2.1 Real forms of del Pezzo surfaces of degree 1 (see [Reference Finashin and Kharlamov7] and references therein)

Recall that the real deformation class of any real del Pezzo surface X of degree 1 is determined by the topology of $X_{\mathbb R}$ . There are 11 deformation classes. The corresponding topological types are shown in the first line of Table 1, where ${\mathbb T}^2$ stands for a 2-torus and $\mathbb {K}$ for a Klein bottle.

Table 1 The root lattices $\operatorname {\Lambda }(X)=K_X^{\perp } \cap \ker (1+\operatorname {conj}_*)$

The lattice $\operatorname {\Lambda }(X)=K_X^{\perp } \cap \ker (1+\operatorname {conj}_*)$ is one of the main deformation invariants which plays a crucial role in the further proofs. These lattices are enumerated in the bottom line of Table 1.

This table is organized according to the so-called Smith type of surfaces, which is denoted by $(M-r)$ and indicates that in the Smith inequality, $\dim H_*(X_{\mathbb R};{\mathbb Z}/2)\leqslant \dim H_*(X;{\mathbb Z}/2)$ , the right-hand side is greater by $2r$ than the left-hand side. The $(M-2)$ -case includes four deformation classes and two of them, encoded with $(M-2)_I$ , are of type I, which means that the fundamental class of $X_{\mathbb R}$ is realizing $w_2(X)=K_X(\mathrm {mod}\, 2)$ in $H_2(X;{\mathbb Z}/2)$ .

Surfaces belonging to the same real Bertini pair have the same Smith type. The real Bertini pairs form 7 pairs of real deformation types. In 3 pairs the deformation types (indicated in the last 3 columns of Table 1) are dual to itself: $X^+$ is deformation equivalent to $X^-$ . The other 4 pairs are shown in the 4 columns marked M, (M-1), (M-2), and (M-3).

Since $\operatorname {conj}_-=\tau _X\circ \operatorname {conj}_+$ and $\tau _X$ acts in $H_2(X)$ as multiplication by $-1$ in $K^{\perp }_X$ , the lattice $\operatorname {\Lambda }(X^-)= K_X^{\perp }\cap \ker (1+\operatorname {conj}_*^-)$ coincides with $\ker (1+\operatorname {conj}_*^+)= K_X^{\perp }\cap \ker (1+\operatorname {conj}_*^+)$ . In particular, the lattices $\operatorname {\Lambda }(X^+)$ and $\operatorname {\Lambda }(X^-)$ are orthogonal complements to each other in $E_8=K_X^{\perp }$ .

2.2 Cremona transformation of Pin-codes

Consider a real blowup model $X\to {\mathbb P}^2$ of a real del Pezzo $(M-r)$ -surface X of degree 1 with r pairs of complex conjugate imaginary exceptional classes $\boldsymbol {\ell }_{8-2k}=-\operatorname {conj}_* \boldsymbol {\ell }_{8-2k-1}, 0\leqslant k\leqslant r-1$ , and $8-2r$ real exceptional classes $\boldsymbol {\ell }_1,\dots ,\boldsymbol {\ell }_{8-2r}$ . By the code of such model we mean a sequence $(a_0,\dots ,a_{8-2r})$ of residues $\pm 1\mod 4$ , where $a_i=\operatorname {\widehat q}_X(\boldsymbol {\ell }_i)$ for $i\geqslant 1$ and $a_0=\operatorname {\widehat q}_X(h)$ with h staying for the class realized by the pull-back of straight lines. The condition $\operatorname {\widehat q}_X(h+\ell _1+\dots +\ell _{8-2r})=q_X(w_1(X_{\mathbb R}))=1$ (see Theorem 1.1.1(2)) imposes the relation

(2.2.1) $$ \begin{align} a_0+\dots+a_{8-2r}=1\,\mod4. \end{align} $$

2.3 Real rulings on del Pezzo surfaces of degree 1

The following corollary is straightforward.

2.4 Signed count in the case of connected M-surfaces

In what follows, for a given surface X, we use notation

$$ \begin{align*} R^{2k}(X)=\{v\in K_X^{\perp}\,|\, v^2=-2k \},\ k\in {\mathbb Z}. \end{align*} $$

If X is real then we consider the real counterpart of the above sets of divisor classes and put

$$ \begin{align*} R_{\mathbb R}^{2k}(X)=\{x\in R^{2k}(X)\,|\,\operatorname{conj}_*(x)=- x\}. \end{align*} $$

As is well known (and indicated in Table 1), if $(X,\operatorname {conj})$ is a maximal real del Pezzo surface of degree 1 and $X_{\mathbb R}$ is connected, then $X_{\mathbb R}= \mathbb {RP}^{2}\#4{\mathbb T}^2$ and $\operatorname {\Lambda }(X)=K_X^{\perp }$ , so that in this case $R_{\mathbb R}^{2}(X)=R^{2}(X)$ is nothing but the set of roots in $K_X^{\perp }=E_8$ . To enumerate the elements of this set and to determine their $\operatorname {\widehat q}_X$ -values, we use the special blowup model given by Proposition 2.4.1, which we call a positive blowup model.

Table 2 Real roots in the case $X_{\mathbb R}= \mathbb {RP}^{2}\#4{\mathbb T}^2$

A straightforward calculation shows that with respect to a positive blowup model the 240 roots that constitute $R_{\mathbb R}^{2}(X)$ in the case $X_{\mathbb R}= \mathbb {RP}^{2}\#4{\mathbb T}^2$ split into 4 types with corresponding values of $\operatorname {\widehat q}_X$ as shown in Table 2 Footnote ¹ . Each type is characterized there by its level equal (up to sign) to the coefficient at h in the basic coordinate expansion.

As a consequence the 240 elements in $\mathcal B_{\mathbb R}^{2}(X)$ split into 7 types with corresponding values of $\operatorname {\widehat q}_X$ as shown in Table 3.

Table 3 ${\mathcal B}_{\mathbb R}^{2}$ -classes in the case $X_{\mathbb R}= \mathbb {RP}^{2}\#4{\mathbb T}^2$

Next, we consider $\mathcal B^4_{\mathbb R}$ . As is known (see [Reference Conway and Sloane6, Table 4.9]), its number of elements is $2160$ . All 2160 elements are listed in Table 4.

Table 4 ${\mathcal B}_{\mathbb R}^{4}$ -classes in the case $X_{\mathbb R}= \mathbb {RP}^{2}\#4{\mathbb T}^2$

Proposition 2.4.2. If $X_{\mathbb R}=\mathbb {RP}^{2}\#4{\mathbb T}^2$ then, for any generic $x\in X_{\mathbb R}$ , we have

$$ \begin{align*}\sum\limits_{A\in {\mathcal C}_{\mathbb R}^4(X,x)}\widetilde{{\frak s}}(A)=112. \end{align*} $$

2.5 Signed count in the case of connected $(M-1)$ -surfaces

If $(X,\operatorname {conj})$ is an $(M-1)$ -surface and $X_{\mathbb R}$ is connected, then $X_{\mathbb R}= \mathbb {RP}^{2}\#3{\mathbb T}^2$ and $\operatorname {\Lambda }(X)= E_7$ (see Table 1). To enumerate the elements of $\mathcal B_{\mathbb R}^{4}$ and to determine their $\operatorname {\widehat q}_X$ -values, we use the special blowup model given by Proposition 2.5.1, which we call a submaximal negative blowup model.

First of all, we observe that among $2160$ classes $v\in K^{\perp } =E_8$ with $v^2=-4$ precisely $\frac {126\times 60}{10}=756$ are real, i.e. belong to $R^{4}_{\mathbb R}(X)$ . Same calculation as in M-case above shows that the corresponding 756 elements of $\mathcal B^{4}_{\mathbb R}(X)$ split into 11 subsets listed in Table 5. In accordance with notation in the proof of Proposition 2.5.1, by $\boldsymbol {\ell }_i$ (with unspecified value of index) there meant the classes of the 6 real exceptional divisors, while $\boldsymbol {\ell }_7,\boldsymbol {\ell }_8$ specify the pair of complex conjugate imaginary ones. Furthermore, each type is accompanied by an indication of its bi-level, that is a pair $(a,b)$ where a is the ${\mathbb Z}/2$ -residue of the coefficient at h and b the number of classes $\boldsymbol {\ell }_1,\dots ,\boldsymbol {\ell }_6$ which enter with odd coefficients in the expansion of the element. Due to the choice of the negative blowup model, for $\beta \in \mathcal B^{4}(X)$ of bi-level $(a,b)$ , the value $\operatorname {\widehat q}_X(\beta )$ (shown in Table 5) is equal to $a+b \mod 4$ . Therefore, in accordance with Corollary 2.3.2(1) and with Table 5 the following holds.

Table 5 ${\mathcal B}_{\mathbb R}^{4}$ -classes in the case $X_{\mathbb R}= \mathbb {RP}^{2}\#3{\mathbb T}^2$

Proposition 2.5.2. If $X_{{\mathbb R}}= \mathbb {RP}^{2}\#3{{\mathbb T}}^2$ then, for any generic $x\in X_{\mathbb R}$ , we have

$$\begin{align*}\sum\limits_{A\in {\mathcal C}_{\mathbb R}^4(X,x)}\widetilde{{\frak s}}(A)=\sum\limits_{\beta\in \mathcal B_{\mathbb R}^{4}(X)} i^{\operatorname{\widehat q}_{X}(\beta)}=84. \\[-52pt] \end{align*}$$

□

3.1 Signed count of curves in ${\mathcal C}_{\mathbb R}^2(X,x)$

Proposition 3.1.1. For every Bertini pair of real del Pezzo surfaces $X^{\pm }$ of degree 1, and any pair of real generic points $x^{\pm }\in X^{\pm }_{\mathbb R}$ , we have

$$ \begin{align*}\begin{aligned} \sum_{A\in {\mathcal C}_{\mathbb R}^{2}(X^+,x^+)} {\frak s}(A)= 2(r^--r^+)r^+, & \qquad \sum_{A\in {\mathcal C}_{\mathbb R}^{2}(X^-,x^-)} {\frak s}(A)= 2(r^+-r^-)r^- , \\ \sum_{A\in {\mathcal C}_{\mathbb R}^{2}(X^+,x^+)\cup {\mathcal C}_{\mathbb R}^{2}(X^-,x^-)} {\frak s}(A) = & - 2(r^+-r^-)^2. \end{aligned} \end{align*} $$

where $r^{\pm }=\operatorname {rk} \operatorname {\Lambda }(X^{\pm })$ .

Proof. Any class $\alpha = -2K-e\in \mathcal B_{\mathbb R}^{2}(X^{\pm })$ gives a real net of curves of arithmetic genus 1 (see Proposition 2.3.1). By fixing a generic basepoint $x\in X_{\mathbb R}^{\pm }$ , we obtain a real pencil, whose other basepoint $x'\ne x$ has to be real (see Corollary 2.3.2). Singular curves from this pencil are irreducible (and thus, rational, with one node), except one curve, which has to be real and gives a splitting $\alpha =(-K)+(-K-e)$ . The first, anticanonical, component is a real curve of genus 1, passing through x, and the second component is a real line. After blowing up the points $x,x'$ we obtain a real fibration $X_{\mathbb R}^{\pm }\#2\mathbb {RP}^{2}\to \mathbb {RP}^{1}$ , so that counting the Euler characteristic of $X_{\mathbb R}^{\pm }\#2\mathbb {RP}^{2}$ by means of this fibration we get the relation (the last equality is due to the Lefschetz fixed-point formula)

(3.1.1) $$ \begin{align} -1+ \sum_{A\in {\mathcal C}_{\mathbb R}^{2}(X,\alpha,x^{\pm})}w(A)=\chi(X_{\mathbb R}^{\pm})-2=r^{\mp}-r^{\pm}-1 \end{align} $$

where $-1$ is the Euler characteristic of the reducible fiber and, in accordance with our notation, $w(A)$ stands for $\chi (A_{\mathbb R})=s_A-c_A$ (where $s_A$ and $c_A$ stand for the number of solitary and cross-point nodes, respectively).

On the other hand, $\sum \limits _{\alpha \in \mathcal B_{\mathbb R}^{2}(X^{\pm })} i^{\operatorname {\widehat q}_{X\pm }(\alpha )}=2r^{\pm }$ due to Proposition 3.4.5 in [Reference Finashin and Kharlamov7] (where it is written in a bit different notation, so that $\operatorname {rk} \Lambda (X)$ becomes $\operatorname {rk} R_{\mathbb R}(X)$ and $\sum \limits _{\alpha \in \mathcal B_{\mathbb R}^{2}(X)} i^{\operatorname {\widehat q}_{X}(\alpha )}$ turns into $\sum \limits _{l\in L_{\mathbb R}(X)} s(l)$ ). Since furthermore ${\frak s}(A)=i^{\operatorname {\widehat q}_{X\pm }(\alpha )}w(A)$ , we conclude that

$$ \begin{align*} \sum_{A\in {\mathcal C}_{\mathbb R}^{2}(X,x^{\pm})} {\frak s}(A)= \sum_{\alpha\in \mathcal B_{\mathbb R}^{2}(X^{\pm})}i^{\operatorname{\widehat q}_{X\pm}(\alpha)} \sum_{A\in {\mathcal C}_{\mathbb R}^{2}(X,\alpha,x^{\pm})}w(A)=2r^{\pm}(r^{\mp}-r^{\pm}). \end{align*} $$

The third identity in the statement is nothing but the sum of the first two.

3.2 On deformation invariance of partial counts

As it could be already observed in the proof of Proposition 3.1.1, the value of a sum

(3.2.1) $$ \begin{align} \sum_{A\in {\mathcal C}_{\mathbb R}^{2}(X,\alpha,x^{\pm})}i^{\operatorname{\widehat q}_{X\pm}(\alpha)} w(A) \end{align} $$

does not depend on a choice of the point x, and moreover is invariant under real deformations of $X^{\pm }$ . The same property holds for $\sum _{A\in {\mathcal C}_{\mathbb R}^{2k}(X,\alpha ,x^{\pm })}i^{\operatorname {\widehat q}_{X\pm }(\alpha )} w(A)$ with $k=0$ and $2$ . In fact, such an invariance property holds in much more general situation.

Proposition 3.2.1. For any $\alpha \in H_2(X)$ , a function $f:{\mathbb Z}\times {\mathbb Z}\times {\mathbb Z}/4\to {\mathbb R}$ , and a $\operatorname {conj} $ -invariant collection $\bf x\subset X$ of $\operatorname {Card}(\bf x)=-\alpha \cdot K_X -1$ points, the sum

$$ \begin{align*} \sum_A f([\alpha]^2,\alpha\cdot K_X, \operatorname{\widehat q}_X(\alpha))w(A) \end{align*} $$

taken over all real rational curves A with $[A]=\alpha $ , $\mathbf {x}\subset A$ , depends only on $\alpha $ , f and the number $\operatorname {Card}(\mathbf {x}\cap X_{\mathbb R})$ of real points, but not on the set $\bf x$ itself. Such a sum is also invariant under real deformations of X.

3.3 Signed count of curves in ${\mathcal C}_{\mathbb R}^4(X,x)$ if $X_{\mathbb R}$ is other than $\mathbb {RP}^{2}\#4{\mathbb T}^2$ and $\mathbb {RP}^{2}\#3{\mathbb T}^2$

In the case-by-case analysis of each of the lattices $\operatorname {\Lambda }(X)$ from Table 1 we use such a root basis on which $\operatorname {\widehat q}_X$ is vanishing identically. For existence of such a basis, see [Reference Finashin and Kharlamov7, Lemma 3.1.2].

3.3.1 Cases $X_{\mathbb R}=\mathbb {RP}^{2} \bot \!\!\!\bot k\mathbb S^2$

If $X_{\mathbb R}=\mathbb {RP}^{2} \bot \!\!\!\bot k\mathbb S^2$ , $0\leqslant k\leqslant 4$ , then $\operatorname {\Lambda }(X)$ is isomorphic to $(4-k)A_1$ . Since each $(-4)$ -vector in $(4-k)A_1$ splits into a sum of generators of a pair of $A_1$ -summands, the number of such vectors is $4\binom {4-k}2$ which is $0$ for k equal 3 and 4. Each $(-4)$ -vector v yields a unique curve $A\in {\mathcal C}_{\mathbb R}^4(X,-2K_X-v, x)$ (see Corollary 2.3.2(1)) and for them the value ${\frak s}(A)=i^{\operatorname {\widehat q}_{X}(v)}$ is $1$ since $\operatorname {\widehat q}_X=0$ on each $A_1$ -summand. Thus, we just count the number $\operatorname {Card}(R_{\mathbb R}^{4})$ of (-4)-vectors in $\operatorname {\Lambda }(X)$ :

$$ \begin{align*}\sum_{A\in {\mathcal C}_{\mathbb R}^4(X,x)}{\frak s}(A)= \operatorname{Card}(R_{\mathbb R}^{4})=4\binom{4-k}2. \end{align*} $$

3.3.2 Cases $X_{\mathbb R}=\mathbb {RP}^{2}\bot \!\!\!\bot \mathbb {K}^2$ and $X_{\mathbb R}=(\mathbb {RP}^{2}\#{\mathbb T}^2)\bot \!\!\!\bot \mathbb S^2$

As it follows from Table 1, if $X^+_{\mathbb R}=\mathbb {RP}^{2}\bot \!\!\!\bot \mathbb {K}^2$ (resp. $X^+_{\mathbb R}=(\mathbb {RP}^{2}\#{\mathbb T}^2)\bot \!\!\!\bot \mathbb S^2$ ) then $X^-_{\mathbb R}=\mathbb {RP}^{2}\bot \!\!\!\bot \mathbb {K}^2$ (resp. $X^-_{\mathbb R}=(\mathbb {RP}^{2}\#{\mathbb T}^2)\bot \!\!\!\bot \mathbb S^2$ ) too, and in all the cases both $\operatorname {\Lambda }(X^{\pm })$ are isomorphic to $D_4$ . Note that $D_4$ can be seen as a sublattice of $4 \langle -1\rangle $ generated by the roots $e_0=(1,1,0,0), e_1=(1,-1,0,0), e_2=(0, 1,-1,0)$ and $e_3= (0,0,1,-1)$ . With respect to this presentation, the (-4)-vectors split into 2 kinds: sixteen vectors $(\pm 1, \pm 1, \pm 1, \pm 1)$ and eight vectors with one coordinate $\pm 2$ and the others $0$ . Thus, the vanishing of $\operatorname {\widehat q}_X$ on $e_0,\dots , e_3$ implies its vanishing on all the (-4)-vectors in $D_4$ . Indeed, it has then to vanish on $e_1+e_3$ and thus on all the sixteen first-kind (-4)-vectors (as they are congruent modulo $2D_4$ to each other), while vanishing of $\operatorname {\widehat q}_X$ on the eight second-kind (-4)-vectors follows from their presentation as a sum of $(\operatorname {mod}\,2)$ -orthogonal first-kind vectors. For each of $X=X^{\pm }$ , this gives (in accordance with Corollary 2.3.2(1))

$$ \begin{align*}\sum_{A\in {\mathcal C}_{\mathbb R}^4(X,x)}{\frak s}(A)= \sum_{v\in R_{\mathbb R}^{4}} i^{\operatorname{\widehat q}_{X^{\pm}}(v)}= 16+8=24. \end{align*} $$

3.3.3 The case of $X_{\mathbb R}=\mathbb {RP}^{2}\#{\mathbb T}^2$

According to Table 1, $\operatorname {\Lambda }(X)$ is $D_4\oplus A_1$ . Each (-4)-vector in $D_4\oplus A_1$ is either one of the (-4)-vectors of $D_4$ (24 choices), or a sum of one root in $D_4$ (24 choices) with one root in $A_1$ (2 choices). On the (-4)-vectors of the first-kind the form $\operatorname {\widehat q}_X$ vanishes, like in the previous case. On the latter sums we have $\operatorname {\widehat q}_X(v_1+v_2)=\operatorname {\widehat q}_X(v_1)$ , and in accordance with [Reference Finashin and Kharlamov7] the signed count of the 2-roots $v_1$ gives $2 \operatorname {rk}|D_4|=8$ , which after that should be multiplied by 2 because of two choices of $v_2$ in $A_1$ . Thus, in accordance with Corollary 2.3.2(1),

$$ \begin{align*}\sum_{A\in {\mathcal C}_{\mathbb R}^4(X,x)}{\frak s}(A)= \sum_{v\in R_{\mathbb R}^{4}} i^{\operatorname{\widehat q}_{X}(v)}= 24+16=40. \end{align*} $$

3.3.4 The case of $X_{\mathbb R}=\mathbb {RP}^{2}\#2{\mathbb T}^2$

According to Table 1, $\operatorname {\Lambda }(X)$ is isomorphic to $D_6$ , which can be seen as a sublattice of $6\langle -1\rangle $ generated by the following roots $\boxed {\begin {matrix} -1&-1&0&0&0&0\\ 1&-1&0&0&0&0\\ 0&1&-1&0&0&0\\ 0&0&1&-1&0&0\\ 0&0&0&1&-1&0\\ 0&0&0&0&1&-1\\ \end {matrix}}$ The vanishing of $\operatorname {\widehat q}_{X}$ on these basic roots implies immediately its vanishing on the twelve (-4)-vectors that contain $\pm 2$ as one coordinate and $0$ as others. The rest of $(-4)$ -vectors are obtained from $(\pm 1,\pm 1,\pm 1,\pm 1,0,0)$ by permutation of coordinates (totally $2^4\binom 62$ ). The $\binom 62$ permutations interpreted as partitions $n_1+n_2+n_3=4$ have either all $n_i$ even ( $6$ possibilities), in which case $\operatorname {\widehat q}_X=0$ , of give two odd summands $n_i$ ( $9$ cases), in which case $\operatorname {\widehat q}_X=2$ .

Thus, in accordance with Corollary 2.3.2(1),

$$ \begin{align*}\sum_{A\in {\mathcal C}_{\mathbb R}^4(X,x)}{\frak s}(A)= \sum_{v\in R_{\mathbb R}^{4}} i^{\operatorname{\widehat q}_{X}(v)}= 12+16(9-6)=60. \end{align*} $$

3.4 Proof of Theorems 1.2.2 and 1.2.1

The results obtained above are summarized in Table 6 which is organized by columns according to Smith types of Bertini pairs and where the rows show the result of the signed count over $A\in {\mathcal C}_{\mathbb R}^{2k}(X,x)$ for $X=X^{\pm }$ in each Bertini pair. For the first 4 columns our convention is that $X^+$ refers to surfaces with connected real locus (see Table 1).

For $k=1$ , the results shown in Table 6 are given by Proposition 3.1.1. For $k=2$ , they are taken from Propositions 2.4.2, 2.5.2 and previous Subsection. For $k=0$ they are due to [Reference Brugallé2] (note that the original Welschinger weight used in [Reference Brugallé2] coincides with our ${\frak s}$ -weight in the case of curves $A\in {\mathcal C}_{\mathbb R}^0(X,x)$ , since their arithmetic genus $g_A=2$ is even, and thus $c_A$ and $s_A$ have the same pairity).

Table 6 $\sum \limits _{A\in {\mathcal C}_{\mathbb R}^{n}}{\frak s}(A)$ for each type of Bertini pairs $X^{\pm }$

To the right from the table, for each $n=0,2,4$ , these values are expressed as a single function of $r=\mathrm {rk}\operatorname {\Lambda }(X^{\pm })$ .

Adding the 3 terms $\sum _{A\in {\mathcal C}_{\mathbb R}^{0}(X,x)}{\frak s}(A)+\sum _{A\in {\mathcal C}_{\mathbb R}^{2}(X,x)}{\frak s}(A)+\sum _{A\in {\mathcal C}_{\mathbb R}^{4}(X,x)}{\frak s}(A)$ for each type of $X=X^{\pm }$ we obtain Theorem 1.2.1.

As we take the sum $\sum _{A\in {\mathcal C}_{\mathbb R}^{2}(X,x)}{\frak s}(A)+2\sum _{A\in {\mathcal C}_{\mathbb R}^{4}(X,x)}{\frak s}(A)$ for $X=X^+$ and add it with the same sum for $X=X^-$ , we obtain Theorem 1.2.2.

4.1 Wall-crossing families

The bi-anticanonical map establishes an isomorphism between the moduli space of Bertini pairs of real del Pezzo surfaces X of degree 1 and that of real non-singular sextics C on the real quadric cone Q based on a non-empty real conic. In particular, such an isomorphism allows not only to identify the real deformation classes of these objects but also to visualize nodal degenerations of the former by means of nodal degenerations of the latter.

Let us say that a germ of a complex analytic family of sextics $C(t)\subset Q$ , $t\in D^2=\{t\in {\mathbb C}\,|\, \vert t\vert <1\}$ form a (simple) nodal degeneration, if $C(t)$ are non-singular sextics for $t\ne 0$ while $C(0)$ is uninodal. We name a nodal degeneration a Morse-Lefschetz family if the total space of the family is smooth (in other words, “wall-intersection” is transverse), and real if the complex conjugation on $Q\subset {\mathbb P}^3$ maps $C(t)$ to $C(\bar t)$ .

By taking the double coverings $X(t)\to Q$ branched along $C(t)$ we obtain a complex analytic family $\{X(t)\}_{t\in D^2}$ where $X(t)$ are del Pezzo surfaces of degree 1 for $t\ne 0$ , while $X(0)$ is uninodal. Such a family will be called a Morse-Lefschetz family of del Pezzo surfaces of degree 1, if the family $C(t)$ is Morse-Lefschetz.

If the family $C(t)$ is real, then the real structure $\operatorname {conj}: Q\to Q$ lifts to a pair of Bertini-dual real structures on the total space of the family $\{X(t)\}_{t\in D^2}$ . In particular, for each $t\in {\mathbb R}\cap D^2$ , the surface $X(t)$ acquires a pair of Bertini dual real structures, whose real loci are denoted $X_{\mathbb R}^{\pm }(t)$ .

A real Morse-Lefschetz family $X(t)$ will be called bi-cone-like if the real node on each of $X_{\mathbb R}^{\pm }(0)$ is modeled locally by equation $x^2+y^2-z^2=0$ (rather than by $x^2+y^2+z^2=0$ ). In terms of the exceptional divisor $E\subset \widetilde X(0)$ of the blowing up $\widetilde X(0)\to X(0)$ this means that the real locus $E^{\pm }_{\mathbb R}\subset \widetilde X^{\pm }_{\mathbb R}(0)$ is non-empty for both Bertini-dual real structures lifted to $\widetilde X(0)$ . In terms of $C(t)$ , this means that the node of $C_{\mathbb R}(0)$ is a cross-point (rather than a solitary point).

In the rest of Section 4 we always suppose that:

1. (1) $\{X(t)\}_{t\in D^2}$ is a real Morse-Lefschetz bi-cone-like family of del Pezzo surfaces of degree 1;

2. (2) $\chi (X_{\mathbb R}(t))<\chi (X_{\mathbb R}(-t))$ for $t>0$ (for a bi-cone-like family this means that the node of $X_{\mathbb R}(0)$ gives one-sheeted perturbation on $X_{\mathbb R}(t)$ with $t>0$ and two-sheeted on $X_{\mathbb R}(-t)$ );

3. (3) $x=\{x(t)\}_{t\in D^2}$ is a $\operatorname {conj}$ -invariant family of basepoints, $x(t)\in X(t)$ , in particular, for $t\in {\mathbb R}$ , $x(t)\in X_{\mathbb R}(t)$ ;

4. (4) the family x is generic, and $x(0)$ is chosen on a real arc issued from the node in a generic direction, not at the node but sufficiently near to it. Such choice (used, in particular, in 4.7.2) is possible due to our assumption that the node is bi-cone-like.

4.2 Merging families of curves

For each $n\in \{0,2,4\}$ and a sufficiently small $\varepsilon>0$ , the family

$$ \begin{align*} \cup_{t\in[-\varepsilon,\varepsilon], t\ne 0}{\mathcal C}^{n}_{\mathbb R}(X(t),x(t)) \end{align*} $$

has a natural structure of semi-analytic set with a proper, having finite fibers, projection to $[-\varepsilon ,\varepsilon ]\smallsetminus 0$ . Their union has a natural compactification by a finite set ${\mathcal C}_{\mathbb R}(X(0),x(0))$ formed by the curves on $X(0)$ obtained as limits of the curves from $\cup_n\cup _{t\in [-\varepsilon ,\varepsilon ], t\ne 0}\,{\mathcal C}^{n}_{\mathbb R}(X(t),x(t))$ . This gives us what we call a wall-crossing diagram: a compact 1-dimensional graph-like set

$$ \begin{align*} {\mathcal C}_{\mathbb R}(X,x)=\cup_n\cup_{t\in[-\varepsilon,\varepsilon]}{\mathcal C}_{\mathbb R}(X(t),x(t)), \end{align*} $$

with a proper map $\pi :{\mathcal C}_{\mathbb R}(X,x)\to [-\varepsilon ,\varepsilon ]$ that sends ${\mathcal C}^{n}_{\mathbb R}(X(t),x(t))$ to t (cf., [Reference Itenberg, Shustin and Kharlamov9, Proposition 4.1]). We are interested only in the germ of this map at ${\mathcal C}_{\mathbb R}(X(0),x(0))$ and treat each of the elements of the latter as vertices, although some of them represent a smooth point with a non-singular projection, in which case we call a vertex inessential. More precisely, it happens if the curve A representing a vertex is not passing through the node of $X(0)$ ; then it extends to a unique smooth family $A(t)\subset X(t)$ , $A(0)=A$ , $x(t)\in A(t)$ , $t\in [-\epsilon ,\epsilon ]$ .

For a fixed real Morse-Lefschetz family $(X(t),x(t))$ , there exists $\epsilon>0$ such that each positive (respectively, negative) semi-branch issued from any vertex $A\in {\mathcal C}_{\mathbb R}(X(0),x(0))$ represents a unique continuous family $A(t)\subset X(t)$ , $A(0)=A$ , $x(t)\in A(t)$ , $t\in [0,\epsilon ]$ (respectively, $t\in [-\epsilon , 0]$ ). We name such germs of semi-branches positive and negative edges. A vertex has branch signature $(p,q)$ if it has degree $m=p+q$ with p positive and q negative incident edges. The set of all vertices of a fixed signature $(p,q)$ is denoted by $\mathcal V_{p,q}$ and the set of all vertices by $\mathcal V=\cup _{p,q\geqslant 0}\mathcal V_{p,q}$ .

The set of all edges of the wall-crossing diagram has a natural partition according to a partition of the sets $\mathcal B^n(X(t))$ , $t\ne 0$ , into

$$ \begin{align*}\mathcal B^{n,k}(X(t))=\{ \alpha\in B^n(X(t))\,|\, \alpha\cdot v=\pm k\},\ \ k\geqslant0,\end{align*} $$

depending on the intersection index $x\cdot v$ with the vanishing class $v\in H_2(X(t))$ of the node of $X(0)$ . Namely, for $n=2, 4$ and every $t\ne 0$ , replacing divisor classes $\alpha \in B^n(X(t))$ by curves A representing $\alpha $ , we obtain a partition

$$ \begin{align*} {\mathcal C}_{\mathbb R}^{n}({X(t)},x(t))= {\mathcal C}_{\mathbb R}^{n,0}(X(t),x(t))\cup{\mathcal C}_{\mathbb R}^{n,1}(X(t),x(t))\cup{\mathcal C}_{\mathbb R}^{n,2}(X(t),x(t)), \end{align*} $$

where

$$ \begin{align*} {\mathcal C}_{\mathbb R}^{n,k}(X(t),x(t))= \{A\in {\mathcal C}^{n}_{\mathbb R}(X(t),x(t))\,|\, [A]\cdot v=\pm k\},\ k\geqslant0 \end{align*} $$

(for $n=0$ , we get a trivial splitting ${\mathcal C}^0_{\mathbb R}(X(t),x(t))={\mathcal C}^{0,0}_{\mathbb R}(X(t),x(t))$ , since $\mathcal B^0(X)$ contains only one element, $-2K_X$ ). Such a partition is preserved under deformations and induces a partition of the edges into (n,k)-types. We denote by $\mathcal E^{n,k}_+(v)$ (resp. $\mathcal E^{n,k}_-(v)$ ) the set of positive (resp. negative) edges of the given $(n,k)$ -type incident to a given vertex $v\in \mathcal V $ , and let $\mathcal E_\pm (v)=\cup _{n,k}\mathcal E^{n,k}_\pm (v)$ .

The value ${\frak s}(A(t))$ does not change with t for the curves $A(t)$ representing an edge $\varepsilon \in \mathcal E$ , so we get a well-defined sign ${\frak s}(\varepsilon)={\frak s}(A(t))$ . To measure how the signed count of edges is effected by a wall-crossing, we define

$$ \begin{align*}\Delta^{n,k}_{p,q}=\sum_{v \in \mathcal V_{(p,q)}} \left(\sum_{\varepsilon \in\mathcal E^{n,k}_+(v)} {\frak s}(\varepsilon)- \sum_{\varepsilon \in\mathcal E^{n,k}_-(v)}{\frak s}(\varepsilon)\right). \end{align*} $$

This Proposition will be proved in Subsection 4.2.1. Before we derive from it the invariance property of the counts from Theorems 1.2.1 and 1.2.2, and make preparations in Sections 4.5 and 4.6.

4.3 Derivation of Theorem 1.2.1

Due to Proposition 3.2.1, our count does not change under real variations of X as long as X is preserved non-singular, and under variations of $x\in X_{\mathbb R}$ . Thus, for proving Theorem 1.2.1, it is sufficient to check the invariance under the wall-crossing, i.e., in real Morse-Lefschetz families of real del Pezzo surfaces of degree 1, $X(t), t\in {\mathbb C}, \vert t\vert \!<\!\!<\!1$ with $\operatorname {conj}(t): X(t)\to X(\bar t)$ . Due to Proposition 4.1.1, we may restrict ourselves only with nodal degenerations whose exceptional divisor E has $E_{\mathbb R}\ne \varnothing $ both for $X^+(0)$ and $X^-(0)$ . Then, Proposition 4.2.1(A) applies and shows even a stronger invariance statement: the invariance of the count (from Theorem 1.2.1) at each vertex of the wall-crossing diagram.

To obtain the value of $\sum _{A\in {\mathcal C}_{\mathbb R}(X,x)}{\frak s}(A)$ , it is enough now to consider a surface X with $X_{\mathbb R}=\mathbb {RP}^{2} +4{\mathbb S}^2$ . Then, $\Lambda (X)=0$ (see Table 1) which implies $\mathcal {B}^2(X)= \mathcal {B}^4(X)=\varnothing $ . Thus the above sum is reduced to the Welschinger invariant of $-2K$ , which is $30$ due to [Reference Brugallé2].□

4.4 Derivation of Theorem 1.2.2

As above, due to Propositions 3.2.1 and 4.1.1 we may restrict ourselves only with nodal degenerations whose exceptional divisor E has $E_{\mathbb R}\ne \varnothing $ both for $X^+(0)$ and $X^-(0)$ . Then, the invariance claimed follows from weighted summation for classes ${\mathcal C}^2_{\mathbb R}$ and ${\mathcal C}^4_{\mathbb R}$ indicated in the rightmost column of Table 7. Namely, it gives $8(r^+-1)-4(r^+-1)-2(r^+-r^-)=2(r^++r^-)-4=12$ coming from $X^+$ and the opposite sum from its Bertini dual, $-2(r^++r^-)+4=-12$ . Alternation of the sign here is due to our convention $\chi (X_{\mathbb R}^+(t))>\chi (X_{\mathbb R}^+(-t))$ for $t>0$ , which implies that $\chi (X_{\mathbb R}^-(-t))>\chi (X_{\mathbb R}^-(t))$ , since $\chi (X^+_{\mathbb R}(t))+ \chi ( X^-_{\mathbb R}(t))=2$ .

To obtain the value of

$$ \begin{align*}\sum_{A\in {\mathcal C}_{\mathbb R}^2(X^+,x^+)\cup{\mathcal C}_{\mathbb R}^4(X^+,x^+)}\widetilde{{\frak s}}(A) +\sum_{A\in {\mathcal C}_{\mathbb R}^2(X^-,x^-)\cup{\mathcal C}_{\mathbb R}^4(X^-,x^-)}\widetilde{{\frak s}}(A), \end{align*} $$

we consider a Bertini pair $X^{\pm }$ where $X^+_{\mathbb R}=\mathbb {RP}^{2}\#4{\mathbb T}^2$ and $X^-_{\mathbb R}=\mathbb {RP}^{2} +4\mathbb S^2$ . The second sum is equal to $0$ , since it is empty because of $\mathcal B^2(X^-)=\mathcal B^4(X^-)=\varnothing $ (cf. the proof above). To treat the first sum, note that by definition of $\widetilde {{\frak s}}$ this sum is equal to $\sum _{A\in {\mathcal C}_{\mathbb R}^2(X^+,x^+)} \mathbf s(A) + 2 \sum _{A\in {\mathcal C}_{\mathbb R}^4(X^+,x^+)} \mathbf s(A)$ . Here, according to Proposition 3.1.1, $\sum _{A\in {\mathcal C}_{\mathbb R}^2(X^+,x^+)} \mathbf s(A)= 2(r^--r^+)r^+= - 128$ . By Theorem 1.2.1 we also have $\sum _{A\in {\mathbb R}^0(X^+,x^+)} \mathbf s(A) + \sum _{A\in {\mathcal C}_{\mathbb R}^2(X^+,x^+)} \mathbf s(A) + \sum _{A\in {\mathcal C}_{\mathbb R}^4(X^+,x^+)} \mathbf s(A) = 30$ while ${\mathcal C}_{A\in {\mathbb R}^0(X^+,x^+)} \mathbf s(A)=46$ (see Table 6). Taking this into account we get $-128 + 2(30-46+128)=96$ .□

4.5 Untwisting of Morse-Lefschetz families

To kill the monodromy in a Morse-Lefschetz family $X(t), t\in {\mathbb C}, \vert t\vert <\!\!<1$ , as above, we consider an associated untwisted family, $X'(\tau ), \tau \in {\mathbb C}, \vert \tau \vert <\!\!<1$ , induced by the base change $t=\tau ^2$ . The total space, a 3-fold $X'=\cup _{\tau } X'(\tau )$ , acquires a node at the nodal point of $X'(0)=X(0)$ . Blowing up at this node, $\widetilde X\to X'$ , leads to a new a family $\widetilde X(\tau ), \tau \in {\mathbb C}, \vert \tau \vert <\!\!<1$ with $\widetilde X(\tau )=X'(\tau )$ for $\tau \ne 0$ and $\widetilde X(0)$ formed by two irreducible components with normal crossing: one component, denoted $\widetilde X^1(0)$ , is the minimal nonsingular model of $X'(0)$ , and the other one, denoted $\widetilde X^0(0)$ , is isomorphic to ${\mathbb P}^1\times {\mathbb P}^1$ . These components intersect each other along a nonsingular rational curve E that represents the exceptional divisor of the blowup $\widetilde X^1(0)\to X'(0)$ and which can be seen in $\widetilde X^0(0)={\mathbb P}^1\times {\mathbb P}^1$ as the diagonal.

By contraction $\widetilde X^0(0)\to E$ along the lines of one of the rulings, we get a smooth family of smooth surfaces with $\widetilde X^1(0)$ as central fiber (see [Reference Atiyah1] for details). A choice of such a contraction provides then natural isomorphisms

$$ \begin{align*} \operatorname{Pic}(\tilde X(\tau))=H_2(\widetilde X(\tau)) \overset{\simeq}{\longrightarrow}H_2(\widetilde X^1(0))=\operatorname{Pic}(\widetilde X^1(0)) \end{align*} $$

that preserve the intersection form. This allows us for simplicity of notation to use the same symbol for all corresponding divisors and homology classes. Although we also have a natural isomorphism between $H_2(\widetilde X(\tau ))$ and $H_2({X}({t}))$ with $t=\tau ^2$ for $\tau \ne 0$ , the composed map

$$ \begin{align*} H_2({X}({t}))\overset{\simeq}{\longleftarrow} H_2(\widetilde X(\tau)) \overset{\simeq}{\longrightarrow}H_2(\widetilde X^1(0)) \overset{\simeq}{\longleftarrow} H_2(\widetilde X({-\tau})) \overset{\simeq}{\longrightarrow}H_2({X}({t})) \end{align*} $$

is not identity but the Dehn twist $x\mapsto x+([E]\circ x)[E]$ . This does not allow us to transport such a simplification of notation to the whole family $X(t)$ with $t\ne 0$ . However, if the Morse-Lefschetz family is equipped with a real structure, then there appear two lifts of the real structure to $\tilde X$ : one of them is coherent with the standard complex conjugation $\tau \mapsto \bar \tau $ , and the other one with $\tau \mapsto -\bar \tau $ . As a rule we privilege the first one, which, for real $t>0$ , identifies $X(t)$ as a real surface with $\tilde X(\tau )$ where $\tau>0$ , $\tau ^2=t$ . When we turn to $X(t)$ with $t<0$ , we use the second lift and identify $X(t)$ with $\tilde X(\tau )$ , $\tau ^2=t$ , where $\tau $ is pure imaginary with positive imaginary part. In such a way we transport the above simplification of notation to $X(t)$ with $t<0$ too. This allows us also to simplify our notation $\mathcal B(X)=\mathcal B^0(X)\cup \mathcal B^2(X)\cup \mathcal B^4(X)$ when applied to $X=\widetilde X(\tau )$ with $\tau \ne 0$ , to $X=\widetilde X^1(0)$ , and to $X=X(t)$ with any real $t\ne 0$ .

Note also that the real structures on $\tilde X(\tau )$ , $\tau>0$ , and on $\tilde X(i \tau ), \tau>0,$ converge to the same real structure on $\tilde X^1(0)$ and to different real structures on $\tilde X^0(0)={\mathbb P}^1\times {\mathbb P}^1$ . Due to assumption (2) in Subsection 4.1, the first one acts on $\tilde X^0(0)={\mathbb P}^1\times {\mathbb P}^1$ as $(p,q)\mapsto (\bar p, \bar q)$ , while the second one as $(p,q)\mapsto (\bar q, \bar p)$ .

4.6 Enumerating of limit splittings

In this subsection we consider a Morse-Lefschetz family $(X(t),x(t))$ as in Proposition 4.2.1. As is known (see [Reference Vakil13, Theorem 4.2] and [Reference Itenberg, Shustin and Kharlamov9, Proposition 4.1]), each family $A(t)\in {\mathcal C}_{\mathbb R}^{2}(X(t),x(t))\cup \, {\mathcal C}_{\mathbb R}^{4}( X(t),x(t))$ , $0\!<\! t\! <\!\epsilon $ (resp., $-\epsilon \! <\!t\!<\!0$ ), in $X(t)$ after a canonical lift $\widetilde A(\tau ): =A(\tau ^2)$ to $\widetilde X(\tau )$ with $0\!<\!\tau \!<\!\sqrt {\epsilon }$ (resp., with $\tau =i \tau ', 0\!<\!\tau '\!<\!\sqrt {\epsilon }$ ) admits a well defined, and unique, extension up to a family $\widetilde A(\tau )$ defined for every $\tau \in {\mathbb C}, \vert \tau \vert <\sqrt {\epsilon }$ . As the first step of the proof of Theorem 1.2.2, we enumerate possible splittings $\widetilde A(0)=D{}+ rE$ , $r>0$ , in the limit $\widetilde A(0)=\lim _{\tau \to 0} \widetilde A(\tau )\subset \widetilde X^1(0)$ . As is known (Loc.cit.), in each of these splittings, due to a generic choice of $x(t)$ , the divisors D are always reduced irreducible rational and intersect E transversally.

Lemma 4.6.1. If $A(t)\in {\mathcal C}^{2}(X(t),x(t))$ , so that $[A(t)]= -2K-e\in \mathcal B^2$ , then the only 3 cases of splitting $\widetilde A(0)=D{}+ rE$ , $r>0$ , are like indicated in the table below. In the third case, we have $e=-[E]$ .

Lemma 4.6.2. If $A(t)\in {\mathcal C}^{4}(X(t),x(t))$ , so that $[A(t)]= -2K-v\in \mathcal B^4$ , then the only 2 cases of splitting $\widetilde A(0)=D{}+ rE$ , $r>0$ , are like indicated in the table below

Lemma 4.6.3. If $A(t)\in {\mathcal C}^{0}(X(t),x(t))$ , so that $[A(t)]= -2K$ , then the only splitting $\widetilde A(0)=D{}+ rE$ , $r>0$ , is

4.7 Proof of Proposition 4.2.1

Note first that our assumption $\chi (X^+_{\mathbb R}(t))<\chi ( X^+_{\mathbb R}(-t))$ for $t>0$ implies that the both rulings of $\widetilde X^{0}(0)$ are real, and thus, the real structure descends from the family $\widetilde X(\tau )$ to the family obtained by contraction of any of the two rulings. When passing from $X^+(t)$ to the Bertini dual family $X^-(t)$ , we do the same, only the direction is changing: $\chi (X^-_{\mathbb R}(-t))<\chi ( X^-_{\mathbb R}(t))$ if $t>0$ .

To treat the semi-branches issued from a vertex we apply the gluing procedure underlying Abramovich-Bertram-Vakil formula (see [Reference Itenberg, Shustin and Kharlamov9, Proposition 4.1] and [Reference Brugallé and Puignau5, Theorem 2.5]). Enumeration of the semi-branches issued from a vertex represented by a curve D is equivalent, by definition, to enumerate curve families $A(t)$ having in the limit $A(0)=D{}+rE$ , $r\geqslant 0$ . To do this we use Lemmas 4.6.1, 4.6.2, and 4.6.3. Due to Lemmas 4.6.1, 4.6.2, and 4.6.3, $r\leqslant 2$ and for each $r\leqslant 2$ there are 2 kind of cases to be considered separately: $D{}\cdot E=1$ and $D{}\cdot E=2$ .

4.7.1 Case $\mathbf {r\leqslant 1}$ and $\mathbf {D{}\cdot E=1}$

At each vertex of such kind we have 2 positive semi-branches. Namely, as it follows from Lemmas 4.6.1, 4.6.2, and 4.6.3, a pair of real curves $A'(t,) A''(t)\in {\mathcal C}_{\mathbb R}^{n,1}(X(t),x(t))$ , $t>0$ , with $n=2$ or $4$ and $[A^{\prime \prime }_{\mathbb R}(t)]=[A^{\prime }_{\mathbb R}(t)]+ [E_{\mathbb R}]$ , merge together in the limit $t\to 0^+$ . The curves $A'(t)$ , $A''(t)$ have the same number of cross-point nodes, while

$$ \begin{align*} q([A^{\prime\prime}_{\mathbb R}(t)])=q([A^{\prime}_{\mathbb R}(t)]+ [E])=q([A^{\prime}_{\mathbb R}(t)])+ q([E])+2= q([A^{\prime}_{\mathbb R}(t)])+2. \end{align*} $$

This implies ${\frak s}(A'(t))+{\frak s}(A''(t)=0$ .

There are no negative semi-branches at such a vertex, since neither $[A'(0)]$ nor $[A''(0)]$ is orthogonal to $[E]$ , while the homology action of $\operatorname {conj}$ after wall-crossing is changed by reflection in $[E]$ .

4.7.2 Case $\mathbf {r\leqslant 2}$ and $\mathbf {D{}\cdot E=2,\, D{}^2=0}$

In this case the curve D generates a real rational ruling of $\widetilde X^1(0)$ . Thus, due to our choice as the base-point $x(0)\in X_{\mathbb R}(0)$ a point which is close to a generic point of $E_{\mathbb R}$ (see assumption (4) in 4.1), the curve D intersects E in 2 real points. Therefore, Lemmas 4.6.1(2) and 4.6.2(2) provide 3 groups of real families of curves sharing such a $D{}$ in the limit $t\to 0^+$ . They correspond to divisor classes of type $-2K-e-[E], -2K-e+[E]$ and $-2K -e$ , $e\in R^2(X(t))$ . The first two classes provide one merging pair of curves $A'(t),A''(t)\in {{\mathcal C}_{\mathbb R}^{4,2}(X(t),x)}$ , $t\geqslant 0$ , and the last class yields another merging pair $\widetilde A'(t),\widetilde A''(t)\in {{\mathcal C}_{\mathbb R}^{2,0}(X(t),x)}$ .

Equality of modulo 2 homology classes $[A'(t)]=[A''(t)]$ , $[\widetilde A'(t)]=[\widetilde A''(t)]$ , implies $\operatorname {\widehat q}([A'(t)])=\operatorname {\widehat q}([A''(t)])$ and $\operatorname {\widehat q}([\widetilde A'(t)])=\operatorname {\widehat q}([\widetilde A''(t)])$ . On the other hand, $A'(t)$ and $A''(t)$ has no singular points at all, while the number of cross-points in $\widetilde A^{\prime }_{\mathbb R}(t),\widetilde A^{\prime \prime }_{\mathbb R}(t)$ is by 1 greater than the number of cross-points in $D_{\mathbb R}$ , that is in $A^{\prime }_{\mathbb R}(0)$ . Therefore,

$$ \begin{align*}{\frak s}(A'(t))={\frak s}( A''(t))=-{\frak s}(\widetilde A'(t))=-{\frak s}(\widetilde A''(t))\end{align*} $$

as it follows from the definition of weights (1.2.2). This implies that the sum ${\frak s}(A'(t))+{\frak s}( A''(t))+{\frak s}(\widetilde A'(t))+ {\frak s}(\widetilde A''(t))$ that corresponds to positive semi-branches issued from D is zero.

The situation with counting by means of the weights (1.2.4) is different. Here we need to treat the branches of type $(4,2)$ and type $(2,0)$ separately. For that, we use a bijection between the set of vertices D under consideration and the set of real rulings with $D\cdot E=2$ , and, in its turn, with the real roots e orthogonal to $[E]$ . Thus, taking the sum over this set of vertices we get

$$ \begin{align*}\sum_{A\in{\mathcal C}_{\mathbb R}^{4,2}(X(t),x(t))}{\frak s}(A)= \sum_{e\in R^2(t),\, e\cdot E=0}\operatorname{\widehat q}(e)=2(r^+-1),\end{align*} $$

and

$$ \begin{align*}\sum_{A\in{\mathcal C}_{\mathbb R}^{2,0}(X(t),x(t))}{\frak s}(A)= -\sum_{e\in R^2(t), e\cdot E=0}\operatorname{\widehat q}(e)=-2(r^+-1).\end{align*} $$

There are no negative semi-branches at this kind of vertices. Indeed, there are no real curves in divisor classes $-2K-e-[E], -2K-e+[E]$ on $X(-t)$ since these classes are not orthogonal to $[E]$ , and there are no negative semi-branches for $-2K -e$ , since, as it was already pointed out in the beginning, all the counted rational curves of class $-2K-e$ share in the limit the curve $D\subset \widetilde X^1_{\mathbb R}(0)$ that intersects $E_{\mathbb R}\subset \widetilde X^1_{\mathbb R}(0)$ at 2 real points, which obstructs gluing D with 2 generators of $\widetilde X^0(0)$ to have a real perturbation in $X(-t)$ (cf. [Reference Brugallé and Puignau5, Theorem 2.5]).

4.7.3 Case $\boldsymbol{r\le 2}$ and $\boldsymbol{D\cdot E=2,\ D^2=0}$

Here, we distinguish two sub-cases: the 2 points of $D\cap E$ are real or imaginary complex conjugate.

First, assume that the both points where $D{}$ meets E are real. Then Lemmas 4.6.1(3) and 4.6.3 provide four families of curves sharing the same divisor $D{}$ in the limit $t\to 0^+$ . Two of them represent divisor classes $-2K-[E]$ , $-2K+[E]$ and the other two belong to the same class $-2K$ . The first two classes provide one merging pair of curves $A'(t),A''(t)\in {{\mathcal C}_{\mathbb R}^{2,2}(X(t),x(t))}$ , $t\geqslant 0$ , and the last class contains another merging pair $\widetilde A'(t),\widetilde A''(t)\in {{\mathcal C}_{\mathbb R}^{0}(X(t),x(t))}$ .

Note that $\operatorname {\widehat q}$ vanishes for all these curves, since

$$ \begin{align*}\operatorname{\widehat q}([-2K])=0, \, q([-2K\pm E])=\operatorname{\widehat q}([E])=0\end{align*} $$

and, thus, ${\frak s}$ for the curves involved coincides with the weights

$$ \begin{align*}w(A'(t))=w(A''(t))=-w(\widetilde A'(t))=-w(\widetilde A''(t)),\end{align*} $$

where alternation is explained by the same reason as in the previous case. This gives ${\frak s}(A'(t))+{\frak s}(A''(t))+{\frak s}(\widetilde A'(t))+{\frak s}(\widetilde A''(t))=0$ for $t>0$ . Since, also as above, there are no negative semi-branches issued from D, we are done with part (A) of Proposition in the case of real $D\cap E$ .

If the 2 points of $D\cap E$ are imaginary complex conjugate, then we have two positive semi-branches $A'(t),A''(t)$ of divisor classes $-2K-[E]$ , $-2K+[E]$ and two negative semi-branches $\widetilde B'(-t),\widetilde B''(-t)$ of divisor class $-2K$ . The curves $\widetilde B'(-t),\widetilde B''(-t)$ acquire an additional, as compared with $A'(t),A''(t)$ , solitary point but have the same number of cross-points. Therefore, we have still a balance between positive and negative semi-branches: ${\frak s}(A'(t))+{\frak s}(A''(t))={\frak s}(\widetilde B'(t))+{\frak s}(\widetilde B''(t)).$

For completing the proof of part (B) of Proposition, it remains to evaluate the sum of ${\frak s}$ -weights over all positive semi-branches in a fixed class $-2K\pm [E]$ . Such a sum can be counted like in (3.1.1), in the proof of Proposition 3.1.1. Each class yields a sum equal to $\chi (X_{\mathbb R}(t))-1=r^+-r^-$ , and both classes together give $2(r^+-r^-)$ .□

5.1 Count of quartics 6-tangent to a sextic on a quadratic cone Q

For a given sextic $C\subset Q$ , a 6-tangent quartic is a transversal intersection $A=Z\cap Q $ of Q with a quadric Z such that the intersection divisor $Z \circ C=A\circ C$ contains each point with even multiplicity. Let us denote by $T(Q,C,z)$ the set of rational irreducible reduced 6-tangent quartics that pass through a fixed point $z\in Q\smallsetminus C$ . Consider, as usual, the double covering $\pi : X\to Q$ branched along C. For a generic $z\in Q\smallsetminus C$ and $\pi ^{-1}(z)=\{x_1,x_2\}$ , each of the induced projection $\pi _k: {\mathcal C}^2(X,x_k)\cup {\mathcal C}^4(X,x_k)\to T(Q,C,z), k\in \{1, 2\},$ is a bijection.

Over ${\mathbb R}$ , we pick a pair of real points $z^{\pm }\in Q^{\pm }_{\mathbb R}=\pi (X^{\pm }_{\mathbb R})$ and consider two sets of real 6-tangent quartics, $T_{\mathbb R}(Q_{\mathbb R}^{\pm },C, z^{\pm })=\{A\in T(Q,C,z^{\pm })\,|\, A \text { is real}\} $ . Let $\pi ^{-1}(z^{\pm })=\{x^{\pm }_1,x^{\pm }_2\}$ . Note that the real locus $A_{\mathbb R}$ of each $A\in T_{\mathbb R}(Q_{\mathbb R}^{\pm },C, z^{\pm })$ lies entirely inside region $Q^{\pm }_{\mathbb R}$ , and, thus, for each choice of $k=1,2$ , the curve A lifts to a unique real rational curve $A_k=\pi _k^{-1}\in {\mathcal C}^2_{\mathbb R}(X^{\pm }, x^{\pm }_k)\cup {\mathcal C}^4_{\mathbb R}( X^{\pm }, x^{\pm }_k)$ , and we obtain bijections

$$ \begin{align*}{\mathcal C}^2_{\mathbb R}( X^{\pm}, x^{\pm}_k)\cup {\mathcal C}^4_{\mathbb R}(X^{\pm}, x^{\pm}_k) \to T_{\mathbb R}(Q^{\pm},C,z^{\pm}).\end{align*} $$

By Theorem 1.1.1(1), ${\frak s}(A_1)={\frak s}(A_2)$ for each $A\in T_{\mathbb R}(Q_{\mathbb R}^{\pm },C, z^{\pm })$ . This allows to distinguish two species of real 6-tangent quartics: hyperbolic if ${\frak s}(A_k)>0$ , and elliptic if ${\frak s}(A_k)<0$ . Recall, that by definition ${\frak s}(A_k)$ takes the values $\pm 1$ .

We also consider a larger set $\widetilde T_{\mathbb R}(Q^{\pm },C,z^{\pm })\supset T_{\mathbb R}(Q^{\pm },C,z^{\pm })$ that in addition to 6-tangent quartics contains those real conics in $Q_{\mathbb R}^{\pm }$ that are 2-tangent to C and pass through $z^{\pm }$ . For every $k=1,2$ , each of these conics A lifts to a unique curve $\widetilde A\in {\mathcal C}^0_{\mathbb R}( X^{\pm }, x_k^{\pm })$ which provides us with a bijection

$$ \begin{align*} {\mathcal C}^0_{\mathbb R}( X^{\pm}, x^{\pm}_k)\cup {\mathcal C}^2_{\mathbb R}( X^{\pm}, x^{\pm}_k)\cup {\mathcal C}^4_{\mathbb R}( X^{\pm}, x^{\pm}_k)\to \widetilde T_{\mathbb R}(Q^{\pm},C,z^{\pm}). \end{align*} $$

Similar to above, we call a real conic $A\in \widetilde T_{\mathbb R}(Q^{\pm },C,z^{\pm })$ hyperbolic if ${\frak s}(\widetilde A)>0$ and elliptic if ${\frak s}(\widetilde A)<0$ . We let also $\widetilde {{\frak s}}(A)=\widetilde {{\frak s}}(A_1)= \widetilde {{\frak s}}(A_2)$ for every $A\in T_{\mathbb R}(Q^{\pm },C,z^{\pm })$ .

Theorems 1.2.1 and 1.2.2 imply then the following result.

5.2 A few speculations on extension to symplectic setting

A natural question is how to extend the results obtained in [Reference Finashin and Kharlamov7] and in this paper to symplectic del Pezzo surfaces of degree 1, that is to rational symplectic 4-manifolds $(X,\omega _X)$ with $K^2_X=1$ . Alas, even for the existence of an appropriate $Pin^-$ -structure, additional conditions will have to be imposed (see Proposition 5.2.1). Namely, we restrict our consideration to those $(X,\omega _X)$ that are equipped with an analog of Bertini involution, that is a symplectic involution $\tau _X: X\to X$ which acts in $H^2(X)$ as a reflection in $K_X$ , and for which there exist $\omega _X$ -tamed $\tau _X$ -invariant generic J-holomorphic structures $J_X$ such that the following Bertini condition is satisfied: all J-holomorphic curves Poincaré dual to $K_X$ have a common point.

Due to these assumptions, the above curves form a symplectic Lefschetz pencil composed of curves of arithmetic genus 1, and then $\tau _X$ can be seen as an extension by continuity of the involution that acts on non-singular elements of the pencil as multiplication by $(-1)$ with respect to the group structure with $0$ in the fixed point of the pencil. Taking the quotient of X by $\tau _X$ , we obtain a symplectic quadratic cone (defined uniquely up to symplectic deformation): an orbifold symplectic manifold Q with one node q (the image of the above fixed piont p) and an orbifold Lefschetz pencil formed by curves of genus 0 passing through the node.

All this can be extended to the real setting, that is to symplectic manifolds equipped with an anti-symplectic involution. So, starting from a real rational symplectic 4-manifold $(X,\omega _X, J_X, \tau _X, \operatorname {conj}_X)$ with $K_X^2=1$ , $(X, \omega _X, J_X, \tau _X)$ satisfying the above Bertini condition, and an anti-symplectic involution $\operatorname {conj}_X$ commuting with $\tau _X$ , we get a real symplectic quadratic cone $(Q,\omega _Q,\operatorname {conj}_Q)$ with a real symplectic quotient map $\pi : X\to Q$ . The real part of Q is then diffeomorphic to the real part of a real algebraic quadratic cone over a real non-empty nonsingular conic, and $\pi _{\mathbb R}: X_{\mathbb R}\smallsetminus p \to Q_{\mathbb R} \smallsetminus q$ naturally turns into a composition of an embedding of $X_{\mathbb R}$ into a real line bundle $\xi $ over $Q_{\mathbb R}\smallsetminus q$ with the projection from the total space of the bundle to its base. The bundle $\xi $ is nontrivial, its Stiefel-Whitney class $w_1(\xi )$ is Poincaré dual to a line-generator of $Q_{\mathbb R}$ . In particular, this gives a natural diffeomorphism between the total space of $\xi $ and $(\mathbb {RP}^{2}\smallsetminus point) \times {\mathbb R}$ .

Thus, applying this construction we get an embedding $X_{\mathbb R}\smallsetminus p \to \mathbb {RP}^{2}\times {\mathbb R}$ defined uniquely up to isotopy and reversing of ${\mathbb R}$ -direction in $\mathbb {RP}^{2}\times {\mathbb R}$ . After that, we proceed as in [Reference Finashin and Kharlamov7]. The 3-manifold $\mathbb {RP}^{2}\times {\mathbb R}$ admits a unique (up to reversing) $\operatorname {Pin}^-$ -structure, the latter induces a unique (up to reversing) $\operatorname {Pin}^-$ -structure on $X_{\mathbb R}\smallsetminus p$ , which in its turn extends to $X_{\mathbb R}$ . Finally, between the two opposite structures we choose that one (which we called monic) whose quadratic function $q_X$ takes value $1$ on $w_1(X_{\mathbb R})$ .

As a conclusion, we obtain, in particular, the following statement.

It looks to us plausible that the following conjecture should be true, and that its proof can be achieved by the methods borrowed from [Reference Shevchishin10], [Reference Shevchishin and Kharlamov11].

If this conjecture will turn out to be true, it will imply that the enumerative results obtained in this paper extend to the symplectic setting described above.