(Cross-posted by math.SE because I'm not sure which platform is suitable)

Given a smooth cubic surface $ X $ (Say over $ mathbb {C} $) as an explosion of $ mathbb {P} ^ 2 $ at the $ 6 $ Points) with Neron-Severi group $ NS (X) $ and canonical divider $ K_X $, the subset $ R = { alpha in NS (X): alpha. K_X = 0, alpha. alpha = -2 } $ is the root system of the Weyl group of $ E_6 $, I understand that there are previous questions about this grid on this site (for example, https://math.stackexchange.com/questions/1477220/27-lines-on-a-cubic-surface?rq=1 and https : // math.stackexchange.com/questions/82199/automorphism-group-of-the-configuration-of-lines-on-a-cubic-surface-and-quadrati?rq=1), but I could not find one Explanation of what happens *Singular* cubic surfaces (on this site or other sources).

Let's take a closer look $ varphi: X & # 39; longrightarrow X $ is a "minimal" resolution of singularities of a singular cubic surface $ X $ with rational colons as singularities. Suppose that $ X subset mathbb {P} ^ 3 $ has rational colons as singularities. For the sake of simplicity we will take $ X $ to have only one singular point (which we consider to be one) $ (1: 0: 0: 0) $) with singularity type $ A_1 $ or $ A_2 $, Then, for which sublattice of the root system described above $ X & # 39; $ Do the components of the extraordinary divider produce this resolution?

Under the above conditions, we can write $ X = (f = 0) $ With $ f (t_0, t_1, t_2, t_3) = t_0 g_2 (t_1, t_2, t_3) + g_3 (t_1, t_2, t_3) $ for some homogeneous polynomials $ g_2 $ and $ g_3 $ of the degree $ 2 $ and $ 3 $ each (listed in many sources, such as in Section 9.2 of Dolgachev's classical algebraic geometry).

In that $ A_2 $ Case I tried to look at $ (- 2) $– Curves resulting from two consecutive explosions of three collinear points (blow up one point and then one point on the extraordinary divider in the next explosion). We repeat this for two collinear triple points. I do not know how to get a subgrid from $ R $ (built for $ X & # 39; $), which follows the above information. I think that is, I have to better understand "usual" generators for the root lattice in the sleek case. For example, what do classes in $ X & # 39; $ come from line classes passing through the singular point in $ X $ look like? Are there any suggestions on how to proceed in this situation (or even in the $ A_1 $ case or other singularity types)?