naf
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Why does the Section Conjecture exclude curves of genus 1?
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27 votes

I think that Grothendieck had already observed that the map from rational points to sections is injective (for curves of genus at least 2 over a number field) and I believe that his proof works even ...

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Is every algebraic smooth hypersurface of affine space parallelizable?
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27 votes

Yes. Suslin has proved that every stably trivial vector bundle of rank $n$ on an affine variety of dimension $n$ over an algebraically closed field is trivial. See: Suslin, A. A. Stably free modules. ...

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Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications
27 votes

Let $E$ be an elliptic curve. The moduli space $M_E$ of line bundles with a connection on $E$ is an $\mathbb{A}^1$ bundle over $Pic^0(E) \cong E$. In particular, $E$ can be recoved from $M_E$ as the ...

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Can one find the hodge number by counting points over finite fields?
22 votes

No, one cannot find the Hodge numbers this way. For an example, consider $X_0$ the Kummer surface associated to a product of supersingular elliptic curves $E_1$ and $E_2$. Recall that this is the ...

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Does isomorphic generic fibre imply isomorphic special fibre for smooth morphisms?
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17 votes

Here is an example showing the answer is no: Start with $Z =\mathbb{P}^2_R$, $R$ an arbitrary dvr. Let $P$ be a section of $Z \to Spec(R)$ and let $W$ be the blowup of $Z$ along the image of the ...

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Brauer group of projective space
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16 votes

I don't think the assumption of characteristic zero simplifies the proof a great deal. However, it does allow us to avoid having to give a more involved proof for the $p$-power torsion (where $p = ...

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Embedding number fields in fields with class number 1
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15 votes

See Proposition 1 on p.231 of Cassels and Frohlich for a proof of the claim in the textbook: The point is that if such an $L$ exists then $K_1L$ is abelian and unramified over $L$ so it is contained ...

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Number of $(-1)$ curves on toric surfaces
14 votes

No, toric surfaces can have only finitely many $(-1)$-curves. Since $(-1)$-curves are rigid, i.e., cannot form a non-trivial family, it follows that any $(-1)$-curve must be contained in the ...

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normal crossing divisor v.s. strict normal crossing divisor
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14 votes

Your definition of normal crossings divisor is, as you say, often called a strict normal crossings divisor. People who use this terminology allow normal crossings divisors to have components which are ...

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Is the group of integer points on a finite-type group scheme over Z finitely presented?
14 votes

It follows from Theorem 6.12 of Borel and Harsh-Chandra, "Arithmetic subgroups of algebraic groups", that $G(\mathbb{Z})$ is finitely generated if $G$ is affine. Perhaps one can combine this with ...

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Which curves can be found on Abelian varieties?
14 votes

For any $g >0$ there exists an abelian surface $A$ containing a smooth curve $C$ of genus $g$; the surface can be assumed to be simple if $g > 1$: For $g=1$, one can just consider $C \times C$....

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Example where you *need* non-DVRs in the valuative criteria
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14 votes

You can probably just take Y to be the spectrum of a valuation ring A which is not a DVR, for example the integral closure of C[[t]] in an algebraic closure of C((t)). In this case any homomorphism ...

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Motivation behind Kac's notation for affine root systems
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13 votes

I think the notation might be explained by the explicit construction of the twisted affine Lie algebras as fixed points of automorphisms of the untwisted ones: the $r$ indicates the order of the ...

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When does a bijective morphism of schemes induce an isomorphism of Hodge structures?
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13 votes

The induced morphism of Hodge structures for any map of varieties will be an isomorphism if and only if the induced map on cohomology (forgetting the Hodge structure) is an isomorphism. This follows ...

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Rational points on smooth compactifications
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12 votes

Yes, this is true. One implication is immediate: if $X$ has a $k((t))$ point then by the valuative criterion of properness there is a map $Spec(k[[t]])$ to any compactification of $X$, so the image ...

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Genus of smooth varieties with small Chow group
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12 votes

As you suspect, the answer to your first question is yes. A. Roĭtman in "Rational equivalence of zero-dimensional cycles". Math. USSR-Sb. 18 (1974), 571--588, generalised Mumford's theorem to show ...

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Picard group of $\mathcal{M}_{0,n}$
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12 votes

Yes. By fixing the three points $\{0,1,\infty\}$ one sees that $M_{0,n}$ is isomorphic to an open subscheme of $\mathbb{A}^{n-3}$ which has trivial Picard group. Since it is smooth, the Picard group ...

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Is every algebraic extension of a field of absolute transcendence degree one a separable extension of a purely inseparable extension?
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12 votes

Yes. One may also replace $\mathbb{F}_p$ by any perfect field $F$. The reason is that any non-perfect algebraic extension $L$ of $k$ has a unique inseparable extension of degree $p$, i.e. $L^{1/p}$, ...

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Can proper-smooth base change be used to show that varieties cannot be lifted to characteristic zero?
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12 votes

There is an example due to Hirokado of a Calabi-Yau threefold in characteristic 3 with third Betti number zero which implies that it cannot be lifted to characteristic zero. See: Hirokado, Masayuki -...

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The Jouanolou trick
12 votes

Jouanolou's trick has been extended to schemes with an "ample family of line bundles" by Thomason; see Weibel: Homotopy Algebraic K-theory, Proposition 4.4. This includes all smooth varieties and more ...

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Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?
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11 votes

It follows from results of Ribet in "On l-adic representations attached to modular forms" (Invent. Math. 28 (1975), 245–275) that the Mumford-Tate conjecture holds for the motives attached to modular ...

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Invariant differential forms on commutative group schemes are closed!?
11 votes

It seems to me that in the proof they only use the fact for abelian varieties: For this you can use the fact that $[n]^*$ acts on invariant $1$-forms by multiplication by $n$ -- this follows from the ...

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Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$
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10 votes

I combine user37314's answer and my comments; the claim is that any smooth projective complex algebraic surface with $\pi_1$ abelian and $\pi_2$ finite has a finite cover which must be an abelian ...

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Picard groups of (fiber) products
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10 votes

It can happen that $Pic(X) = Pic(Y) = Pic(Z) = 0$ but $Pic(W) \neq 0$! For example, let $f: \mathcal{E} \to Z$ be a non-isotrivial family of elliptic curves, where $Z$ is a smooth rational curve. ...

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Galois groups at closed points from Galois group at generic point?
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10 votes

What you need is the Hilbert irreducibility theorem. This implies that the Galois group for "most" rational points (i.e. outside a thin subset) is the full symmetric group. More generally one can ...

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Does the derived category of coherent sheaves determine the hodge theory?
10 votes

For varieties of dimension 1, 2 and 3 it is known that derived equivalent varieties have the same Hodge numbers. As far as I know, this is open (though believed to be true) for higher dimensional ...

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Extending a vector bundle to a torsion free sheaf
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10 votes

No, this is not even true for line bundles. For an example, let $X = \mathbb{P}^2$ and $Y$ a smooth curve in $X$ of genus $>0$ (over an algebraically closed field). Since $X$ is smooth, any line ...

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A Jacobian with a good reduction, which is simple : how is the reduction of the curve?
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9 votes

Yes, this is true. One can argue as follows: Since $J(C)$ has good reduction, your other hypothsism implies that the special fibre of its Neron model at $p$ is an absolutely simple abelian variety. ...

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l-adic vs complex Perverse Sheaves
9 votes

No. If $L$ is a local system on a smooth variety $X$ of dimension $d$ then $L[d]$ is perverse. As suggested by BBD, if we take $L$ to be a rank $n$ local system of $\overline{\mathbb{Q}}_l$ vector ...

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A question about quotient singularity
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9 votes

In general, for a finite group $G$ acting faithfully on a smooth variety $X$, whether or not the quotient is smooth is determined by the Chevalley-Shephard-Todd theorem: For $x \in X$, let $G_x\...

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