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21 votes

If $X\times X$ is rational, must $X$ also be rational?

For the first question I am not that pessimistic. At least there are candidates as follows: Recall that $Z$ is stably rational if there is $n\ge0$ such that $Z\times\mathbf A^n$ is rational. Now ...
Friedrich Knop's user avatar
15 votes
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When is the map $H^0(X,mK_X) \times H^0(X,nK_X) \to H^0(X,(m+n)K_X)$ surjective?

I assume you mean $H^0(X, K_X)^{\otimes m}$ rather than $\oplus_{i=1}^m H^0(X, K_X)$. If $X$ is a smooth projective connected complex curve of genus $g \geq 2$, then the map $$H^0(X, K_X)^{\otimes m} \...
js21's user avatar
  • 7,199
15 votes
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Does a resolution of a rational singularity have rationally connected fibers?

No. For instance the cone over an Enriques surface (with respect to any projective embedding) has rational singularity, but Enriques surface is not rationally connected.
Sasha's user avatar
  • 37.5k
15 votes
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Is there a classification of minimal algebraic threefolds?

It depends what you mean by classification. The key results for surfaces IMO are: 1) Any surface $S$ of general type has a canonical model given by $S_{can}:={\rm Proj} R(K_S)$ and a unique minimal ...
Hacon's user avatar
  • 2,382
13 votes
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Are stably rational surfaces all rational?

The result is true in all characteristics. See O. Zariski, Illinois J. Math. 2(1958), 303-315.
Laurent Moret-Bailly's user avatar
13 votes

Are Du Val singularities smoothable?

Du Val singularities are hypersurface singularities, hence they can be smoothed --- just replace the defining equation $F(x,y,z) = 0$ by the equation $F(x,y,z) = \epsilon$.
Sasha's user avatar
  • 37.5k
12 votes

Singular cohomology and birational equivalence

Here is some of the general philosophy of birational invariants, at least those coming from (co)homology (I don't think this approach quite works for homotopical invariants.) Philosophy. If $f \colon ...
R. van Dobben de Bruyn's user avatar
12 votes
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Automorphism groups of Hirzebruch surfaces

For $n \ge 2$ the surface $\mathbb{F}_n$ is the blowup of the weighted projective plane $\mathbb{P}(1,1,n)$ at its singular point. Because of that $$ Aut(\mathbb{F}_n) \cong Aut(\mathbb{P}(1,1,n)). $$ ...
Sasha's user avatar
  • 37.5k
12 votes
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Complete intersections in toric varieties

Any smooth projective toric variety is rational, in particular simply connected. Then, by the Lefschetz hyperplane theorem for global complete intersections, if $\dim X \geq 3$ is a smooth complete ...
Francesco Polizzi's user avatar
12 votes

Applications of derived categories to "Traditional Algebraic Geometry"

The global Torelli Theorem for cubic fourfolds says the following. Let $X_1 \subset \mathbb{P}^5$ and $X_2 \subset \mathbb{P}^5$ be smooth cubic fourfolds. The fourfolds $X_1$ and $X_2$ are ...
12 votes
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Degree of secant varieties of Veronese varieties

The secant variety $Sec_k(V^n_2)$ is the variety parametrizing $(n+1)\times (n+1)$ symmetric matrices modulo scalar of rank at most $k$ that is of corank at least $n+1-k$. Then by Proposition 12(b) in ...
Puzzled's user avatar
  • 8,872
12 votes
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diagonal cubic hypersurfaces

Yes, this is true over $\mathbb{C}$, and rather easy. You can assume your equation is $\sum x_i^3=0$. For convenience, let me call the coordinates $x_0,\ldots ,x_m;y_0,\ldots ,y_m$. Then your ...
abx's user avatar
  • 37.4k
11 votes
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Reference request on birational invariance of Chow group of zero cycles of degree zero

A reference for birational equivalence of $CH_0$ is Fulton's Intersection Theory [1], Example 16.1.11. In the example, he makes the assumption that $k$ is algebraically closed, but he never uses it. ...
R. van Dobben de Bruyn's user avatar
11 votes
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Liftable rational varieties

Two such examples were given by Achinger and Zdanowicz [AZ17], both of which satisfy a whole bunch of other good properties (e.g. their classes in the Grothendieck ring of varieties are polynomials in ...
R. van Dobben de Bruyn's user avatar
11 votes

Reference request: birational automorphism group is finite

EDIT. An alternative reference is Husemoller, Dale H.: Finite automorphism groups of algebraic varieties, Finite groups, Santa Cruz Conf. 1979, Proc. Symp. Pure Math. 37, 611-619 (1980). ZBL0466....
Francesco Polizzi's user avatar
11 votes
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Normal bundle of a linear subspace

For instance, if $X$ is a smooth 4-dimensional quadric in $\mathbb{P}^5$ and $H = \mathbb{P}^2$, the normal bundle fits into the exact sequence $$ 0 \to N_{H/X} \to \mathcal{O}(1)^{\oplus 3} \to \...
Sasha's user avatar
  • 37.5k
10 votes

Crepant resolutions of cDV singularities?

Background. The threefold compound du Val singularities have been introduced by Miles Reid in the 1980s [R1, R2, R3]. Their geometric description is that a general hyperplane section through the ...
Evgeny Shinder's user avatar
10 votes
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Singular cohomology and birational equivalence

Basically, any invariant $T$ which satisfies the following purity property will turn into a birational invariant for proper smooth varieties (aka complex compact manifolds): let $X$ be a smooth ...
ACL's user avatar
  • 12.8k
10 votes

Quadrics in the Grothendieck ring

Edit. Following Remy van Dobben de Bruyn's excellent suggestion, I clarified the use of "irreducible quadrics of dimension $0$." Daniel Loughran's observation about Chevalley-Warning is the key to ...
10 votes

Is an open subscheme of a rationally connected variety, rationally connected?

If I have understood the OP's definitions correctly, the answer is no for rational chain connectedness. Let $E$ be a smooth genus $1$ curve in $\mathbb{P}^2$ and let $X \subset \mathbb{P}^3$ be the ...
David E Speyer's user avatar
10 votes
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$K_X+B \equiv 0$ implies $K_X + B \sim_\mathbb{Q} 0$?

For lc pair or slc pair, it is true. This is Gongyo’s result. See [J. ALGEBRAIC GEOMETRY 22 (2013) 549–564]. BTW, the relative version is also true, which is not a trivial generalization of the ...
Chen Jiang's user avatar
  • 1,084
10 votes
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Singular curves of genus 1

There's no problem over finite fields, but there is a problem over fields that have a nontrivial Brauer class. If you take a genus $0$ curve that's not rational (say a plane quadric), it will always ...
Will Sawin's user avatar
  • 138k
10 votes
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Equivalent definitions of Kodaira dimension

For $(1) \iff (3)$, we have the following chain of identities: $\operatorname{trdeg} \operatorname{Frac} \left( \bigoplus_{m ≥ 0} H^0(X, mL) \right)$ is equal to $\max \operatorname{trdeg}(F)$ where ...
Will Sawin's user avatar
  • 138k
10 votes
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Field extensions over which algebraic varieties cannot acquire points

Let $K$ be a finite type field extension of $k$ which corresponds to a rational function field of an algebraic variety $V$ for which the rational points are not Zariski dense. Let $U$ be the ...
Will Sawin's user avatar
  • 138k
10 votes
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Alterations and smooth complete intersections

As you guessed, there are cohomological obstructions. Indeed, if $f \colon Y \twoheadrightarrow X$ is a surjective morphism of smooth projective varieties and $H$ is a Weil cohomology theory (with ...
R. van Dobben de Bruyn's user avatar
9 votes
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Anti-canonical divisor of a Fano variety

If you want to consider smooth (weak) Fano variety, then Fukuda has effective estimation of the birationality of anti-canonical systems for any dimension (but not optimal), see [S. FUKUDA, A note on ...
Chen Jiang's user avatar
  • 1,084
9 votes
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Chern classes of a vector bundle

As $\mathcal{O}_{\mathbb{P}^2}$ is trivial, then multiplicativity of Chern classes in exact sequences implies: $$ c_*(\mathcal{E}) = c_*(\mathcal{I}_p(-1)). $$ We can compute $c_*(\mathcal{I}_p(-1))$ ...
Strawberry's user avatar
9 votes
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Does a torus action with isolated fixed points imply rational?

Yes. This follows from the Białynicki-Birula decomposition (see Theorem 4.4 in the original paper).
Piotr Achinger's user avatar

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