\mathrm{GL}(n)$-orbits
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Mat{Mat}$Consider the general linear group
$$
\GL(n) = \left\lbrace
\left(\begin{array}{cc}
A & C \\
M & B
\end{array}\right) \text{ with } A\...
user168611
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Picard group of a cone over the elliptic curve
Let E be an smooth elliptic curve in a projective plane.
Suppose that X is the projective cone over E in a projective space of dimension three.
What is the Picard group of X ?
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Are algebras with invertible linear duals always Frobenius?
Let $A$ be a finite dimensional algebra over a ground field $k$. The linear dual $A^* = Hom_k(A,k)$ is naturally an $A$-$A$ bimodule. I am interested in those algebras such that $A^*$ is an invertible ...
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Picard groups of determinantal varieties
Consider a general $4\times 4$ matrix:
$$
X:=\left(
\begin{array}{cccc}
X_0 & X_1 & X_2 & X_3 \\
X_4 & X_5 & X_6 & X_7 \\
X_8 & X_9 & X_{10} & X_{11} \\
X_{12} &...
user124234
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Very general quartic hypersurface in $\mathbb{P}^3$ has Picard group $\mathbb{Z}$
I am looking for a reference from which I can cite the following statement:
The Picard group of a very general quartic hypersurface $X\subset\mathbb{P}^3$ is generated by the class of a hyperplane ...
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