Let $S$ be a smooth del Pezzo surface of degree $d$ and $K_S^*$ the anticanonical class. It is well known that the set of classes

$$R(S)={alphain H^2(S,mathbb Z)|alpha^2=-2,alphacdot K_S^*=0},$$

together with intersection pairing is isomorphic to a root system $R_d$ of rank $9-d$, where $R_d$ from $d=1$ to $6$ is $E_8,E_7,E_6,D_5,A_4,A_2times A_1$. Moreover, Each root can be written as $(L_1)-(L_2)$, where $L_1,L_2$ are two disjoint lines on $S$.

I’m particularly interested in the $d=3$ case, where $S$ is a cubic surface, and the corresponding root system $E_6$ has 72 roots. I’d like to know

**Question 1:** Is there a notion of the root system on singular del Pezzo surfaces? Is there any related reference?

Now let $S$ be a normal del Pezzo surface of degree $d$ with at worst ADE singularities. Then there is a minimal resolution $pi:S’to S$, where $S’$ is a weak del Pezzo surface, which arises blowing up $9-d$ bubble points on $mathbb P^2$. Moreover, $pi$ is given by anticanonical map $|K_{S’}|$ and contracts all effective $(-2)$ curves. Since $S’$ is deformation equivalent to a smooth del Pezzo surface, the root system of $S’$ is also isomorphic to $R_d$. According to Dolgachev’s book, section 8.4 and 9.2.3, and also see Martin Bright’s answer, the set of effective $(-2)$ curves on $S’$ correspond to a sub root system $R_esubseteq R_d$. Such sub-root systems are classified. Moreover, the rank $r=text{rank}(R_e)$ coincides with the number of the $(-2)$-effective curves.

To me, the roots on singular $S$ should be a degeneration of the root system $R_d$. It should be certain “complement” of the sub-root system $R_e$ of the $(-2)$ curves occur in the resolution. However, the orthogonal complement does not seems to the correct notion, since strict transform of a line through the singular point will have nontrivial intersection with the exceptional curves, and there should be “roots” as differences of lines through the singularity.

The notion of root system $R(S)$ on singular $S$ that I have in mind is a quotient:

**Definition 1:** Let $R(S)$ be the quotient set $R_d/R_e:=R_d/sim$, where the equivalence relation on $R_d$ is $r_1sim r_2$ iff $r_1-r_2in R_e$.

Such a quotient is just a set and does not have a root system structure in general. (Although the quotient seems to behave well for type $A$ root systems, so for $dge 5$ cases, the quotients correspond to the Dynkin diagram by removing the sub-Dynkin diagram corresponding to $R_e$).

For example, for a cubic surface with an $A_1$ singularity, $R(S)=E_6/A_1$ has 50 “roots”; for a quartic del Pezzo surface with an $A_1$ singularity, $R(S)=D_5/A_1$ has 26 “roots”. As far as I know, 50 and 26 are not the number of roots of any root systems (even reducible).

The reason for me to consider this quotient is that the class group $text{Cl}(S)$ arises as a surjection $text{Pic}(S’)twoheadrightarrow text{Cl}(S)$ with kernel generated by the set of effective $(-2)$ curves. However, I haven’t seen any reference ever discussed such quotient.

**Question 2**: Is there a Lie theory interpretation of the quotient in Definition 1?

Any suggestions and comments are welcome!