Timeline for Rigid curves, and the "richness" of their homology class

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when toggle format	what		by	license	comment
Jan 25, 2015 at 15:12	comment	added	Walter Neff		I think this is known, at least in many cases. You might want to google 'curve counting for quintic threefolds' and the paper and P. Candelas, X. C. de la Ossa, P. S. Green, and L. Parkes. A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory.
Jan 23, 2015 at 20:05	comment	added	Brenin		Yes, that is a great example, thanks! As an aside: Is it true (it seems to be, according to your answer) that there is a rational curve $C\subset X$ of degree $d$ for every $d$?
Jan 22, 2015 at 14:19	comment	added	user47305		Ah, that'll do it! (though note that every other curve on $X$ is proportional in $H_2$ as well). Example 6.1 in Oguiso's arxiv.org/pdf/1206.1649v3.pdf is another: there you get a ray on $H_2$ that contains a finite set of rational curves, and nothing else. Actually, that example simultaneously shows that neither implication holds. Some of the curves have normal bundle $\mathcal O(-1) \oplus \mathcal O(-1)$, some have $\mathcal O \oplus \mathcal O(-2)$ (if memory serves), and they're all in a ray in $H_2$.
Jan 22, 2015 at 10:22	history	answered	Walter Neff	CC BY-SA 3.0