# bijection of moduli space of equivariant holomorphic embeddings

Consider the moduli space $\mathcal{M}$ of equivariant holomorphic embeddings of closed oriented Riemann surfaces into a generic quintic three-fold $X$ in $\mathbb{P}^4,$ of given degree $d \in H_2(X,L;\mathbb{Z})$ (i.e. $f_*([\Sigma])=d$) and genus $\hat g.$

Here equivariance means the following: we can write every surface (2 real dimensions) (with boundaries, possibly non-oriented) as $\Sigma = \hat\Sigma / \Omega$ where $\hat\Sigma$ is closed oriented of genus $\hat g$ and $\Omega$ is an orientation reversing involution; moreover we define $\sigma:\mathbb{P}^4 \to \mathbb{P}^4$ given by $(x_1:x_2:x_3:x_4:x_5) \mapsto (\overline{x}_2:\overline{x}_1:\overline{x}_4:\overline{x}_3:\overline{x}_5)$ and call $L$ the pointwise fixed subset of $X$ under $\sigma.$ Equivariance means $\hat f \circ \Omega = \sigma \circ \hat f$ (assuming that $f:\Sigma \to X/\sigma$ always admits a lift $\hat f:\hat \Sigma \to X$)

I'd like to know whether, for every map containing a crosscap (i.e. $\mathbb{RP}^2$) which develops a node on top of the lagrangian $L,$ this map admits another smoothing to a disk, and viceversa, thus giving a bijection between the two kinds of behavior.

A local model should be $\mathbb{P}^1\ni(u:v) \mapsto (x:y:z) \in \mathbb{P}^2$ given by $$x= au^2, \quad y=av^2, \quad z=uv$$ which is mapped to the conic $xy-a^2z^2,$ which is real if $a^2 \in \mathbb{R}.$ This admits two different equivariant smoothings \begin{aligned}[3] &a \in \mathbb{R} \quad &(u:v)& \sim (\overline{v}:\overline{u}) \quad &\text{disk} \\ &a \in \mathrm{i}\mathbb{R} \quad &(u:v)& \sim (\overline{v}:-\overline{u}) \quad &\text{crosscap}\end{aligned}

Does this extend to a local model for higher genera?

This is expected to hold only if $d$ contains an even ''piece'' $d_i,$ because only this can come from a trivial element in $H_1(L;\mathbb{Z})$ due to exactness in $$H_2(X;\mathbb{Z}) \to H_2(X,L;\mathbb{Z}) \to H_1(L;\mathbb{Z})$$

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