Is there an explicit description of (the moduli space of) equivariant holomorphic embeddings from the 2-sphere $S^2$ to the quintic $Q$, such that $f_*[S^2]=2 \in H_2(Q,L;\mathbb{Z})?$

I'd like to know a phenomenology of what one can find, e.g., that only two or three kind of maps would have to be considered, for example the ones giving rise to an open surface, or to an unoriented surface with a crosscap.

Take e.g. the Fermat quintic
$$ Q = \left\{ (x_1:x_2:x_3:x_4:x_5) \in \mathbb{CP}^4 \left| \quad \sum_{i=1}^5 x_i^5 = 0 \right.\right\},$$
the antiholomorphic involution is taken to be
$$\sigma: \mathbb{CP}^4 \ni (x_1:x_2:x_3:x_4:x_5) \mapsto (\overline{x}_2:\overline{x}_1:\overline{x}_4:\overline{x}_3:\overline{x}_5),$$
$L$ is the fixed locus of $Q$ under $\sigma$ and $\Omega: S^2 \to S^2$ is an antiholomorphic involution; equivariant means $f\circ\Omega=\sigma\circ f.$

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Perhaps, a simpler question is to find first all the (classes of) embeddings, without reference to equivariance, and then check which ones are well-behaved with respect to equivariance. Being maps from $\mathbb{CP}^1$ into $Q$ they should be somewhat more rigid then the general genus $g$ case.

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I think another way of phrasing this is to provide an explicit check for Clemens conjecture with $d=2,$ which is proved for small $d$ by
[Katz][1] (see also [Katz2][2])

Note. (Correct me if I'm wrong) Here $d$ can be thought more or less equivalently either as the homogeneous degree of the map or as the class in $H_2(Q;\mathbb{Z}).$

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Attempt of answer. I take there are two classes $\Omega_\pm:(u:v)\mapsto(\overline{v}:\pm\overline{u}).$
Thus, the prototypes of embeddings are
$$ x_1= u^2-v^2,\quad x_2=-(u^2-v^2),\quad x_3=uv,\quad x_4=-uv,\quad x_5=0$$
for the $-$ case, and
$$ x_1= u^2-v^2,\quad x_2=-(u^2-v^2),\quad x_3=uv,\quad x_4=uv,\quad x_5=-2^{1/5}uv$$
for the $+$ case. In both cases the jacobian matrix has rank two, so these are embeddings.

Another thing to note is that, for the *generic* quintic, we expect from [Klemm][3] $n^{g=0}_{d=2}=609250;$ from this examples, it seems that Fermat quintic is not generic in the sense of Clemens conjecture, since we have continuous families of maps. This is a bit puzzling to me.

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Perhaps a better way to phrase my original question: take a generic but concrete quintic, for which we know $n^0_2$ to be finite, and then try to classify the various (sub-)families of equivariant maps with regard to their compatibiliy with $\Omega_+$ or $\Omega_-.$


  [1]: http://www.numdam.org/numdam-bin/item?id=CM_1986__60_2_151_0
  [2]: http://arxiv.org/abs/alg-geom/9312009
  [3]: http://arxiv.org/abs/hep-th/0612125