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5 votes
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Does a nonzero section induce Isomorphism of Locally Free Sheaves?

Suppose $X = \operatorname{Spec} k$ (for any field $k$), and $E = F = k^n$. Then $s$ is just a linear map $$s: k^n \to k^n.$$ But being a non-zero linear map is not the same as being an isomorphism.
red_trumpet's user avatar
  • 8,952
2 votes
Accepted

Computing torsion subgroup of elliptic curve

$x^3+2x^2+1$ is a perfectly good way to write this and everything still works. Smooth reduction is a synonym for good reduction (good reduction means the reduction is smooth). By the Jacobian criteria,...
KReiser's user avatar
  • 66.1k
2 votes

Compute $Spec(\mathbb{Z}_{(p)})\times_{\mathbb{Z}}Spec(\mathbb{Z}_{(q)})\cong Spec(\mathbb{Q})$

The product of affine schemes is the spectrum of the tensor product, so the question is equivalent to the ring-theory question of whether $$ \mathbb{Z}_{(p)} \otimes_{\mathbb{Z}} \mathbb{Z}_{(q)} = \...
hunter's user avatar
  • 30.8k
2 votes
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Do elliptic K3 surfaces have line bundle of relative degree $1$?

The existence of $L$ of relative degree 1 is in fact equivalent to the existence of a section. Indeed, the pushforward $f_*L$ is a line bundle on $\mathbb{P}^1$ and the zero locus of the adjunction ...
Sasha's user avatar
  • 17.4k
2 votes
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Is Jacobian $Pic^0(X_t)$ invariant in some sense for some good family?

The varieties $Pic^0(X_t)$ are dependent of the choice of fiber. For an example, the Torelli Theorem says that a smooth complex curve is determined by its Jacobian, so it is easy to come up with ...
Brian Nugent's user avatar
1 vote

Is Jacobian $Pic^0(X_t)$ invariant in some sense for some good family?

The Jacobians of the fibers in the family are certainly equidimensional (and therefore diffeomorphic, since all complex tori of the same dimension are diffeomorphic), but they are not biholomorphic to ...
hunter's user avatar
  • 30.8k
1 vote
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Abhyankar and Sathaye conjecture

If $R[f_1,f_2,\dots,f_n] = R[x_1,\dots,x_n]$ then $f_1, f_2, \dots, f_n$ behave just like variables: every element of $R[f_1,f_2,\dots,f_n]$ can be written in a unique way as a polynomial in $f_1, f_2,...
Magdiragdag's user avatar
  • 15.1k
1 vote

Parameter Space of Smooth Cubic Surfaces over $\mathbb{C}$ is Connected

The result is true, ignoring any conditions on $Z=\Bbb P^{19}\setminus U$ being a manifold. Here is the outline of a proof. Let $k=\Bbb R$ or $\Bbb C$. Then for any affine $k$-variety $X$ and closed ...
KReiser's user avatar
  • 66.1k
1 vote
Accepted

Is $H^1(G_{\Bbb{Q}_p},E[2])=0$ for good prime of $E/\Bbb{Q}$?

No - I don't think this is ever true. For an easy example, take any prime $p$ that splits completely in $\mathbb Q(E[2]))$. Then $H^1(G_{\mathbb Q_p}, E[2]) \cong H^1(G_{\mathbb Q_p}, \mathbb Z/2\...
Mathmo123's user avatar
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1 vote

Example of quasi-compact, non-quasi seperated scheme where qcqs fails?

Here is an example of quasi-compact non-quasi-separated scheme $X$ and a global section $f\in \Gamma(X,\mathcal{O}_X)$ such that the natural morphism $\Gamma(X,\mathcal{O}_X)_f\to\Gamma(X_f,\mathcal{O}...
YJ_cat's user avatar
  • 316

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