It is a well known fact that a smooth cubic surface in $P^3$ contains 27 lines. One proof proceeds by moving through the parameter space $U$ of smooth cubics until one reaches an cubic that can be understood by simple algebra, such as the Fermat cubic $V(x_0^3 + x_1^3 + x_2^3 + x_3^3)$. This is formalized by introducing an incidence correspondence $X$ of lines on cubics in $U \times G(1,3)$; X turns out to be a $P^{15}$ bundle over $G(1,3)$, and the projection of $X$ onto $U$ can be shown via the Jacobian criterion to be etale and finite. From there one is essentially done.

Fix some $N > 3$.

If one considers the Hilbert scheme of degree 3 surfaces in $P^N$, it is plausible that one can repeat the flavor of this argument. However, it's not clear that the Hilbert polynomials of the smooth integral degree 3 surfaces are the same (it's also not clear to me that they will be different), so it is possible that the Hilbert scheme will have multiple components. However, fixing or giving geometric meaning to the extra coefficient in the Hilbert polynomial will solve this problem.

Moreover, this Hilbert scheme may stop being smooth, and the incidence correspondence may not be so easily analyzable. I don't know how to compute the dimensions of these Hilbert schemes - I recall one can compute $H^0$ of the normal bundle of a given cubic surface to find the tangent space on the Hilbert scheme at that cubic. In the case of a complete intersection, it's possible to describe the normal bundle using adjunction, but computing global sections doesn't seem straightforward.

I am wondering what is known in this direction. Let me some specific questions:

1) How many components does the Hilbert scheme of degree 3 surfaces have? What if we pass to the (open?) subscheme of smooth, irreducible surfaces? What are their dimensions? Are they smooth?

2) How many lines are on a smooth irreducible degree 3, cubic surface in $P^N$? What are the possibilities? (Irreducibility imposed to rule out the disjoint union of a plane and a degree 2 hypersurface in $P^5$, which has infinitely many lines.)

3) Can someone give an example of a smooth, irreducible degree 3 surface in $P^N$ ($N > 3$) that is not the blow up of $P^2$ at 6 points? How many lines does it have? Is there an example with no lines?

[4) Is this suitable for MO?]


1 Answer 1


If $X\subset \mathbb P^n$ is a smooth and non-degenerated variety then $$ \deg X \ge 1 + \mathrm{codim} X $$ as you can learn from Varieties of Minimal Degree by Eisenbud and Harris.

Thus to understand smooth cubic surfaces, it suffices to consider cubic surfaces in $\mathbb P^3$ and $\mathbb P^4$. Smooth degree $3$ surfaces in $\mathbb P^4$ are rational normal scrolls (loc. cit. Theorem 1), and as such contains infinitely many lines.


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