Timeline for $K_0$-equivalence of varieties
Current License: CC BY-SA 3.0
25 events
| when toggle format | what | by | license | comment | |
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| S Nov 29, 2017 at 11:32 | history | bounty ended | CommunityBot | ||
| S Nov 29, 2017 at 11:32 | history | notice removed | user95222 | ||
| Nov 26, 2017 at 1:55 | vote | accept | CommunityBot | ||
| Nov 25, 2017 at 23:42 | answer | added | dhy | timeline score: 6 | |
| Nov 25, 2017 at 8:48 | comment | added | dhy | Call it $X$. By the Lefschetz hyperplane theorem $X$ will have Picard rank $1$ and $4$th Betti number equal to $2$ (assuming large degree in a large enough Grassmannian.) Under that assumption, X will be its own canonical model. Therefore $X$ can't be birational to a product of two varieties (or, taking canonical models of both, $X$ would need to have Picard rank at least $2$.) And it can't be birational to one of your specified varieties - the only general type (of large dimension) possibility is a hypersurface in projective space, also its own canonical model, but with $H^4$ rank $1$. | |
| Nov 25, 2017 at 8:40 | comment | added | dhy | Does the following argument work? By Larsen-Lunts, $K_0(Var)/[\mathbb{A}^1]$ is the free abelian group on stable birational classes. It thus suffices to show that there is some variety which is not stably birational to a variety in class $P$. In fact, as the MRC base of any variety in $P$ is in $P$ (up to birational equivalence), it suffices to show that there is a general type variety not birational to a variety in P. To do this, consider a high degree hypersurface in a Grassmannian. (cont.) | |
| S Nov 25, 2017 at 8:10 | history | bounty started | CommunityBot | ||
| S Nov 25, 2017 at 8:10 | history | notice added | user95222 | Draw attention | |
| S Nov 23, 2017 at 2:00 | history | suggested | user97068 | CC BY-SA 3.0 |
Edit to make question clearer.
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| Nov 23, 2017 at 1:59 | review | Suggested edits | |||
| S Nov 23, 2017 at 2:00 | |||||
| Nov 23, 2017 at 1:54 | history | edited | user95222 | CC BY-SA 3.0 |
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| S Nov 23, 2017 at 1:51 | history | suggested | user97068 | CC BY-SA 3.0 |
I edited the question so as to make it more understandable.
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| Nov 23, 2017 at 1:51 | review | Suggested edits | |||
| S Nov 23, 2017 at 1:51 | |||||
| Nov 23, 2017 at 1:40 | comment | added | user95222 | My question was meant to ask if one can find a more handy generating class for the Grothendieck ring of $k$-varieties, instead of just all smooth projective varieties over $k$. Smooth projective varieties are known to be generators of $K_0(\text{Var}_k)$. I was merely asking if there is any serious reason why the class $\mathcal{P}$ cannot be a class of generators, why, and what is it that is missing to it in order to become one | |
| Nov 23, 2017 at 1:35 | comment | added | user97068 | Side comment. I came to this question earlier and can't avoid noting it has been down voted. If you're going to down vote a question, at least make sure you explain why, out of courtesy. It really looks to me the OP is asking a meaningful question, though somewhat confusingly phrased. Unless you have a full argument as to why those classes of varieties, and the transformations allowed to construct the class $\mathcal{P}$ out of them, are not sufficient to generate the full $K_0(\text{Var}_k)$, then you should probably refrain from just down voting, until somebody provides a good answer. | |
| Nov 23, 2017 at 0:33 | comment | added | R. van Dobben de Bruyn | The answer to (2) should be no, but it might not be so easy to write down an actual counterexample. (Maybe it's possible to write down a variety with a big irreducible sub-Hodge structure that cannot come from any of these varieties?) | |
| Nov 22, 2017 at 20:41 | review | Close votes | |||
| Nov 23, 2017 at 12:12 | |||||
| Nov 22, 2017 at 20:28 | comment | added | abx | A toric variety is rational. If (2) holds with $Y$ toric, $X$ is unirational. Similarly, most of your questions imply strong restrictions on $X$. | |
| Nov 22, 2017 at 20:16 | history | edited | user95222 | CC BY-SA 3.0 |
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| Nov 22, 2017 at 20:05 | history | edited | user95222 | CC BY-SA 3.0 |
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| Nov 22, 2017 at 19:59 | history | edited | user95222 | CC BY-SA 3.0 |
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| Nov 22, 2017 at 19:52 | history | edited | user95222 | CC BY-SA 3.0 |
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| Nov 22, 2017 at 19:46 | history | edited | user95222 | CC BY-SA 3.0 |
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| Nov 22, 2017 at 19:39 | history | edited | user95222 | CC BY-SA 3.0 |
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| Nov 22, 2017 at 19:31 | history | asked | user95222 | CC BY-SA 3.0 |