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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 $X' = Y' \times_Y X$...
iteo's user avatar
  • 39
5 votes
0 answers
262 views

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 ...
QYB's user avatar
  • 51
4 votes
1 answer
221 views

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 \...
John Baez's user avatar
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2 votes
1 answer
176 views

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?
prochet's user avatar
  • 3,442
3 votes
0 answers
152 views

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 ...
spiderchips's user avatar
1 vote
0 answers
60 views

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{...
Li Li's user avatar
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0 answers
182 views

Smoothness of Picard scheme when $H^2(\mathcal{O}_{X_s})$ on fibers vanish

A question about the proof of Proposition 5.19 in Kleiman's notes on Picard scheme. Let $X$ be a $S$-scheme. Then the claim is that: Assume that Picard scheme $\operatorname{Pic}_{X/S}$ exists and ...
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1 vote
0 answers
49 views

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}...
SeoyoungK's user avatar
2 votes
1 answer
157 views

$K_0((k[x]/(x^2))[y])$

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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4 votes
1 answer
179 views

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 $K_0(R)$ ...
user443060's user avatar
3 votes
1 answer
233 views

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 ...
Puzzled's user avatar
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3 votes
0 answers
63 views

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\...
GreginGre's user avatar
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0 answers
39 views

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\...
Syu Gau's user avatar
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1 vote
0 answers
268 views

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 ...
JKDASF's user avatar
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0 answers
122 views

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(...
user443060's user avatar
0 votes
1 answer
329 views

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 ...
John Z.'s user avatar
  • 53
12 votes
2 answers
1k views

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\...
Neil Strickland's user avatar
2 votes
1 answer
152 views

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{...
TCiur's user avatar
  • 557
2 votes
1 answer
545 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 ...
user267839's user avatar
  • 5,938
1 vote
1 answer
113 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 ...
Luca Francone's user avatar
1 vote
0 answers
206 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 $s_0\in S$. ...
Puzzled's user avatar
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4 votes
0 answers
154 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 $I\subseteq A$, ...
nariri's user avatar
  • 392
10 votes
1 answer
440 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 $k$-...
Antoine Labelle's user avatar
2 votes
1 answer
471 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 ...
LOCOAS's user avatar
  • 363
2 votes
0 answers
81 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 ...
taf's user avatar
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3 votes
0 answers
121 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 (...
Somatic Custard's user avatar
3 votes
0 answers
108 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 $\text{Cl}(X)$ ...
Jonathan Love's user avatar
6 votes
1 answer
946 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 ...
user147163's user avatar
1 vote
0 answers
57 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 ...
a196884's user avatar
  • 323
2 votes
1 answer
439 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 ...
Puzzled's user avatar
  • 8,902
2 votes
1 answer
217 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=\...
yors's user avatar
  • 195
3 votes
2 answers
377 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 ...
Asvin's user avatar
  • 7,716
3 votes
1 answer
362 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 ...
pozio's user avatar
  • 599
1 vote
1 answer
246 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} &...
user avatar
3 votes
1 answer
157 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 $((A,B),Z)\mapsto AZB^{T}$. ...
user avatar
4 votes
1 answer
409 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 ...
Arno Fehm's user avatar
  • 2,041
2 votes
1 answer
392 views

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\...
user avatar
10 votes
2 answers
1k views

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\...
user avatar
2 votes
0 answers
86 views

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{...
hennlu's user avatar
  • 323
2 votes
1 answer
294 views

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 ...
Hans's user avatar
  • 2,905
5 votes
1 answer
689 views

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 ...
Kim's user avatar
  • 505
2 votes
1 answer
386 views

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 ?
Rraa's user avatar
  • 21
7 votes
2 answers
2k views

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 $R$ ...
Kevin Buzzard's user avatar
11 votes
1 answer
223 views

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 ...
Chris Schommer-Pries's user avatar
9 votes
1 answer
336 views

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 ...
Matthias Ludewig's user avatar
7 votes
0 answers
228 views

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 ...
SashaP's user avatar
  • 7,037
1 vote
0 answers
195 views

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 ...
Daniel Loughran's user avatar
2 votes
0 answers
105 views

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 $P$ ...
user45397's user avatar
  • 2,303
5 votes
2 answers
484 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 $...
Puzzled's user avatar
  • 8,902
15 votes
1 answer
741 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 ...
John Baez's user avatar
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