Olivier Benoist
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Is there a complex surface into which every Riemann surface embeds?
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84 votes

The answer is negative. Suppose for contradiction that $S$ is such a surface, and let me first assume that it is smooth and projective. Fix $g\geq 24$. Then the coarse moduli space of genus $g$ ...

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Kodaira dimension of symmetric products of curves
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23 votes

Let $C$ be a smooth projective connected complex curve of genus $\geq 2$. Let me show that $C^{(d)}$ is of general type if $1\leq d\leq g-1$. Equivalently, one needs to show that the image $W_d$ of $...

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Class number measuring the failure of unique factorization
23 votes

Here is a partial answer. In a UFD, the following statement is true : "either an element is prime or you can write it as a nontrivial product". In a Dedekind ring with finite class number h, it is ...

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Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$
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22 votes

It has been conjectured by Euler that this equation has no solutions in positive integers when $n\geq 4$. When $n=4$, this was disproved by Elkies in the paper [Elkies, On A4+B4+C4=D4] in a very ...

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A line bundle that does not admit a G-linearisation
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17 votes

Let me give an example showing that the normality hypothesis is necessary. Let $Y=\mathbb{P}^1$ with natural $G=\mathbb{G}_m$-action. Let $X$ be the $G$-variety obtained by glueing transversally the ...

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Do varieties with ample canonical bundle have finite automorphism group in small characteristic?
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16 votes

A much more general result is proven in Martin-Deschamps, Lewin-Menegaux : Applications rationnelles séparables dominantes sur une variété de type général. Théorème 2 : if $X$ and $Y$ are smooth and ...

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Is an algebraic space over a DVR, whose special fibre and generic fibre are schemes, actually a scheme?
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16 votes

The answer is negative. Let me give a counter-example by modifying the example of a singular complex algebraic surface that is not a scheme given by Knutson in [Algebraic Spaces, p.21-22]. Let me ...

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On condition when the push-forward of coherent sheaf is locally free
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15 votes

I believe that this statement is not true. Take $Y=Spec(k[t]/t^2)$ and $X=\mathbb{P}^1_Y$. On $X$, extensions of $\mathcal{O}$ by $\mathcal{O}(-2)$ are parametrized by : $$Ext^1_X(\mathcal{O},\mathcal{...

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Characteristic zero and characteristic $p$ in algebraic geometry
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15 votes

Here are two examples. The moduli space of dimension $g$ principally polarized abelian varieties $A_g$ contains complete codimension $g$ subvarieties in any positive characteristic $p$ (for instance, ...

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Bialynicki-Birula decomposition of a non-singular quasi-projective scheme.
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15 votes

It is a theorem of Sumihiro (Equivariant completion, Corollary 2) that a normal variety over an algebraically closed field with an action of a torus is covered by invariant affine open subsets. Here, ...

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Blow up along codimension one closed subscheme
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14 votes

I think it's not true : Let $X=Spec(A)$ with $A=k[x,y,z]/(x^2-y^2-z^2)$ be a quadratic cone. Let $Y$ be a line through the origin of the cone : its ideal is $I=(z,x-y)$. We calculate : $$X'=Proj_{...

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injectivity of the pull-back via a finite map
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13 votes

Let $f:X\to Y$ be a finite surjective map of degree $d$ between smooth projective varieties of dimension $n$. Then $f$ is flat (apply EGA IV 2 Prop 6.1.5), so Example 1.7.4 of Fulton's Intersection ...

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Counter-example to faithfully flat descent
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12 votes

I think the following should be a counter-example. Let $X=\text{Spec}(\mathbb{C}[T])$ be the affine line and $X'=\bigsqcup_{x\in\mathbb{C}}\text{Spec}(\mathbb{C}[T]_{(T-x)})$ be its faithfully flat ...

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Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?
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12 votes

The new edition of SGA3 by Philippe Gille et Patrick Polo provides a connected example due to Raynaud. It is in SGA3 Exposé VIA Exemples 1.3.2 (2) and you may find it at http://www.math.jussieu.fr/~...

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Finite map from quasi-projective variety
11 votes

It is true if $Y$ is normal. Let me explain why. I will say that a variety has the Chevalley-Kleiman (CK) property if every finite subset is contained in an affine open. By Corollary 48 of Kollar's "...

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If the direct image of f preserves coherent sheaves on noetherian schemes, how to show f is proper?
10 votes

If $f:X\to Y$ is separated of finite type between noetherian schemes and $f_*$ preserves coherence, then $f$ is proper. Here is a proof that follows the geometric idea (given in the comments of Piotr ...

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Complex structures on a K3 surface as a hyperkähler manifold
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10 votes

These $2$-spheres are called 'twistor lines'. They indeed cover the moduli space (in the non-polarized case) : more precisely, any two points of the moduli space may be linked by a chain of twistor ...

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Localizing at the primitive polynomials?
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10 votes

A prime ideal of $S^{-1}R[X]$ is the extension of a unique prime ideal of $R$, so that the morphism $Spec(S^{-1}R[X])\to Spec(R)$ is a bijection, and even an homeomorphism. All the extensions of ...

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When is a submanifold of $\mathbf R^n$ given by global equations?
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9 votes

An example of a compact $16$-dimensional submanifold of $\mathbb{R}^{30}$ with trivial normal bundle that is not defined by such global equations may be found in [Akbulut-King, Submanifolds and ...

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A question on infinitely many closed points on a smooth projective variety and their behavior under embeddings
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9 votes

Let me explain why this statement cannot be true in general. I will give a counterexample where $X$ is a complex K3 surface. By a result of [Beauville and Voisin, On the Chow ring of a K3 surface], ...

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Closed points of field extension of k-scheme under projection
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9 votes

No, it is not ! For instance, if $k=\bar{\mathbb{Q}}$, $K=\mathbb{C}$ and $X$ has dimension $\geq 1$, there is exactly one closed point of $X_K$ above each closed point of $X$. But there is only ...

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Spectrum and scheme of the commutative group-algebra of an abelian group.
9 votes

What follows does not answer your precise question, but is very related to it and may be of interest to you. I consider the case where $G$ is finite, but not necessarily abelian. Then there are ...

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flatness criterion on normal bases
9 votes

No, the smoothness is really needed ! To construct a counter-example, let us choose $X=\mathbb{A}^2$. Consider the action of $\mathbb{Z}/2\mathbb{Z}$ by $(x,y)\mapsto (-x,-y)$. The quotient of $X$ ...

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Parameter space for complete intersections and their discriminant
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9 votes

The description of the Hilbert scheme of complete intersections (obtained by taking in an iterative way open subsets of grassmannian bundles, as explained in the answer above) may be found in part 2.2 ...

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Is reflexivity an open condition?
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9 votes

This locus is indeed open. I will explain why using Kollar's "Hulls and Husks" (arXiv:0805.0576). More generally, this article studies in great detail when taking the double dual commutes with base ...

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Hn(X, OX) = 0 for X birational to a regular affine variety?
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8 votes

Let me give a counter-example, where $f$ is moreover proper (see Piotr Achinger's comment), but $Y$ is not normal. Let $X$ be the affine plane over a field $k$ and let $\tilde{Y}$ be the blow-up of ...

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Irreducible divisors containing an arbitrary closed set
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8 votes

EDIT : Olivier Wittenberg pointed out to me that a positive answer to the question follows from Theorem 1 of [Altman-Kleiman, Bertini theorems for hypersurface sections containing a subscheme]. I keep ...

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Existence of points on varieties which avoid a given number field.
8 votes

I think this is a consequence of (variants of) Hilbert's irreducibility theorem. Let me explain why. Suppose that $C$ is a geometrically integral curve defined over a number field $K$. Let $K'/K$ be a ...

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Secant variety of an irreducible non-degenerate projective curve
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7 votes

Suppose that the secant variety of your curve is a surface $S$. This implies that $S$ is covered by lines in such a way that there is a $1$-dimensional family of such lines through every general ...

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Can a non-trivial action of a connected group on a reduced scheme be trivial on a dense open?
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7 votes

Let me work over an algebraically closed field $k$, let $X$ be a reduced $k$-scheme of finite type and let $G$ be a smooth connected $k$-group scheme acting on $X$, and acting trivially on an open ...

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