# Questions tagged [picard-group]

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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 ...
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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{... 2 votes 1 answer 519 views ### 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 ... 1 vote 1 answer 90 views ### 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 ... 2 votes 0 answers 159 views ### 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. ... 4 votes 0 answers 139 views ### 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, ... 10 votes 1 answer 410 views ### 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-... 2 votes 1 answer 360 views ### 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 ... 2 votes 0 answers 71 views ### 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 ... 3 votes 0 answers 96 views ### 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 (... 3 votes 0 answers 83 views ### 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) ... 5 votes 1 answer 647 views ### 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 ... 1 vote 0 answers 54 views ### Class groups and zeta functions for maximal orders in CSAs I'm looking for references for certain algebraic objects in the context of maximal orders in (finite dimensional) central simple algebras over algebraic number fields. Does anyone know of any good ... 3 votes 1 answer 341 views ### 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 ... 2 votes 1 answer 185 views ### 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=\... 2 votes 2 answers 288 views ### 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 ... 3 votes 1 answer 349 views ### 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 ... 1 vote 1 answer 195 views ### 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} &... 147 views

### Picard group of $(SL(n)\times SL(m))$-orbits

Let $\mathbb{P}^N$ be the projective space of $n\times m$ matrices with complex entries modulo scalar. Consider the $(SL(n)\times SL(m))$-action on $\mathbb{P}^N$ given by $((A,B),Z)\mapsto AZB^{T}$. ... 368 views

### 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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### 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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### 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 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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### 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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### 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-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 ...
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### Albanese morphism induces an isomorphism on global $1$-forms

Let $X$ be a smooth projective variety over a field $k$ of characteristic zero equipped with a point $e\in X(k)$. There is Albanese morphism $a:X\to \mathrm{Alb}\,X$ which is initial among pointed ...
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### Picard scheme of family of quartic surfaces

Recall that a quartic surface in $\mathbb{P}^3_\mathbb{C}$ has $N = 35$ coefficients. Let $U$ be the open subset of $\mathbb{P}^{N-1}$ parametrising smooth quartic surfaces and let $Q \to U$ be the ...
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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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### 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 ...
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### 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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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 $h^{1,1}(X)$?