Timeline for An analogue of Brill-Noether for hypersurfaces?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 19, 2017 at 18:20 | comment | added | Nati | Let us continue this discussion in chat. | |
| Mar 19, 2017 at 17:02 | comment | added | Jason Starr | There are obstructions to deforming individual curves. However, there is a powerful, foundational theorem in algebraic geometry -- the "Generic Flatness Theorem" -- that guarantees that for every finite type morphism whose target is Noetherian and reduced, there exists a dense open subset of the target over which the morphism is flat. | |
| Mar 19, 2017 at 15:05 | comment | added | Nati | @JasonStarr Again, I believe there should be some kind of deformation sequence that shows this. It is probably obvious for an algebraic geometer... but I'm not sure how to do it. | |
| Mar 19, 2017 at 5:00 | comment | added | Nati | What I'm worried about is the possibility of having a curve contained in a hypersurface/complete intersection and being unable to extend it to nearby curves. So I was wondering if that's a deformation theory problem (i.e., we deform the abstract curve, then we need to make sure that there exists a possibility to lift it to the deformation of the embedded curve, then we need to check the hypersurface/complete intersections deforms as well, and that there exists such a deformation that contains a deformation of the embedded curve). | |
| Mar 19, 2017 at 4:56 | comment | added | Nati | Here is what I'm asking (btw it is completely possible that for an algebraic geometer this is trivial): I pick a general curve with a general $\mathfrak{g}^r_d$ with $r \geq 3$ which is contained in a smooth complete intersection of multidegree $(e_1,\ldots,e_m)$. What conditions do I need to impose so that any other general curve with a general $\mathfrak{g}^r_d$ is contained in a smooth complete intersection of multidegree $(e_1,\ldots,e_m)$. I also want to make sure that these complete intersections can vary smoothly (i.e., in a flat family over an open subset of $\mathcal{M}_g$). | |
| Mar 19, 2017 at 3:36 | comment | added | Jason Starr | I do not understand what you are asking. By generic flatness, every finite type morphism with target $\mathcal{G}_d^r$ (or $\mathcal{W}_d^r$ if you prefer) is flat over a dense Zariski open of the unique irreducible component that dominates $\mathcal{M}_g$. | |
| Mar 19, 2017 at 1:12 | comment | added | Nati | @Jason: thanks again! But if I understand correctly - your answer explains when a specific curve is contained in a hypersurface of degree $e$, but what about the deformation theory? i.e. this is probably a naive question, but why is it obvious you can choose these hypersurfaces to vary in a flat family? | |
| Mar 18, 2017 at 22:28 | comment | added | Jason Starr | Your question appears to be asking the following: for a general curve $C$, and for a general $\mathfrak{g}^r_d$ on that curve, say $\phi:C\to \mathbb{P}^r$ (let's assume $r\geq 3$ so that $\phi$ is an embedding), for the ideal sheaf $I$ of $\phi(C)\subset \mathbb{P}^r$, for which $e>0$ is $h^0(\mathbb{P}^r,I(e))$ positive? Of course if $e> 2(g-1)/d$, then $h^0(\mathbb{P}^r,I(e)) \geq \binom{r+e}{r} - (ed + 1 -g)$. | |
| Mar 17, 2017 at 20:45 | comment | added | Jason Starr | Yes, it is true if you add "linearly nondegenerate" to the hypotheses for a general member of $\mathcal{H}_{d,g,r}$. | |
| Mar 17, 2017 at 20:43 | comment | added | Nati | corrected. Though out of curiousity - as you said in the previous question: it is true if I add "smooth, embedded, linearly nondegenerate curves", right? | |
| Mar 17, 2017 at 20:41 | history | edited | Nati | CC BY-SA 3.0 |
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| Mar 17, 2017 at 20:32 | comment | added | Jason Starr | It is not true that Brill-Noether theory implies that when there exists at least one component $\mathcal{H}_{d,g,r}$ dominating $\mathcal{M}_g$, then this component is unique. I already addressed this in a previous MO answer: mathoverflow.net/questions/178822/… | |
| Mar 17, 2017 at 19:56 | history | asked | Nati | CC BY-SA 3.0 |