Timeline for Root lattices and (resolutions of) singular cubic surfaces
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Mar 12, 2021 at 8:23 | comment | added | AG learner | Dear Martin, thank you for the comment. The surjectivity $\pi:Pic(X')\to Cl(X)$ is a very useful observation. This means that the root system on $X$ should be a quotient of the 72 roots on $X'$. However $\pi$ does not seem to preserve the intersection pairing, for example. If $l_1,l_2$ are two lines through an $A_1$ signularity, then $l_1\cdot l_2=\frac12$ is a fraction, according to the formula in Sakai's paper, but their liftings should have self-intersection an integer. | |
| Mar 9, 2021 at 11:07 | comment | added | Martin Bright | @AGlearner, I don't know of this being studied specifically, but you can look at it like this: the Weil divisors are unaffected by removing the singular locus of X (of codimension 2), so Cl(X) = Pic(U) where U is the complement of the singular locus, isomorphic to the complement of the exceptional divisors in X'. So Cl(X) is a quotient of Pic(X'), and presumably this respects the intersection pairings. | |
| Mar 9, 2021 at 9:14 | comment | added | AG learner | Dear Martin, thank you for the nice answer. I'm wondering do you know of any study on the "root system" on the singular cubic surface $X$? For example, there is still an intersection pairing on Weil divisors as long as $X$ is normal (mathoverflow.net/q/90372). So it still makes sense to talk about the set of $(-2)$-classes on $X$. I'm thinking that since the root system of $X'$ is still $E_6$, could the "root system" of $X$ by any chance is some complement of the sub-root systems corresponding to the rational singularities? | |
| Aug 23, 2019 at 18:55 | vote | accept | modnar | ||
| Aug 23, 2019 at 11:32 | history | answered | Martin Bright | CC BY-SA 4.0 |