# Questions tagged [picard-group]

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### Picard group of normalization

Let $X$ be a projective variety with at worst (analytic) normal crossings singularities and $\pi:\tilde{X} \to X$ be the normalisation. Is there a "nice" description relating the picard group of $X$ ...
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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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### Finite Picard group

Does there exist a connected scheme, smooth, proper, and positive-dimensional over $\mathbb{C}$ with finite Picard group? Note that Picard group has cardinality$>1$. Also note that this can not ... 161 views

### Explicit algebraic cycles

Fix a smooth sextic curve curve $C = \{f_6(x,y,z) = 0\}$ in $\mathbb{P}^2$, and consider the double cover $X_{f_6}$ defined by $z^2 = f_6$ in the appropriate weighted projective space. This is known ...
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### Given an embedding of $X$ into $\mathbb{P}^n_K$, do you get an induced embedding of any twist of it into $\mathbb{P}^n_K$?

Let $X$ be a projective algebraic curve over some number field $K$, and let $\varphi:X\hookrightarrow \mathbb{P}^n_K$ be an embedding of it (defined over $K$) into some projective space. Now let $X'$ ...
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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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### 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 ...
64 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 ...
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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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1 vote
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### 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(...
1 vote
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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 vote
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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 ...
1 vote
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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 vote
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### The Anderson dualizing spectrum and $E$-theory
Hopkins-Lurie have shown that the fiber of $gl_1 E \to L_n gl_1 E$ (shown by Ando-Hopkins-Rezk to be coconnected and torsion) looks very very similar to a shift of the Anderson dual of the sphere (in ...
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 ...