21
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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 ...
- 14.5k
15
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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} \...
- 7,139
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.
- 34.8k
15
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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 ...
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13
votes
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Is every complex rational algebraic variety simply connected for the Euclidean topology?
I want to mention the positive direction. Let $X$ be a smooth, projective variety over $\mathbb{C}$, resp. over an algebraically closed field of arbitrary characteristic. Let $Z\subset X$ be a ...
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.
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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$.
- 34.8k
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 ...
- 24.4k
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)).
$$
...
- 34.8k
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 ...
- 63.9k
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
...
- 6,770
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 ...
- 35.7k
11
votes
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Generalization of the rigidity lemma in birational geometry
EDIT: I've just realized that this holds under somewhat weaker assumptions. It is not necessary that the fibers of $g$ are connected.
EDIT#2: Apparently, in my previous edit I weakened the conditions ...
- 41.6k
11
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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. ...
- 24.4k
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 ...
- 24.4k
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....
- 63.9k
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 \...
- 34.8k
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 ...
- 12.6k
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 ...
- 145k
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 ...
- 1,054
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 ...
- 126k
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 ...
- 126k
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 ...
- 126k
9
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 ...
- 2,240
9
votes
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pull back of an ample line bundle under a blow up
Let me start with a little nitpicking:
What on Earth do you mean by $\mu^*L-\epsilon E$ when you said that $L$ was a line bundle? You cannot add a line bundle and a divisor! So, let's assume that you ...
- 41.6k
9
votes
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A Bertini-type result for hypersurfaces containing a subvariety
For non-smooth $Z$ the answer is in general no.
Take $P=\mathbb{P}^4$ and let $Z \subset \mathbb P^4$ be a surface with a non-normal double point $p$ (i.e. a singularity locally analytically ...
- 63.9k
9
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 ...
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 ...
- 1,054
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