Genus decreasing formula for higher genus Welschinger invariants

Abstract

We study the changes experienced by higher genus Welschinger invariants of real del Pezzo surfaces in families undergoing a nodal degeneration and obtain a genus decreasing wall-crossing formula.

Appendix: Curves on del Pezzo surfaces

We recall some basic facts from [30, Sect. 2 and Appendix A].

Let X be a smooth projective rational surface and $D\in {\text {Pic}}(X)$. Let ${{\mathscr {M}}}_{g,n}(X,D)$, $g\ge 0$, be the set of isomorphism classes of pairs $(f:\varSigma _g\rightarrow X, {\varvec{p}})$, where $\varSigma _g$ is a Riemann surface of genus g, f is a holomorphic map such that $f_*\varSigma _g\in |D|$, and ${\varvec{p}}=(p_1,\ldots ,p_n)$ is a sequence of distinct points in $\varSigma _g$. We denote by ${\overline{{\mathscr {M}}}}_{g,n}(X,D)$ the space of stable maps of genus g with n marked points. More precisely, ${\overline{{\mathscr {M}}}}_{g,n}(X,D)$ consists of isomorphism classes of pairs $(f:{\hat{\varSigma }}_g\rightarrow X, {\varvec{p}})$ which satisfy the following conditions:

• ${\hat{\varSigma }}_g$ is either a Riemann surface of genus g or a connected reducible nodal curve of arithmetic genus g;

• f is a holomorphic map such that $f_*{\hat{\varSigma }}_g\in |D|$;

• ${\varvec{p}}=(p_1,\ldots ,p_n)$ is a sequence of distinct smooth points in ${\hat{\varSigma }}_g$, and each component C of ${\hat{\varSigma }}_g$ of genus $g'$ which is contracted by f contains at least $3-2g'$ special points.

The points in ${\varvec{p}}$ are called marked points. Marked points together with the nodal points in ${\hat{\varSigma }}_g$ are called special points. An isomorphism between two pairs $(f_1:{\hat{\varSigma }}_g\rightarrow X, {\varvec{p}})$ and $(f_2:{\hat{\varSigma }}'_g\rightarrow X, {\varvec{p'}})$ is an isomorphism $\phi :{\hat{\varSigma }}_g\rightarrow {\hat{\varSigma }}'_g$ such that $f_1=f_2\circ \phi$ and $p_i'=\phi (p_i)$. According to [13], ${\overline{{\mathscr {M}}}}_{g,n}(X,D)$ is a projective scheme. There are two natural maps:

$$\begin{aligned} \varPhi :\quad {\overline{{\mathscr {M}}}}_{g,n}(X,D)\rightarrow |D|,&\quad [f:{\hat{\varSigma }}_g\rightarrow X, {\varvec{p}}]\mapsto f_*{\hat{\varSigma }}_g,\\ ev:\quad {\overline{{\mathscr {M}}}}_{g,n}(X,D)\rightarrow X^n,&\quad [f:{\hat{\varSigma }}_g\rightarrow X, {\varvec{p}}]\mapsto f({\varvec{p}}). \end{aligned}$$

The intersection dimension ${\text {idim}}{\mathscr {V}}$ of any subscheme ${\mathscr {V}}\subset {\overline{{\mathscr {M}}}}_{g,n}(X,D)$ is defined as the maximum over the dimensions of all irreducible components of $(\varPhi \times ev)({\mathscr {V}})$. Denote by

$$\begin{aligned} \begin{aligned} {{\mathscr {M}}}^{br}_{g,n}(X,D)=\{[f:{\hat{\varSigma }}_g\rightarrow X, {\varvec{p}}]\in {\mathscr {M}}_{g,n}(X,D)| {\hat{\varSigma }}_g \text { is smooth, and} \\ f \text { is birational on to }f({\hat{\varSigma }}_g)\},\\ {{\mathscr {M}}}^{im}_{g,n}(X,D)=\{[f:{\hat{\varSigma }}_g\rightarrow X, {\varvec{p}}]\in {{\mathscr {M}}}^{br}_{g,n}(X,D)| f \text { is an immersion}\}. \end{aligned} \end{aligned}$$

Let $\overline{{\mathscr {M}}^{br}_{g,n}}(X,D)$ denote the closure of ${{\mathscr {M}}}^{br}_{g,n}(X,D)$ in ${{\overline{{\mathscr {M}}}}}_{g,n}(X,D)$. Denote by

$$\begin{aligned} {{\mathscr {M}}}'_{g,n}(X,D)=\{[f:{\hat{\varSigma }}_g\rightarrow X, {\varvec{p}}]\in {\overline{{\mathscr {M}}}}^{br}_{g,n}(X,D)| {\hat{\varSigma }}_g \text { is smooth}\}. \end{aligned}$$

Suppose X is a del Pezzo surface which is sufficiently generic in its deformation class and $-DK_X>0$. It follows from [30, Lemma A.3], ${\text {idim}}{{\mathscr {M}}}'_{g,0}(X,D)\le -DK_X+g-1$. This inequality indicates that $-DK_X+g-1$ is the upper bound of the dimensions of all irreducible components of $\varPhi ({{\mathscr {M}}}'_{g,0}(X,D))$ in |D|. In the rest of this section, we use notation $n=-DK_X+g-1$. From [30, Lemma A.3], we know that if ${{\mathscr {M}}}^{br}_{g,0}(X,D)$ is not empty, $\dim {{\mathscr {M}}}^{br}_{g,0}(X,D)\le n$ and ${{\mathscr {M}}}^{im}_{g,0}(X,D)$ is an open dense subset of ${{\mathscr {M}}}^{br,n}_{g,0}(X,D)$, where ${{\mathscr {M}}}^{br,n}_{g,0}(X,D)$ is the union of the components of ${{\mathscr {M}}}^{br}_{g,0}(X,D)$ of dimension n. From the proof of [30, Lemma A.3], we know $\dim {{\mathscr {M}}}'_{g,0}(X,D){\setminus }{{\mathscr {M}}}^{br,n}_{g,0}(X,D)<n$. One can choose n generic points ${\varvec{w}}$ in X such that the curves in $\varPhi ({{\mathscr {M}}}'_{g,0}(X,D))\subset |D|$ passing ${\varvec{w}}$ are actually in $\varPhi ({{\mathscr {M}}}^{im}_{g,0}(X,D))$. Therefore, any irreducible curve $C\in |D|$ of genus g, passing through ${\varvec{w}}$ are contained in $\varPhi ({{\mathscr {M}}}^{im}_{g,0}(X,D))$. Suppose $[f:{\hat{\varSigma }}_g\rightarrow X]\in {{\mathscr {M}}}^{im}_{g,0}(X,D)$ is a parametrization of C, and denote by ${\varvec{p}}=f^{-1}({\varvec{w}})\subset {\hat{\varSigma }}_g$. Since ${\varvec{w}}$ is generic in X, ${\varvec{p}}$ consists of n points.

In summary, if the del Pezzo surface X is sufficiently generic in its deformation class and ${\varvec{w}}$ is generic, the set

$$\begin{aligned} {\mathscr {C}}^{\mathbb {C}}(X,D,g,{\varvec{w}}):=\{[f:\varSigma _g\rightarrow X, {\varvec{p}}]\in {\mathscr {M}}_{g,n}(X,D)|f({\varvec{p}})={\varvec{w}}\} \end{aligned}$$

(3.4)

is finite and consists of immersed curves. Moreover, the elements of ${\mathscr {C}}^{\mathbb {C}}(X,D,g,{\varvec{w}})$ are exactly the parametrizations of elements in the set of irreducible immersed curves $C\in |D|$ of genus g, passing through ${\varvec{w}}$. Hence, in this case it makes no difference to consider immersed curves in a given divisor class or the parametrized curves, even though these two settings are not equivalent in general.