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It is well known that if you have a smooth quartic surface $X\subset \mathbb{P}^3$, it may or may not have lines in it. Indeed, $X$ has the following options, 64 (the maximal number), 32, 16, or none.

Between the space of all such quartics in $\mathbb{P}^3$ which is $|\mathcal{O}_{\mathbb{P}^3}(4)|=\mathbb{P}^{34}$, those with at least one line in it form a divisor $D$. Therefore, intersections of such a divisor with curves in $\mathbb{P}^{34}$ may give enumerative information about points(quartics) in the intersection.

Is all the enumerative information about such quartics encoded in $D$?

I'd like to know more about such a divisor $D$. So, Could someone recommend free-references of this online?

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  • $\begingroup$ Do you want to restrict to smooth surfaces maybe? Otherwise of course you can have more lines... Also your question is a little vague. What would "all the enumerative information" being "encoded" in D mean? Is there a specific enumerative question you're interested in? $\endgroup$
    – mdeland
    Commented Feb 9, 2010 at 14:15
  • $\begingroup$ Sure, the smooth case, thks. On the other hand, what I meant is that a result in enumerative geometry may have an interpretation in terms of the "intersection theory" of such a divisor $D$. $\endgroup$
    – Csar Lozano Huerta
    Commented Feb 9, 2010 at 14:58
  • $\begingroup$ I guess you know the article arxiv.org/PS_cache/arxiv/pdf/0705/0705.1653v2.pdf ? $\endgroup$
    – Dmitri Panov
    Commented Sep 13, 2010 at 17:11

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