Questions tagged [picard-group]
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148 questions
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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 $F$...
- 3,031
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1
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1k
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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 ...
- 845
2
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2
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1k
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Blowing down -1 curves
After a good look for anything in the way of an explicit example, and harassing the algebraic geometers in the department, I still am unable to find an answer to my question, so any light will be very ...
- 153
2
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1
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411
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Picard/cohomology lattice of surfaces of low degree in $\mathbb P^3$
Let $S_{d>3}\subset\mathbb{P}^3_{\mathbb{C}}$ be a smooth surface of degree $d$. What is known (where to read?) about the Picard/cohomology lattice for small d?
e.g. for $d=4$ the cohomology ...
- 2,232
2
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1
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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 ...
- 167
2
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1
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512
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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 ...
- 405
2
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1
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226
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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=\...
- 195
2
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1
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398
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Picard group of $\mathrm{GL}(n)$-orbits
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Mat{Mat}$Consider the general linear group
$$
\GL(n) = \left\lbrace
\left(\begin{array}{cc}
A & C \\
M & B
\end{array}\right) \text{ with } A\...
user168611
2
votes
2
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322
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Normalization of a Noetherian local domain and line bundles on the punctured spectrum
Let $A$ be a Noetherian local domain ($2$-dimensional if needed) such that its punctured spectrum $U$ is regular, and let $A'$ be the normalization of $A$.
1) Is it possible for $A'$ to have ...
- 825
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487
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A question on effective divisors
Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. Then ...
- 8,998
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1
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186
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Finite étale cover of factorial ring
Let $A$ be a regular factorial ring.
Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?
- 3,472
2
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1
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158
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Reference for torsion-freeness of the group of correspondences on a smooth projective variety
In Beauville's "Variétés de Prym et jacobiennes intermédiaires", Proposition 3.5, it is claimed that $\textrm{Corr}(T)$ is torsion-free for a smooth projective variety $T$. Here $$\textrm{...
- 679
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563
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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 ...
- 5,986
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1
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438
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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 ...
- 1,981
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145
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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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0
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242
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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 $s_0\in S$.
...
- 8,998
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83
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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 ...
- 448
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Map from the stack of coherent sheaves on a curve to the Grothendieck group
Let $X$ be a smooth, projective curve. We let $Coh(X)$ be the stack of coherent sheaves on $X$. Its Grothendieck group is $Pic(X)\times\mathbf{Z}$. Is the map
$$
Coh(X)\rightarrow Pic(X)\times \mathbf{...
- 333
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0
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112
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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$ ...
- 2,323
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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'$ ...
- 155
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0
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138
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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) ...
- 7,746
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304
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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'...
- 203
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0
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507
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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 ...
- 2,323
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0
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188
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Is Pic( G((z)) ) = $\mathbb{Z}$?
There are a fair number of papers by Beauville, Laszlo, Sorger, Kumar and others on the geometry of $LG/L^+G = G((z))/G[[z]]$ where $G$ is a simply connected and simple group over $\mathbb{C}$. In ...
- 3,968
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515
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A motivic complex
By definition, Voevodsky's motivic complex (an object of his $DM^{eff}_-$) is a complex of sheaves with transfers whose cohomology sheaves are homotopy invariant. Now, I consider the complex (of ...
- 16.9k
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2
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325
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Jacobian of a curve and field extension
Let $K$ be a field of characteristic zero and $X_K$ a smooth projective curve on $K$. Denote by $\bar{K}$ the algebraic closure of $K$ and $X_{\bar{K}}$ the base change of $X_K$ to $\bar{K}$. Under ...
- 503
1
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1
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258
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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
1
vote
1
answer
201
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 ...
- 1,103
1
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1
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119
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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 ...
1
vote
1
answer
145
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divisors on $\overline{\mathcal{M}}_{g,n}$ that are trivial on certain $F$-curves
Inside the moduli space of curves $\overline{\mathcal{M}}_{g,n}$ one can distinguish two classes of $F$-curves isomorphic to $\mathbb{P}^1$: those of type $\overline{\mathcal{M}}_{0,4}$, and those of ...
- 3,779
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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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0
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62
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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{...
- 439
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0
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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}...
- 11
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0
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39
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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\...
- 467
1
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0
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330
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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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58
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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 ...
- 323
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200
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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 ...
- 21.4k
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206
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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 $h^{1,1}(X)$?
- 2,323
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0
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52
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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
$$
(\mathbb{C}^N \backslash U)/(\...
- 1,155
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0
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60
views
Reference needed for exact sequence of ACM curves with homogeneous coordinate ring.
Let $C\subset\Bbb{P}^n$ be an ACM (arithmetically Cohen-Macaulay) curve with homogeneous coordinate ring $R$. Then there is an exact sequence
$$0\to \text{Tor}_i(R,\Bbb{C})_k\to H^1(C,\wedge^{i+1}M_L(...
- 765
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203
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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 ...
- 1,981
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132
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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 ...
- 101
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780
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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 ...
- 1,103
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0
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213
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Elegant definition for the scheme parametrizing $g_d^r$'s on a curve
Let $X$ be a smooth projective curve over $k=\bar{k}$, and $Pic^d$ the $d$-part of the Picard group of $X$ (isomorphism classes of line bundles of degree $d$ on $X$).
I'd like to define a scheme $G_d^...
- 251
1
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0
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178
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$G_m$-cohomology of a motif (that corresponds to a stack?)
As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety
I am in the following situation: $G$ is an algerbraic group, and X is a smooth $G$-...
- 16.9k
1
vote
0
answers
330
views
Abel-Jacobi map for regular fibered surfaces.
Let $f:C\to S$ be a regular fibered surface where $S=Spec(R)$, $R=dvr$. Assume $C$ has smooth geometrically integral generic fibre $C_K$. We also assume the existence of a section $x\in C(S)$. Let $...
- 11
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1
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405
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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 ...
- 53
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0
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382
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Hypersurfaces with Picard group generated by classes of lines on the same plane
For which values of $d$ is the following possible: There exist a smooth hypersurface $X$ in $\mathbb{P}^3$ of degree $d$ with Picard number $d$, containing $d-1$ lines $l_1,...,l_{d-1}$ on the same ...
- 1,070