Questions tagged [picard-group]
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148 questions
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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^*\...
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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 ...
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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 ...
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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 $\...
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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 \...
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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 ...
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Does every relative curve have a Picard scheme?
More precisely:
Let $X \to S$ be a smooth proper morphism of schemes such that the geometric fibers
are integral curves of genus $g$. Must the fppf relative Picard functor
$\operatorname{\bf ...
- 5,019
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Severi's theorem of base and Hilbert polynomial
Let $X$ be a smooth projective variety over $\mathbb{C}$ satisfying $H^1(\mathcal{O}_X)=0$. Fix $i:X \to \mathbb{P}^n$ a closed immersion and let $\mathcal{O}_X(1)$ be the corresponding very ample ...
- 66.3k
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Relative Picard functor for the Zariski topology
I'm trying to understand better the relative Picard functor, as defined, for example, in Kleiman's article.
Let $X \to S$ be a smooth projective morphism of schemes whose geometric fibres are ...
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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 ...
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Does a semistable curve descend to a regular base?
Let $f\colon X \rightarrow S$ be a semistable curve of genus $g \ge 0$. Being a semistable curve means that $f$ is a morphism of schemes such that
$f$ is proper, flat, and of finite presentation;
The ...
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1
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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 ...
CommunityBot
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4
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1
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Jacobian of a semistable curve
My question is about the proof of Example 8 in section 9.2 of the book "Neron models." There we have a semistable curve $X$ over an algebraically closed field $K$ and we let $\pi\colon \widetilde{X} \...
- 1,103
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Fiberwise vanishing of $H^2$ and formal smoothness of the Picard functor
My question is about the proof of 8.4/2 in "Neron models." The claim is that if $f\colon X \rightarrow S$ is a proper flat morphism of finite presentation such that $H^2(X_s, \mathscr{O}_{X_s}) = 0$ ...
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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 ...
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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 ...
CommunityBot
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Picard scheme of varieties over imperfect fields
Let $k$ be a field and $X$ a proper $k$-scheme. It is a theorem of Murre and Oort that the Picard functor is representable by a $k$-group scheme $\operatorname{Pic}_{X/k}$ which is locally of finite ...
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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 ...
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Injectivity under flat base change of the Picard group on smooth projective curves
Let $K$ be a field of characteristic $0$, $X_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X_{\bar{K}} \to X_K$, induces via the ...
- 38k
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Picard group generated by effective divisors: counterexample?
Let $X$ be an integral variety defined over an algebraically closed field $k$ of characteristic 0 with finitely generated Picard group $Pic(X)$ and such that $k[X]^\times=k^\times$ (i.e. the only ...
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Picard group of infinite direct product of DVRs trivial
Let $R = \prod_{n\in\mathbf{N}}R_n$ be an infinite direct product of discrete valuation rings $R_n$. Why is $\mathrm{Pic}(R) = 0$?
- 5,019
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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 ...
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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^...
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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 ...
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2
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$Pic$ of the stack of elliptic curves vs. $Pic$ of the coarse space
There's a natural map $f:\overline{\mathcal{M}}_{1,1}\to \overline{M}_{1,1}\cong \mathbb{P}^1$ from the stack of elliptic curves to the coarse space. Both spaces have $Pic=\mathbb{Z}$ hence $f^*:\...
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Seeking Noetherian normal domain with vanishing Picard group but not a UFD
Once again, the question says it all.
My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...
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1
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Relatively numerically trivial divisor
Hi,
Let $f : X \rightarrow Y$ a projective morphism of quasi-projective algebraic varieties over $\mathbb{C}$. Assume that $X$ is smooth, that $Y$ is normal and that:
$$\textbf{R} f_* \mathcal{O}_X ...
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Line bundles on a pointless curve
Let $X$ be a smooth projective curve over a field $k$. In chapter 8 of the book Neron Models by Bosch et al., there is a general result (namely Proposition 4) which implies that if $X$ admits a ...
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$Pic(X)/l=0$ in terms of $H^*_{et}(X,\mu_{l^n})$?
I would like to calculate Picard groups of certain schemes over fields; I'm mostly interested in the question whether $Pic(X)$ is infinitely $l$-divisible, i.e. whether $Pic(X)/l=0$, $l$ is a prime ...
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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 ...
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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 ...
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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 ...
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$\psi$ class in $\overline{M}_{0,n}$
Basic question, but I found no reference.
Is the $\psi$ class the only one which is not a boundary class in the PIcard group of the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$? Or can it ...
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How can one check that two line bundles on $\overline{M}_{0,n}$ coincide?
Let $X$ be the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$. Suppose I have two (big) line bundles $L$ and $L'$ on $X$ and that I want to show that they are the same element of $Pic(X)$. Of ...
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Picard group of $\mathcal{M}_{0,n}$
Let $\mathcal{M}_{0,n}$ be the complement of the boundary of the Mumford-Knudsen compactification of the moduli space of genus zero, n-pointed curves.
Is $Pic(\mathcal{M}_{0,n})$ trivial?
- 3,908
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On generators of the Picard group of a projective smooth surface over a finite field
Let $X$ be a smooth projective surface over a finite field $k=\mathbb{F}_q$. Let us first review the proof of the finite generation of $Pic(X)$ (notice that the proof is valid for any smooth ...
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What is the Picard group of a Schubert variety in the affine Grassmannian?
I'm not sure I have a lot more to say than the title. Let $G$ be your favorite simple algebraic group over $\mathbb{C}$, and let $$\overline {\mathrm{Gr}}_\lambda= \overline{G(\mathbb{C}[[t]])\cdot t^...
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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 $G$-...
CommunityBot
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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 ...
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Two questions about finiteness of ideal classes in abstract number rings
Let us say that an abstract number ring is an integral domain $R$ which is not a field, and which has the "finite norms" property: for any nonzero ideal $I$ of $R$, the quotient $R/I$ is finite.
(I ...
CommunityBot
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Is the ring of all cyclotomic integers a Bezout domain?
My previous question about the theorem (apparently due to Dedekind -- thanks, Arturo Magidin!) that the ring $\overline{\mathbb{Z}}$ of all algebraic integers is a Bezout domain got me thinking about ...
CommunityBot
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An example where $Pic(X) = H^0(k,Pic(\overline{X}))$?
Let $X$ be a geometrically integral smooth projective variety over a number field $k$. Then if $X$ is everywhere locally soluble, we have $Pic(X) = H^0(k,Pic (\overline{X}))$, where $\overline{X}=X \...
CommunityBot
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Picard Groups of Moduli Problems
First, yes, I've seen Mumford's paper of this title. I'm actually interested in specific ones, and looking for really the most elementary/elegant proof possible.
I'm told that for $g\geq 2$ it is ...
- 12.4k
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What does the Riemann-Hurwitz formula tell us on the Picard variety
Let $f:X\longrightarrow Y$ be a finite separable morphism of smooth projective integral curves over an algebraically closed field.
Then we have a linear equivalence of Weil divisors on $X$: $$ K_X=f^\...
- 66.3k
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Picard groups of non-projective varieties
As far as I know, the main representability result for the relative Picard functor $Pic_{X/k}$, for a noeth. sep. scheme of finite type over a field $k$ is:
If $X$ is proper then $Pic_{X/k}$ is ...
- 1,606
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330
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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 $...
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Torsion-freeness of Picard group
Let $X$ be a complex normal projective variety.
Is there any sufficient condition to guarantee the torsion-freeness of Picard group of $X$?
One technique I sometimes use is following:
If $X$ can be ...
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For a line bundle L on a smooth projective variety X, what is meant by Pic^L(X)
Hi everyone,
Let $X$ be a smooth projective variety over a field $k$ and let $L$ be a line bundle on $X$. I'm reading the article Heights for line bundles on arithmetic varieties and there one ...
CommunityBot
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