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The following doubtlessly naive heuristic suggests to me that there might be some generalizations. I don't know whether, at one extreme, the story is classical, or at the other extreme, the heuristic just fails.

Consider a generic hypersurface $S$ of degree $d$ in ${\Bbb P}^n$. The intersection of $S$ with a generic plane $P$ should form a curve of degree $d$, hence a curve of genus $g=(d-1)(d-2)/2$.

The possible planes $P$ range over a Grassmannian ${\rm Gr}(n+1,3)$ of dimension $3(n-2)$.

One thus gets a morphism from ${\rm Gr}(n+1,3)$ to the moduli space of curves of genus $g$, which has dimension $3(d-1)(d-2)/2 - 3$ (unless $d=3$).

If the numbers work out right, one can try to make $n$ large enough, but not too large, so that one gets a 0-dimensional set of planes where the intersection gives rise to curves with some desired amount of degeneration. With enough degeneration perhaps, the original curves of genus $g$ will acquire components of smaller genus. So one might get interesting configurations of comparatively low genus curves (not necessarily all of the same genus). For example, with $S$ of degree $4$ one might look for a configuration of genus 1 and 2 curves. (Personally, I don't know enough about compactifying moduli spaces even to guess the details at this point.)

In any case, with the 27-lines on the cubic surface, elliptic curves would seem to degenerate into a finite set of lines sharing common points, making this classic object an example of the heuristic above.

All that said, I'll make my question the broad one in the title.

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Enumerative geometry. – Chandan Singh Dalawat Apr 11 '12 at 4:56
Isn't "enumerative geometry" both more general and less refined than what I'm asking. On one hand, it includes the counting of objects that don't necessarily live inside the given variety (bitangents, etc.) On the other hand, the 27 lines are more interesting than just the number 27. – David Feldman Apr 11 '12 at 5:09
I can't tell what structural property about the 27 lines you are hoping to generalize. A generic plane containing one of the 27 lines will yield a curve that is a union of a line with a conic, not a triple of lines (with multiplicity). Are you looking for a phrase like "intersection theory on moduli spaces"? – S. Carnahan Apr 11 '12 at 9:21
One possible answer to the question in the title (though not the one in the body) is given by so-called generalised del Pezzo varieties, as described by Dolgachev--Ortland (Asterisque 165). Briefly, the relevant combinatorial structure is a root lattice associated to a Dynkin diagram of a certain type (three legs, one of length 2), and the symmetry group of the configuration then comes from the Weyl group of this root system (that's intentionally vague, because I don't remember all the details). – user5117 May 24 '12 at 18:06
So for example in the case of the cubic surface, the relevant lattice is E_6, and the number 27 comes from the orbit-stabiliser theorem as |W(E_6)|/|W(E_5)|. One can calculate the number of (-1)-curves on other del Pezzo surfaces in the same way. Apart from the Asterisque volume cited above, this story is also described (at least the surface case) in Dolgachev's book on classical algebraic geometry, available at – user5117 May 24 '12 at 18:10

1 Answer 1

I agree with @Scarnahan that you do not make clear what you want to generalize. Perhaps I am too naive but e.g. you can also make the question not from the point of view of the surface with the lines but rather you could make the question: Is there an analogue of $P^2$ namely the well known construction of $ r \le 9 $ points in the projective plane with a special geometric configuration then by blowing up the plane in these points with the given configuration you(e.g. Manin's book on Cubic forms, Elsevier. ) blow them up and obtain the cubic surface and the exceptional divisors turn out to be the lines. So you could make the question if there is an analogon say for $P^3$ and instead of points take codimension one hypersurfaces with special configuration of lines, etc; You can play the same game for $P^n$ but now you have $r $ hyperplanes with a special geometrical configuration and the blow up with center at these hyperplanes? Depending on r you have to study the linear system defined by the blown up $P^n$. This should be found in the classical literature as e.g. Baker's principles of geometry. It is also well known that there are no surfaces of degree $n$ that are normal in $P^n$ for $ n \ge 9$ and for $ n \leq 9 $ the surfaces with such property arise as projections from a point ending with the cubic surface with the 27 lines. (See e.g. Semple and Roth's Introduction to algebraic geometry, Oxford University Press, spec. Chap. 7 ).

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