Timeline for Components of Kontsevich moduli space of stable maps and reducible curves
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 23, 2016 at 10:16 | comment | added | Jason Starr | "Is it sufficient to say ..." To what are you referring? As my example shows, it is not true that every complete curve intersects the boundary of $\overline{M}_{0,0}(\mathbb{P}^n,e)$. In fact, there exists a (projective) contraction of the boundary of $\overline{M}_{0,0}(\mathbb{P}^n,e)$ discovered independenty by Anca and Andrei Mustata, by Adam Parker, and by Izzet Coskun, Joe Harris and myself. So, even for Fano complete intersections, a general point of $\overline{M}_{0,0}(X,e)$ is contained in a curve that does not intersect the boundary. | |
| Mar 23, 2016 at 9:59 | comment | added | user3001 | Is it sufficient to say that $\bar{M}_{0,0}(X,e)\subset \bar{M}_{0,0}(\mathbb P^n,e)$ and that $\bar{M}_{0,0}(\mathbb P^n,e)$ is irreducible with divisors (hence with nonempty intersection with a complete curve) corresponding to maps with reducible domains? | |
| Mar 20, 2016 at 21:04 | comment | added | Jason Starr | On the other hand, it seems almost certain (based on parameter counts) that for every Fano complete intersection in $\mathbb{P}^n$, for every $e>1$, every irreducible component of $\overline{M}_{0,0}(X,e)$ intersects the locus parameterizing reducible rational curves. Maybe the question has a positive answer if $\mathcal{O}_{\mathbb{P}^n}(1)$ generates the Picard group of $X$. | |
| Mar 19, 2016 at 18:46 | comment | added | Jason Starr | There are silly examples where you embed $X$ by a positive integer multiple $e>1$ of an ample invertible sheaf. Then the degree of every curve is divisible by $e$. So even the "minimal" curves have degree $\geq e$. | |
| Mar 19, 2016 at 17:14 | history | asked | user3001 | CC BY-SA 3.0 |