28 votes

is the Hodge conjecture birationally invariant?

Of course, abx is completely correct in saying that the truth of the Hodge conjecture is not a birational invariant. That said, something slightly weaker is true: if $X$ and $Y$ are $K$-equivalent, ...
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  • 20.5k
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 ...
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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} \...
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  • 7,119
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 ...
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  • 2,247
14 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.
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  • 31.5k
13 votes
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Pseudo-automorphisms on Fano varieties

You don't need your variety, say $X$, to be Fano, only $\mathrm{Pic}(X)=\mathbb{Z}$. A pseudo-automorphism $u$ of $X$ induces an automorphism of $\mathrm{Pic}(X)$, which must be the identity. Let $L$...
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  • 34.2k
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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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 $F(x,y,z) = \epsilon$.
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  • 31.5k
12 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 ...
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 ...
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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 ...
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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 ...
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  • 6,643
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 ...
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  • 34.2k
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 ...
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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. ...
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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 ...
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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....
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10 votes
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Why is the inverse of a bijective rational map rational?

Since you are talking about rational maps, I assume you mean "open dense" in the Zariski topology, so that $X$ and $Y$ are algebraic varieties. Therefore we have a particular case of the following ...
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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 ...
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  • 12.4k
10 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 $$ Aut(\mathbb{F}_n) \cong Aut(\mathbb{P}(1,1,n)). $$ ...
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  • 31.5k
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 ...
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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 ...
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  • 1,044
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 ...
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  • 114k
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 ...
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  • 114k
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 ...
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  • 114k
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 ...
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9 votes

Pseudo-automorphisms on Fano varieties

This is also true for every smooth Fano variety $X$, with any Picard number. One can see it using Mori dream spaces: $X$ is a Mori dream space (by BCHM), and hence has (up to isomorphisms) only ...
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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 ...
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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 ...
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