Timeline for Pushforward and pullback on the level of Chow varieties
Current License: CC BY-SA 4.0
9 events
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| Aug 26, 2022 at 15:18 | comment | added | Jason Starr | A generic curve is liftable to some curve, but not to a curve with the specified degree. In the example of the double cover of the projective plane by a smooth quadric surface branched over a plane conic, the inverse image of a generic line is a smoot plane conic contained in the smooth quadric surface. However, only the inverse images of lines tangent to the conic are unions of a pair of lines. | |
| Aug 26, 2022 at 14:39 | comment | added | user127776 | @JasonStarr Thanks. But in your example instead of lines if you consider curves (dimension 1 cycles), you can pull back every cycle in the plane to its cover. It might have many irreducible components but at least one of these should surject onto the line in the plane. So it seems a generic curve is liftable. By the flat I don't mean surjective flat the image could be a proper open. In your example the image is not open but I think if one considers curves instead of line it might become open. | |
| Aug 26, 2022 at 11:11 | comment | added | Jason Starr | If you like at a refined pushforward map to the space of cycles in the target satisfying the specified contact conditions, that often is flat. This is one of the ingredients for relating “relative” Gromov-Witten invariants to Gromov-Witten theory of cyclic covers. | |
| Aug 26, 2022 at 10:55 | comment | added | Jason Starr | The morphism $f_*$ sometimes is flat, but usually it is not. Consider cyclic branched covers of the projective plane $\mathbb{P}^2$ and the parameter space of lines on the cover. Every line on the cover maps to a line in the projective plane, but typically a line in the projective plane "lifts" to a line in the cover if and only if it satisfies some "contact condition" with respect to the branch divisor in $\mathbb{P}^2$. Thus, for a double cover of the plane branched over a plane conic (i.e., for a quadric surface), only the tangent lines to the conic lift. | |
| Aug 26, 2022 at 0:14 | history | edited | user127776 | CC BY-SA 4.0 |
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| Aug 25, 2022 at 10:53 | comment | added | Jason Starr | The only reason that I wrote "Chow scheme" is because the argument also works for the Chow scheme, and not merely for the Chow variety. There is a functor of points view for the Chow variety as well, cf., the first chapter of "Rational curves on algebraic varieties" by J'anos Koll'ar. | |
| Aug 25, 2022 at 0:44 | comment | added | user127776 | @JasonStarr Thanks I'll take a look. So it seems it becomes easier by working with Chow scheme rather than variety right? The fact the variety structure can be extended to scheme structure makes it slightly easier since I think in that case it just follows from the functor of points view and the regular pushforward/pullback of cycles. | |
| Aug 25, 2022 at 0:31 | comment | added | Jason Starr | Yes, there are pushforwards and pullbacks on the level of Chow schemes. The basic idea is explained in Mumford's "Lectures on curves on an algebraic surface." | |
| Aug 24, 2022 at 22:59 | history | asked | user127776 | CC BY-SA 4.0 |