Cubic surfaces and configurations of 6 points

Question

A smooth cubic surface $X\subset \mathbb{P}^3$ is isomorphic to $\mathbb{P}^2$ blown up at six points, so there should be a rational map

${\rm Hilb}^6\mathbb{P}^2\dashrightarrow H^0(\mathbb{P}^3,\mathscr{O}_{\mathbb{P}^3}(3))//PGL_4$

Given a 1-parameter family $Z_t$ of length six subschemes of $\mathbb{P}^2$, where $Z_t$ is reduced for $t\neq 0$, specializing to $Z_0$, what happens to the singularities of the corresponding 1-parameter family $X_t$ of cubic surfaces? For example, I know that $X_0$ having an $A_1$ singularity could be from 3 points becoming collinear or 6 points lying on a conic (contributing to a (-2) curve getting collapsed under the canonical map), but I don't know of references for other singularities.

Answer 1

Any smooth cubic surface of $\mathbb{P}^3$ is the blow-up of six points of $\mathbb{P}^2$ such that no three are collinear and no six are on a conic. More generally, if $\sigma\colon S\to \mathbb{P}^2$ is the blow-up of six points, maybe some infinitely near, such that $−K_S$ is nef (or equivalently such that all curves of negative self-intersection are smooth rational (−1)-curves or (−2)-curves), the anticanonical map $\eta\colon S \to\mathbb{P}^3$ is a birational morphism to a normal cubic surface, which contracts all $(−2)$- curves of $S$ (it is an isomorphism if and only if $−K_S$ is ample, which means that $S$ is del Pezzo).

Conversely, all normal cubic surfaces except cones over smooth cubic curves are obtained in this way. Indeed such a surface $S$ admits only double points, is rational (project from a singularity) and satisfies $H^1(S, \mathcal{O}_S) = H^2(S, \mathcal{O}_S) = 0$ (use the exact sequence $0 \to\mathcal{O}(−3) → \mathcal{O} → \mathcal{O}(S) → 0$); hence by [ I. Dolgachev, Classical algebraic geometry: a modern view (Cambridge University Press, Cambridge, 2012, Proposition 8.1.8(ii)] $S$ admits only rational double points. Alternatively, one can use the classification of singularities on cubic surfaces in [J. W. Bruce and C. T. C. Wall, ‘On the classification of cubic surfaces’, J. London Math. Soc. (2) 19 (1979) 245–256., § 2], which does not rely on a cohomological argument. Then the minimal resolution $\hat{S} \to S$ is a weak del Pezzo surface and is the blow-up of six points of $\mathbb{P}^2$ by [Dolgachev (as above), Theorem 8.1.13]. The singularities you can obtain follow from this description. You can for instance get $A_1$, $\ldots$, $A_5$, $E_6$, $D_5$.

For more details, see [M. Demazure, Surfaces de del Pezzo, II, III, IV, V, Lecture Notes in Mathematics 777 (Springer, Berlin, 1980) 21–69.] and [Dolgachev (as above), § 8.1].