# Tag Info

### 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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### 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 ...
• 13.5k
Accepted

• 31.5k

### 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 ...
• 138k
Accepted

### $K_X+B \equiv 0$ 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 ...
• 1,044
Accepted

### 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 ...
• 114k
Accepted

### 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 ...
• 114k
Accepted

### 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 ...
• 114k

### 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 ...
• 2,200

### 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 ...
Accepted

### 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 ...
• 40.8k
Accepted

### 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 ...
• 61.8k

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