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Timeline for Numerical invariants of symmetric products of curves

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Dec 27, 2015 at 9:12 vote accept rita
Dec 27, 2015 at 9:12 vote accept rita
Dec 27, 2015 at 9:12
Dec 27, 2015 at 0:06 answer added Yusuf Mustopa timeline score: 4
Dec 26, 2015 at 18:07 comment added rita I am interested in $n<g$.
Dec 26, 2015 at 17:15 comment added abx ... same for $n\geq 2g-1$, since $C(n)$ is a $\mathbb{P}^m$-bundle over $JC$.
Dec 26, 2015 at 15:58 comment added abx For $n=g$ the exceptional divisor of the birational contraction $C(g)\rightarrow JC$ is a canonical divisor, it is certainly not nef.
Dec 26, 2015 at 14:35 history edited rita CC BY-SA 3.0
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Dec 26, 2015 at 14:35 comment added rita @JasonStarr: thank you. In fact $C$ general is what I am interested in, and $g$ large with respect to $n$ is also fine.
Dec 26, 2015 at 12:52 comment added Jason Starr Most of what I know about this topic, I learned from Yusuf Mustopa. If $n$ is at least as large as the gonality of $C$ (e.g., certainly $n\geq (g+2)/2$ suffices), and if $C(n)$ has Picard rank $2$ (as it does if $C$ is sufficiently general in moduli), then we can easily compute the nef cone: the pullback of the theta divisor by the Abel map gives one ray, and the "pairwise difference" morphism to the product of Kummers of the Jacobian gives a second contraction (contracting the small diagonal). So we should be able to determine where the canonical divisor class is relative to the two rays.
Dec 26, 2015 at 12:28 comment added rita yes, I mean nef&big canonical bundle
Dec 26, 2015 at 12:22 comment added Jason Starr When you write "minimal", does that mean that the variety is a "minimal model", i.e., the canonical bundle is nef?
Dec 26, 2015 at 12:06 history asked rita CC BY-SA 3.0