In this section we relate the classifications of pairs in terms of singularity theory and the equations defining them. We have divided our lemmas in four groups: classification of singular cubic surfaces, classification of pairs (S, D) with singular boundary D, classification of pairs (S, D) where S is singular at a point \(P\in D\) and classification of pairs (S, D) invariant under a \({\mathbb {C}}^*\)-action. We will denote homogenous polynomials of degree d in \(n+1\) variables as \(f_d(x_0,\ldots ,x_{n}), g_d\), etc. Recall that pairs (S, D) and \((S',D')\) are projectively equivalent if and only if they are conjugate to each others by elements of \(\mathrm {Aut}({\mathbb {P}}^3)\).
Lemma 6
([4, Lemma 3]) Let \(F=x_0x_1x_3+f_3(x_0,x_1,x_2)\), \(P=(0,0,0,1)\), \(Q=(0,0,1,0)\), \(H=\{x_3=0\}\cong \mathbb P^2_{(x_0,x_1,x_2)}\) and \(H_i=\{x_i=x_3=0\}\subset H\) for \(i=0,1\).
-
1.
The singularities of \(\{F=0\}\) other than that at P correspond to the intersection of \(C=\{x_0x_1=0\}\subset H\) and \(C'=\{f_3=0\}\) at points R other than Q. Indeed, if \(\mathrm {mult}_{R}(C\cdot C')=k\), then R is an \({\varvec{A}}_{k-1}\) singularity.
-
2.
If \(f_3(0,0,1) \ne 0\), then P is an \({\varvec{A}}_2\) singularity. Let \(k_i=\mathrm {mult}_{Q}(H_i\cdot C')\). If both \(k_0\) and \(k_1\) are both at least 2, then \(\{F=0\}\) has non-isolated singularities. Otherwise P is an \({\varvec{A}}_{k_0+k_1+1}\) singularity for \(\{ k_0,k_1 \}=\{1,1 \}\), \(\{1,2\}, \{1,3\}\).
Lemma 7
A pair (S, D) is such that S has an \({\varvec{A}}_2\) singularity at a point \(P\in D\) or a degeneration of one if and only if P is conjugate to (0, 0, 0, 1) and simultaneously (S, D) is projectively equivalent to the pair defined by equations
$$\begin{aligned} x_3f_2(x_0,x_1)+f_3(x_0,x_1,x_2)=0, \qquad {} l_1(x_0,x_1,x_2)=0. \end{aligned}$$
Proof
Without loss of generality, we may assume \(P=(0,0,0,1)\). By Lemma 6, S has (a degeneration of) an \({\varvec{A}}_2\) singularity at P if and only if it is given by the equation \(x_0x_1x_3+f_3(x_0,x_1,x_2)=0\). Any quadric \(f_2(x_0,x_1)\) can be transformed to \(x_0x_1\) or to a degeneration of \(x_0x_1\) (e.g. \(x_0^2\)) by a change of coordinates preserving \(x_2\) and \(x_3\). The lemma follows because a hyperplane section D contains P if and only if D is given by a linear form \(l_1(x_0,x_1,x_2)\).
Lemma 8
A surface S has an \({\varvec{A}}_3\) singularity or a degeneration of one if and only if it is projectively equivalent to:
$$\begin{aligned} \{x_3f_2(x_0,x_1)+x_2^2f_1(x_0,x_1)+x_2g_2(x_0,x_1)+g_3(x_0,x_1)=0\}. \end{aligned}$$
Proof
By Lemma 6, we may assume \(S=\{x_0x_1x_3+f_3(x_0,x_1,x_2)=0\}\) and \(P=(0,0,0,1)\). Moreover, the singularity is of type \({\varvec{A}}_k\) with \(k\geqslant 3\) if and only if \(f_3(0,0,1)=0\). Therefore \(f_3(x_0,x_1,x_2)=x_2^2f_1(x_0,x_1)+x_2g_2(x_0,x_1)+g_3(x_0,x_1)\).
Lemma 9
A surface S has an \({\varvec{A}}_4\) singularity or a degeneration of one if and only if it is projectively equivalent to \(\{x_3x_0l_1(x_0,x_1)+x_0x_2^2+x_2g_2(x_0,x_1)+g_3(x_0,x_1)=0\}\).
Proof
By Lemma 6, the surface S is defined by the equation
$$\begin{aligned} x_0x_1x_3+f_3(x_0,x_1,x_2)=0, \end{aligned}$$
where \(f_3(x_0x_1x_2)=x_2^2f_1(x_0,x_1)+x_2g_2(x_0,x_1)+g_3(x_0,x_1)\), \(k_0=\mathrm {mult}_{Q}(H_0\cdot C')\geqslant 2\) and \(k_1=\mathrm {mult}_{Q}(H_1\cdot C')\geqslant 1\) if and only if P is (a degeneration of) an \({\varvec{A}}_4\) singularity, where \(C'\) is the curve given in Lemma 6. Notice that
$$\begin{aligned} k_i=\mathrm {mult}_Q(H_i\cdot C')=\dim _{{\mathbb {C}}}\left( \frac{{\mathbb {C}}[x_0,x_1]}{\langle x_i,f_1+g_2+g_3\rangle }\right) . \end{aligned}$$
Therefore \(k_0\geqslant 2\) if and only if \(f_1(0,1)=0\). Hence, \(f_1=x_0\). The lemma follows from noticing that \(x_0x_1x_3\) is projectively equivalent to \(x_0x_3l_1(x_0,x_1)\) by an element of \(\mathrm {Aut}({\mathbb {P}}^3)\) fixing \(x_0, x_2, x_3\).
The proof of the next lemma is similar to the proof of Lemma 6, so we omit it.
Lemma 10
A surface S has an \({\varvec{A}}_5\) singularity or a degeneration of one if and only if it is projectively equivalent to
$$\begin{aligned} \{x_3x_0l_1(x_0,x_1)+x_0x_2f_1(x_0,x_1,x_2)+f_3(x_0,x_1)=0\}. \end{aligned}$$
In Figure 2 we see that the only non-trivial degenerations of a \({\varvec{D}}_4\) singularity in a cubic surface which are not a \(\tilde{{{\varvec{E}}}_6}\) singularity are \({\varvec{D}}_5\) and \({{\varvec{E}}}_6\) singularities. Hence the next lemma follows at once from [4, Case C].
Lemma 11
A surface S has a \({\varvec{D}}_4\) singularity or a degeneration of one if and only if it is projectively equivalent to
$$\begin{aligned} \{x_3x_0^2+f_3(x_0,x_1,x_2)=0\}. \end{aligned}$$
Lemma 12
A surface S has a \({\varvec{D}}_5\) singularity or a degeneration of one if and only if it is projectively equivalent to
$$\begin{aligned} \{f_3(x_0,x_1)+x_2g_2(x_0,x_1)+x_0x_2^2+x_0^2x_3=0\}. \end{aligned}$$
Proof
By Lemma 11 and Figure 2, we may assume that S is given by \(x_3x_0^2+f_3(x_0,x_1,x_2)\) since \({\varvec{D}}_5\) is a degeneration of \({\varvec{D}}_4\). Let \(H=\{x_3=0\}\), \(C=\{x_3=f_3(x_0,x_1,x_2)=0\}\subset H\) and \(C'=\{x_3=x_0=0\}\subset H\). We can rewrite \(f_3=x_2^2g_1(x_0,x_1)+x_2g_2(x_0,x_1)+g_3(x_0,x_1)\). By [4, Lemma 4], the point \(P=(0,0,0,1)\) is (a degeneration of) a \({\varvec{D}}_5\) singularity if and only if \(C\cap C'\) consist of at most two points. The equation of \(S\cap H\subset H\) localized at \(Q=(0,0,1,0)\) is \(g_1(x_0,x_1)+g_2(x_0,x_1)+g_3(x_0,x_1)=0\), and \(C\cap C'\) consists of at most two points if and only if
$$\begin{aligned} \dim _{{\mathbb {C}}}\left( \frac{{\mathbb {C}}[x_0,x_1]}{\langle x_0,g_1+g_2+g_3\rangle }\right) \geqslant 2. \end{aligned}$$
The latter is equivalent to taking \(g_1=ax_0\), which by rescaling \(x_2\) gives the result.
Lemma 13
The unique cubic surface S with a \({{\varvec{E}}}_6\) singularity or a degeneration of one such surface is projectively equivalent to
$$\begin{aligned} \{x_3x_0^2+x_0x_2l_1(x_0,x_1,x_2)+f_3(x_0,x_1)=0\}. \end{aligned}$$
Proof
Using the same notation as in Lemma 12 and following [4, Lemma 4], S is defined by \(x_3x_0^2+x_2^2g_1(x_0,x_1)+x_2g_2(x_0,x_1)+g_3(x_0,x_1)=0,\) and has (a degeneration of) an \({{\varvec{E}}}_6\) singularity if and only if
$$\begin{aligned} \dim _{{\mathbb {C}}}\left( \frac{{\mathbb {C}}[x_0,x_1]}{\langle x_0,g_1+g_2+g_3\rangle }\right) \geqslant 3. \end{aligned}$$
The latter is equivalent to take \(g_1=x_0\) and \(g_2=x_0l_1(x_0,x_1)\).
Remark 1
(see [4, Case E]) A surface S has an isolated \(\widetilde{{\varvec{E}}}_6\) singularity if and only if S is the cone over a smooth plane cubic curve given by \(f_3(x_0,x_1,x_2)=0\).
Consider a pair (S, D) and a point \(P\in D\subset S\). By choosing coordinates appropriately we can suppose that \(P=(0,0,0,1)\) and \((S,D)=(\{F=0\},\{F=H=0\})\) for F and H given as
$$\begin{aligned} F=x_0f_2(x_0,\ldots ,x_3)+x_3^2f_1(x_1,x_2)+x_3g_2(x_1,x_2)+f_3(x_1,x_2),\quad H=x_0. \end{aligned}$$
(1)
Lemma 14
A pair (S, D) has D with an \({\varvec{A}}_2\) singularity at a point P or a degeneration of one if and only if (S, D) is projectively equivalent to the pair defined by equations:
$$\begin{aligned} x_0f_2(x_0,x_1,x_2,x_3)+x_3x_1^2+f_3(x_1,x_2)=0,\qquad x_0=0. \end{aligned}$$
(2)
Proof
Without loss of generality we can suppose (S, D) is given by (1). The equation of (a degeneration of) a plane cubic curve in \(\{x_0=0\}\) with an \({\varvec{A}}_2\) singularity at P is given by \(x_1^2x_3+f_3(x_1,x_2)=0\), where the curve has an \({\varvec{A}}_2\) singularity at P if and only if \(x_2^3\) has a non-zero coefficient in \(f_3\). Therefore D is as in the statement if and only if in (1) we take \(f_1=0\) and \(g_2=x_1^2\).
Lemma 15
A pair (S, D) has D with an \({\varvec{A}}_3\) singularity at P or a degeneration of one if and only if (S, D) is projectively equivalent to the pair defined by \(x_0f_2(x_0,x_1,x_2,x_3)+x_1(x_2^2+x_1l_1(x_1,x_2,x_3))=0\) and \(x_0=0\).
Proof
We may assume that the equations of (S, D) are as in (1) and \(P=(0,0,0,1)\). By restricting to \(\{x_0=0\}\cong {\mathbb {P}}^2\) and localizing at P, the equation for D is \(f_1(x_1,x_2)+g_2(x_1,x_2)+f_3(x_1,x_2)\) and by choosing coordinates appropriately we may assume that \(L=\{x_1=0\}\) and \(C=\{x_2^2+x_1l_1(x_1,x_2)=0\}\) are a line and a conic intersecting at P, where l is a polynomial of degree 1, not necessarily homogeneous. Therefore \(D|_{x_0=0}\) has equation \(x_1(x_2^2+x_1l_1(x_1,x_2,x_3))\) so \(f_1\equiv 0\), \(g_2\equiv ax_1^2\), \(f_3=x_1x_2^2 + x_1l_1(x_1,x_2,0)\) and the result follows.
By similar arguments, one can prove the next two results:
Lemma 16
A pair (S, D) has D with a \({\varvec{D}}_4\) singularity at P or a degeneration of one if and only if (S, D) is projectively equivalent to the pair defined by equations \(x_0f_2(x_0,x_1,x_2,x_3)+f_3(x_1,x_2)=0\) and \(x_0=0\).
Lemma 17
A pair (S, D) has D non-reduced if and only if it is projectively equivalent to the pair defined by equations:
$$\begin{aligned} x_0f_2(x_0,x_1,x_2,x_3)+x_1^2f_1(x_1,x_2,x_3)=0,\qquad x_0=0. \end{aligned}$$
Lemma 18
A pair (S, D) has \(D=L+C\) where L is a line and C is a conic such that \(3L\in |-K_S|\) if and only if it is projectively equivalent to the pair defined by equations:
$$\begin{aligned} x_0f_2(x_0,x_1,x_2,x_3)+ax_1^3=0,\qquad l_1(x_0,x_1)=0. \end{aligned}$$
where L and 3L are projectively equivalent to \(\{x_0=x_1=0\}\) and \(=\{x_0=0\}|_S\), respectively. This surface has a point \(Q\in L\subset \mathrm {Supp}(D)\) such that S has a singularity at Q that is not of type \({\varvec{A}}_1\).
Proof
Suppose (S, D) as in the statement. Without loss of generality, we may suppose that the equation of S is as in (1), \(D=\{x_0+b x_1=0\}\) and let \(D':=\{x_0=0\}\). Clearly \(L\subset \mathrm {Supp}(D')\cap \mathrm {Supp}(D)\) and \(D=D'\) if and only if \(b=0\). In this case, the equation of \(D=D'\) in \(\{x_0=0\}\cong \mathbb {P}^2\) is given by \(x_3^2f_1(x_1,x_2)+x_3g_2(x_1,x_2)+f_3(x_1,x_2)=0\) and \(3L\in |-K_S|\) if and only if \(f_1=g_2\equiv 0\) and \(f_3=ax_1^3\). If \(b\ne 0\), then \(x_1=-\frac{x_0}{b}\). Take \(x_0=0\) in (1). The equation of \(D'=\{x_0=0\}|_S\) is \(x_3^2f_1+x_3g_2+f_3=0\) and \(D'\equiv 3L\) if and only if \(f_1=g_2=0\) and \(f_3=x_1^3\). But then, the equation of D in \(\{x_0+bx_1=0\}\) is \(x_1(bf_2+x_1^2)\) and \(C=\{bf_2+x_1^2=x_0+bx_1=0\}\). It is a well known fact that the line L contains a point Q at which S is singular and Q is not of type \({\varvec{A}}_1\) (see [19, p. 227]).
Lemma 19
Given a pair (S, D), S is singular at a point \(P\in D\) and D is an \({\varvec{A}}_2\) singularity at P or a degeneration of one if and only if (S, D) is projectively equivalent to the pair defined by equations:
$$\begin{aligned} x_3x_0l_1(x_0,x_1,x_2)+x_3x_1^2+f_3(x_1,x_2)+x_0f_2(x_0,x_1,x_2)=0,&x_0=0. \end{aligned}$$
(3)
Proof
Without loss of generality we can assume \(P=(0,0,0,1)\). Then, the equation of S can be written as (see [4, Section 2, pp. 247–252])
$$\begin{aligned}&x_3h_2(x_0,x_1,x_2)+h_3(x_0,x_1,x_2)\\&\quad =a_0x_3x_1^2+x_0f_2(x_0,x_1,x_2)+f_3(x_1,x_2)+x_1x_3g_1(x_0,x_2)+x_3g_2(x_0,x_2). \end{aligned}$$
By comparing with the equation in Lemma 14, D has (a degeneration of) an \({\varvec{A}}_2\) singularity at P if and only if \(g_1(x_0,x_2)=ax_0\) and \(g_2(x_0,x_2)=bx_0^2+cx_0x_2\). The lemma follows.
The proof of the next lemma is similar to that of Lemma 19.
Lemma 20
Given a pair (S, D), S is singular at a point \(P\in D\) and D has an \({\varvec{A}}_3\) singularity at P or a degeneration of one if and only if (S, D) is projectively equivalent to the pair defined by equations:
$$\begin{aligned} x_0^2l_1(x_0,x_1,x_2,x_3)+x_0f_2(x_1,x_2) +x_0x_3g_1(x_1,x_2) +x_1^2h_1(x_1,x_2,x_3) +x_1x_2^2=0, \end{aligned}$$
\(x_0 =0\).
Lemma 21
Let (S, D) be a pair that is invariant under a non-trivial \({\mathbb {C}}^*\)-action. Suppose the singularities of S and D are given as in the first and second entries in one of the rows of Table 4, respectively. Then (S, D) is projectively equivalent to \((\{F=0\},\{F=H=0\})\) for F and H as in the third and fourth entries in the same row of Table 4, respectively. In particular, any such pair (S, D) is unique. Conversely, if (S, D) is given by equations as in the third and fourth entries in a given row of Table 4, then (S, D) has singularities as in the first and second entries in the same row of Table 4 and (S, D) is \({\mathbb {C}}^*\)-invariant. Furthermore the element \(\lambda \in \mathrm {SL}(4, {\mathbb {C}}^*)\), as defined in Lemma 2, given in the fifth entry of the corresponding row of Table 4 is a generator of the \({\mathbb {C}}^*\)-action.
Table 4 Some pairs (S, D) invariant under a \({\mathbb {C}}^*\)-action
Proof
There is a unique surface S with three \({\varvec{A}}_2\) singularities [4, p. 255] which corresponds to the equation in Table 4. When a surface S has singularities \({\varvec{A}}_4+{\varvec{A}}_1\), \({\varvec{A}}_5+{\varvec{A}}_1\), \({\varvec{D}}_4\), \({\varvec{D}}_5\) or \({{\varvec{E}}}_6\), and a \(\mathbb C^*\)-action, the equation for F follows from [8, Table 3]. If S has singularities \({\varvec{A}}_3+2{\varvec{A}}_1\), then [8, Table 3] gives that S has equation \(x_3f_2(x_0,x_1)+x^2_2l_1(x_0,x_1)=0\), where \(x_0x_1\) has a non-zero coefficient in \(f_2\), since otherwise S is singular along a line. Hence, after a change of coordinates involving only variables \(x_0\) and \(x_1\) and rescaling \(x_3\), we obtain the desired result. It is trivial to check that each one-parameter subgroup \(\lambda \) in the corresponding row of Table 4 leaves S invariant, and therefore \(\lambda \) is a generator of the \({\mathbb {C}}^*\)-action.
Given H, denote \(D_H=\{F=H=0\}\subset S\). We need to show that for (S, D) with prescribed singularities, \(D_H=D\) if and only if H is as stated in Table 4. Verifying that for F and H as in the table, the pair (S, D) has the exepected singularities is straight forward and we omit it. We verify the converse.
Suppose that S has three \({\varvec{A}}_2\) singularities. Then we may assume that \(F=x_0x_1x_3+x_2^3\) and the singularities correspond to \(P_1=(1,0,0,0)\), \(P_2=(0,1,0,0)\) and \(P_3=(1,0,0,0)\). There are only three lines \(L_1,L_2,L_3\) in S [4, p. 255], which correspond to \(\{x_2=x_i=0\}\) for \(i=0,1,3\), respectively. Clearly any two of these intersect at each of the points \(P_j\). Moreover \(D_H=D=\sum L_i\) and D has an \({\varvec{A}}_1\) singularity at each \(P_i\), as stated in Table 4.
Suppose that S has an \({{\varvec{E}}}_6\) singularity at a point P and D has an \({\varvec{A}}_2\) singularity at a point \(Q\ne P\) and (S, D) is \({\mathbb {C}}^*\)-invariant. Without loss of generality, we can now assume that \(F=x_0^2x_3+x_0x_2^2+x_1^3\), \(H=\sum a_ix_i\) for some parameters \(a_i\) and \(P=(0,0,0,1)\). Since \({{\overline{\lambda }}}_4\) is a generator of the \({\mathbb {C}}^*\)-action, then \({{\overline{\lambda }}}_4(t)\cdot H=a_0t^{11}x_0+a_1t^3x_1+a_2t^{-1}x_2+a_3t^{-13}x_3\). Therefore \(D_H\) is \({\mathbb {C}}^*\)-invariant if and only if \(H=x_i\) for some \(i=0,\dots ,3\). Notice that this happens every time the entries of \(\lambda \) are distinct. If \(H=x_0\), then \(D_H\) is a triple line. If \(H=x_1\), then \(D_H\) is the union of a conic and a line, and therefore \(D_H\) does not have an \({\varvec{A}}_2\) singularity. If \(H=x_2\), then \(D_H\) has an \({\varvec{A}}_2\) singularity at P. If \(H=x_3\), then \(D_H\) has an \({\varvec{A}}_2\) singularity at \(Q=(1,0,0,0)\ne P\) and \(D_H=D\).
Suppose S has a \({\varvec{D}}_5\) singularity at a point P, D has an \({\varvec{A}}_3\) singularity at a point \(Q\ne P\) and (S, D) is \({\mathbb {C}}^*\)-invariant. There is a unique pair satisfying these conditions. Reasoning as in the previous case, we may assume \(F=x_0^2x_3+x_0x_2^2+x_1^2x_2\), \(H=x_i\) for some \(i=0,\dots ,3\) and \(P=(0,0,0,1)\). It follows from the equations that \({{\overline{\lambda }}}_6\) generates the \({\mathbb {C}}^*\)-action. If \(H=x_0\) or \(H=x_2\), then the support of \(D_H\) contains a double line. If \(H=x_2\), then \(D_H\) has an \({\varvec{A}}_3\) singularity at P. If \(H=x_3\), then \(D_H\) has an \({\varvec{A}}_3\) singularity at \(Q=(1,0,0,0)\ne P\) and \(D_H=D\).
Suppose S has an \({\varvec{A}}_5\) singularity at a point P and an \({\varvec{A}}_1\) singularity at a point Q, D has an \({\varvec{A}}_2\) singularity at Q and (S, D) is \(\mathbb C^*\)-invariant. We may assume \(\lambda _6\) generates the \(\mathbb C^*\)-action, \(F=x_0x_2^2+x_0x_1x_3+x_1^3\), \(H=x_i\) for some \(i=0,\dots ,3\), \(P=(0,0,0,1)\) and \(Q=(1,0,0,0)\). If \(H=x_0\) then \(D_H\) is a triple line. If \(H=x_1\), then \(D_H\) has a double line in its support. If \(H=x_2\), then \(D_H\) has two \({\varvec{A}}_1\) singularities. If \(H=x_3\), then \(D_H\) has an \({\varvec{A}}_2\) singularity at \(Q=(1,0,0,0)\ne P\) and \(D_H=D\).
Suppose S has an \({\varvec{A}}_4\) singularity at a point P and an \({\varvec{A}}_1\) singularity at a point Q, D has an \({\varvec{A}}_3\) singularity at Q and (S, D) is \(\mathbb C^*\)-invariant. We may assume \(\lambda _5\) generates the \(\mathbb C^*\)-action, \(F=x_0x_1x_3 + x_0x_2^2+x_1^2x_2\), \(H=x_i\) for some \(i=0,\dots ,3\), \(P=(0,0,0,1)\) and \(Q=(1,0,0,0)\). If \(H=x_0\) or \(H=x_1\) then \(D_H\) contains a double line in its support. If \(H=x_2\), then \(D_H\) has three \({\varvec{A}}_2\) singularities and if \(H=x_3\), then \(D_H\) has an \({\varvec{A}}_2\) singularity at Q and \(D_H=D\).
Suppose S has a \({\varvec{D}}_4\) singularity at a point P, D has a \({\varvec{D}}_4\) singularity at a point \(Q\ne P\) and (S, D) is \({\mathbb {C}}^*\)-invariant. We may assume the generator of the \({\mathbb {C}}^*\)-action is \(\lambda _9\), \(F=x_0^2x_3 + x_1^3 +x_2^3 \) and \(P=(0,0,0,1)\). If \(D_H\) is \(\lambda _9\)-invariant, either \(H=x_i\) for some \(i=0,\dots ,3\) or \(H=x_1-ax_2\) for \(a\ne 0\). If \(H=x_0\), then \(D_H\) has a \({\varvec{D}}_4\) singularity at P. If \(H=x_1\) or \(H=x_2\), then \(D_H\) has an \({\varvec{A}}_2\) singularity. If \(H=x_1-ax_2\) with \(a\ne 0\), then \(D_H=\{x_0^2x_3+\left( 1 + \frac{1}{a}\right) x_1^3=0, x_2=\frac{x_1}{a}\}\) has an \({\varvec{A}}_2\) singularity. If \(H=x_3\), then \(D_H\) has a \({\varvec{D}}_4\) singularity at \(Q=(1,0,0,0)\ne P\) and \(D_H=D\).
Suppose S has an \({\varvec{A}}_3\) singularity at a point P, two \({\varvec{A}}_1\) singularities at points \(Q_1\) and \(Q_2\), \(D=2L+L'\) where L is a line containing \(Q_1\) and \(Q_2\) and \(L'\) is a line such that \(P, Q_1,Q_2\not \in L'\). Furthermore, suppose (S, D) is \({\mathbb {C}}^*\)-invariant. We may assume that \(\overline{\lambda }_3\) is the generator of the \({\mathbb {C}}^*\)-action, \(F=x_0x_1x_3+x_1x_2^2+x_0x_2^2\), \(P=(0,0,0,1)\), \(Q_1=(1,0,0,0)\), \(Q_2=(0,1,0,0)\) and \(L=\{x_2=x_3=0\}\). Moreover, if \(D_H\) is \({{\overline{\lambda }}}_3\)-invariant, either \(H=x_i\) for some \(i=0,\dots ,3\) or \(H=x_0-ax_1\) for \(a\ne 0\). If \(H=x_0\) or \(H=x_1\), then \(D_H\) does not contain L in its support. If \(H=x_2\) or \(H=x_0-ax_1\), then \(D_H\) is reduced. If \(H=x_3\), then \(D_H=2L+L'\), where \(L'=\{x1+x_0=x_3=0\}\). Since \(P, Q_1,Q_2\not \in L\), then \(D_H=D\).